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# The photosynthetic pathways of plant species surveyed in Australia’s national terrestrial monitoring network

## Abstract

The photosynthetic pathway of plants is a fundamental trait that influences terrestrial environments from the local to global level. The distribution of different photosynthetic pathways in Australia is expected to undergo a substantial shift due to climate change and rising atmospheric CO2; however, tracking change is hindered by a lack of data on the pathways of species, as well as their distribution and relative cover within plant communities. Here we present the photosynthetic pathways for 2428 species recorded across 541 plots surveyed by Australia’s Terrestrial Ecosystem Research Network (TERN) between 2011 and 2017. This dataset was created to facilitate research exploring trends in vegetation change across Australia. Species were assigned a photosynthetic pathway using published literature and stable carbon isotope analysis of bulk tissue. The photosynthetic pathway of species can be extracted from the dataset individually, or used in conjunction with vegetation surveys to study the occurrence and abundance of pathways across the continent. This dataset will be updated as TERN’s plot network expands and new information becomes available.

 Measurement(s) oxidative photosynthetic carbon pathway Technology Type(s) digital curation • elemental analysis isotope ratio mass spectrometry Factor Type(s) date • geographic location Sample Characteristic - Organism Viridiplantae Sample Characteristic - Environment terrestrial natural environment Sample Characteristic - Location Australia

Machine-accessible metadata file describing the reported data: https://doi.org/10.6084/m9.figshare.14134994

## Background & Summary

The photosynthetic pathway of plants has a substantial impact on species productivity, abundance, and geographic distribution1,2,3. There are three primary photosynthetic pathways. C3 photosynthesis is the most common pathway. Plants that use this pathway include cool season grasses, most shrubs, and nearly all trees4,5. C4 plants include warm-season grasses, many sedges, and some forbs and shrubs6. Finally, Crassulacean acid metabolism (CAM) plants most commonly include epiphytes and succulents7. C3 plants have no special adaptations to prevent photorespiration, an energetically expensive process that occurs when the enzyme rubisco binds with oxygen to produce 2-phosphoglycolate8,9,10. The rate of photorespiration increases with increasing temperature11, restricting the photosynthetic capacity of C3 plants in warm environments. In contrast, C4 and CAM plants possess a series of biochemical, anatomical, and physiological adaptations that concentrate and isolate CO2 with rubisco, helping to eliminate photorespiration6,12. Consequently, C4 and CAM plants more easily live in hot or arid habitats3,13.

Global warming is expected to alter the competitive advantage of plants with different photosynthetic pathways14,15,16, changing species distributions and community composition, and leading to significant bottom-up effects on the structure, diversity and function of terrestrial communities17,18,19. Thus, the ecology and evolution of these different pathways has become a focus of recent botanical research20,21,22. Australia is an ecologically diverse continent that includes a wide variety of habitats and climatic zones23,24,25, making it an ideal environment to examine trends in C3, C4 and CAM distribution23,26. However, the photosynthetic pathway of numerous Australian species has not been assessed, and nationally systematic, compatible, and comparable vegetation surveys have not been historically available. The absence of these fundamental data severely limits national terrestrial research capacity.

Here we provide a dataset that lists the photosynthetic pathways of 2428 species found across Australia. These species were recorded at 541 vegetation survey plots established between 2011 and 2017 (inclusive; Fig. 1). These plots were established by the Terrestrial Ecosystem Research Network (TERN), Australia’s national terrestrial monitoring organisation. TERN is funded by the National Collaborative Research Infrastructure Strategy (NRCIS) and observes, records, and measures critical terrestrial ecosystem parameters and conditions for Australia over time. TERN Ecosystem Surveillance is one of three major branches within TERN, and is responsible for a nation-wide plot survey program. At each plot, TERN records vegetation composition and structural characteristics, and collects a range of soil and plant samples27,28. TERN data and resources are made freely accessible to scientists around the globe. The photosynthetic pathway dataset presented here was originally created by TERN to help facilitate research examining the distribution and abundance of C4 vegetation in Australia. This dataset will continue to be curated and updated as TERN increases its network of survey plots, and as new research investigates the photosynthetic pathways of terrestrial species.

Photosynthetic pathways were primarily assigned using peer-reviewed literature. We also measured the stable carbon isotope (δ13C) values of 540 species that had no recorded pathway. Tissue samples for δ13C analysis were acquired from plant specimens collected during TERN plot surveys. Using these techniques, we identified 2048 C3, 346 C4, 17 C3-CAM, and 7 C3-C4, 7 CAM, and 4 C4-CAM species across all plots. C4 species were found in 14 families and 84 genera. Most C4 species were Poaceae (228; 65.8%), followed by Cyperaceae (38; 10.9%) and Chenopodiaceae (25; 7.2%). CAM and CAM-facultative species were mainly found in Aizoaceae, Portulacaceae, and Crassulaceae. 14 genera included multiple photosynthetic pathways, specifically Tetragonia (Aizoaceae), Alternanthera (Amaranthaceae), Heliotropium (Boraginaceae), Polycarpaea (Caryophyllaceae), Tecticornia (Chenopodeceae), Cleome (Cleomaceae), Cyperus (Cyperaceae), Euphorbia (Euphorbiaceae), Aristida, Eragrostis, Neurachne, Panicum (Poaceae), and Tribulus (Zygophyllaceae). While data can be extracted for individual species, genera, or families, this dataset was designed to be used in conjunction with other TERN products. For example, photosynthetic pathway assignments can be directly combined with matching species records in TERN AusPlots vegetation surveys to obtain data on plant distribution, growth form, height and cover. These records can also be combined with other TERN plot data and products, including climate, soil, and landscape rasters. We expect this dataset will enable work examining patterns in plant occurrence, richness, and abundance, and ecosystem function at local to national scales.

## Methods

The methods used to create this dataset are presented in the following order:

1. 1.

The TERN plot-based methodologies used to survey and identify plant species, and preserve plant specimens for stable isotope analysis

2. 2.

The procedures used to assign species a photosynthetic pathway using peer-reviewed literature

3. 3.

The procedures used to assign species a photosynthetic pathway using stable carbon isotope (δ13C) analysis.

### TERN plot survey protocols, species identification, and sample collection

Plant species were identified at 541 one-hectare plots systemically surveyed by TERN between 2011 and 2017 (inclusive). Most TERN plots are located within the Australian rangelands (Fig. 1a). The Australian rangelands encompass 81% of the Australian landmass, and are characterised by vast spaces with highly weathered features, old and generally infertile soils29, highly variable rainfall, and diverse and variable plant and animal communities30. These areas have traditionally been underrepresented in Australian environmental monitoring programs, which typically focus on more mesic environments and areas closer to large population centres30. TERN’s AusPlots Rangelands method27,28 and location selection strategy was originally designed to address this underrepresentation by targeting these environments and developing and implementing survey methods that were consistent across the whole of the rangelands. Over time the network has expanded to include sampling in all the major terrestrial environments across the country, including alpine, heathland, and the subtropical systems of the east coast. The dominant vegetation types surveyed at the time of this work were woodlands and savannahs, tussock and hummock grasslands, and shrublands (including chenopod shrublands; Fig. 1b). Climate in TERN plots varies from monsoonal tropics in the north, arid deserts in the centre, to winter-dominant rainfall in the south.

The AusPlots Rangeland method27,28 consists of numerous survey modules designed to collect a wide suite of data on soil and vegetation attributes, as well as site contextual information (e.g. erosion, recent fires, etc.). These modules were conceived to provide the data level necessary to study plant community composition and structure, while also ensuring consistency in the collection of samples and data on vegetation, land, and soil characteristics. A complete description of TERN plot survey protocols is detailed in the TERN AusPlots Rangeland manual27,28. Only the protocols most relevant to plant surveys, identification, and specimen preservation are documented here.

TERN survey plots are 1 ha (100 × 100 m) permanently established sites located in a homogenous area of terrestrial vegetation (Fig. 2). Plots are usually surveyed only once, with an intention to revisit once per decade. Plots are surveyed as seasonal conditions permit, with the aim being to maximise the quality of the plant material collected and facilitate accurate herbarium identifications. Survey teams consist of between 2- and 6 people. A full complement of 6 people would include 1 to 2 people performing the vegetation survey modules, 1 to 2 people performing the soil survey modules, and the remaining team members undertaking other components of the Ausplots Rangelands method, such as recording site contextual information. The duration of each survey is variable and dependent on the density and diversity of the vegetation. Plot selection and orientation avoids major anthropogenic influences (such as roads, cattle yards, fences, bores, etc.). Ten transects (100 m long) are laid out within each plot in a grid pattern. Parallel transects running north to south are spaced 20 meters apart located at 10, 30, 50, 70, and 90 m both north and east from the SW corner (Fig. 2). Each plot is given a unique alphanumeric identifier that indicates the location of the plot, specifically its state (e.g. Western Australia, South Australia, Northern Territory, etc.) and Interim Biogeographic Regionalisation for Australia (IBRA) version 7 bioregion31, and a sequential number based on the number of plots in that bioregion. The date of the survey and GPS co-ordinates are also recorded for each plot.

Recording, collection, and identification of vascular flora is undertaken by specially trained members of the field survey team. One ground observer is tasked to perform line intercept transects. This ground observer records the species and substrate at each point (1 m) along each transect, resulting in survey data at 1010 points per plot. These point-intercept data are collected to calculate species cover (%) and other metrics. A second ground observer collects specimens of each vascular plant species in the plot, with enough material to fill an A3 size herbarium sheet (Fig. 3a,b). These members of the survey team work together to ensure the presence of each vascular plant species is recorded and enough specimens are collected. Each specimen ideally contains flowers or buds, leaves, fruit, and bark (for trees) to help enable identification. Each specimen is then tagged with a unique alphanumeric voucher barcode. All field and voucher data are recorded using a purpose-built app on a tablet to streamline data and sample collection32. The voucher specimen is ultimately delivered to a local herbarium for identification.

Subsamples of each voucher specimen are collected from the main voucher sample to enable stable isotope and molecular analysis (Fig. 3c). These subsamples are ideally free from disease, insect, or fungal contamination. The subsample is placed in a synthetic gauze ‘teabag’ and given its own unique alphanumeric barcode, referred to as the ‘primary genetic barcode’, which is linked to the date, plot, state, and voucher specimen from which it was collected. All teabags for a plot are then sealed in an air-tight, plastic container with 1 cm depth of silica granules (Fig. 3d). The container is stored in a cool location out of direct light for the duration of the survey. Upon return from the field, teabags are stored in dark conditions at room temperature at TERN facilities at the University of Adelaide (Adelaide, Australia). The silica granules are changed regularly until the samples are dehydrated and then replaced as necessary to keep the samples dry.

### Photosynthetic pathway assignment

All TERN plant data were processed in the R statistical environment33 using the ausplotsR package34,35. The ausplotsR package was created by TERN to enable the live extraction, preparation, visualisation, and analysis of TERN Ecosystem Surveillance monitoring data. A list of all vascular plant species at each TERN plot was extracted using the get_ausplots function. This produced an initial list of 4002 unique records. Scientific names for each record are provided by herbaria and are the most commonly used names in the state where the voucher specimen was collected. However, scientific names sometimes vary between states due to jurisdictional differences in taxonomy and nomenclature. TERN Ecosystem Surveillance uses the scientific names as determined by the herbaria as the point of truth in all its analysis and data sets. State herbaria identify species to the lowest possible taxonomic level. Specimens that were only identified to the family or genus level were excluded from the photosynthetic pathway dataset. Hybrids were also excluded from the final species list. Varieties and subspecies were assumed to have the same photosynthetic pathway36, therefore photosynthetic pathways were assigned to the species (i.e. Genus species) rank. This process of elimination generated a final list of 2613 unique species.

To assign each species a photosynthetic pathway, scientific names were first cross-referenced against well-known plant trait databases including Kattge, et al.24, Osborne, et al.36, and Watson and Dallwitz37. We then conducted literature searches of the remaining unassigned species via Google Scholar with combinations of the key words “C3”, “C4”, “CAM”, “photosynthesis” and “photosynthetic pathway”. We used a total of 34 peer-reviewed sources to assign species photosynthetic pathways (Table 1). If species-specific information was not available, but the species belonged to a genus known to be exclusively C3, C4 or CAM it was assigned to that pathway (e.g. Acacia spp., Eucalyptus spp. are presumptive C3). Using these combined strategies, 1888 species were assigned a photosynthetic pathway. Discrepancies between sources were rare (total of 5). In cases where species were assigned different photosynthetic pathways by different sources, the photosynthetic pathway from the source that provided the best direct evidence to support the assignment was selected. If it was not possible to assign a photosynthetic pathway using published sources or presumptive reasoning, then that species was selected for stable carbon isotope analysis.

### The stable carbon isotope values of C3, C4, and CAM plants

The stable carbon isotope values of C3 plants range from −37‰ to −20‰ δ13C (mean = ~−27‰), while the values of C4 plants range from −12‰ to −16‰ δ13C (mean = ~−13‰)38,39. Therefore, for species where either a C3 or C4 pathway was possible (e.g. Poaceae), plants with δ13C values <−19‰ were designated C3, and plants with δ13C values >−19‰ were designated C426. Full CAM plants, or plants in which CAM is strongly expressed, have isotope values of >−20‰, and thus can be distinguished from C3 plants using δ13C39,40. However, CAM photosynthesis almost always co-exists with the C3 pathway (C3-CAM)12. The isotope values of C3-CAM plants are correlated with the proportion of carbon that is obtained during light and dark periods. As a result, C3-CAM δ13C values are highly variable (approximately −13‰ to −27‰) and are dependent upon the species, its developmental stage, and/or the time of day and conditions during which the plant was sampled40,41,42. For example, the CAM pathway is often upregulated during periods of stress, such as drought43,44. Therefore, although the δ13C of wild plant samples can be used to indicate CAM potential, stable isotope values are not a reliable way to distinguish CAM and C4, identify CAM when it is weakly expressed, or a definitive method to discriminate C3 and C3-CAM plants41,42. To confirm the presence of CAM, additional measures of other physiological and biochemical variables are usually required45. With this limitation in mind, for genera with previously confirmed C3-CAM potential, we followed past authors and tentatively denoted plants with a δ13C value >−20‰ as CAM, −21‰ to −24‰ as potentially C3 + CAM, and plants <−24‰ as C340,45,46.

#### Isotope analysis

540 species were selected for stable isotope analysis. The remaining 184 unassigned species were not included in δ13C analysis because no suitable tissue samples were available. TERN plant tissue samples were identified and selected using the ausplotsR package. Each species record is associated with a full list of the available silica-dried tissue samples. One sample was selected for stable isotope analysis based on overall condition and availability (i.e. the amount of sample available from a given plot).

A 2 g subsample of material was taken from each silica-dried tissue sample. Each subsample was placed in an Eppendorf tube with two small ball bearings and pulverised for approximately one minute at 30 htz using a Retsch Mixer Mill. If samples had not homogenised during this initial process, samples were transferred to a stainless-steel ball-mill grinder and were ground for a further one minute at 30 htz. Sample preparation procedures were performed at the Mawson Analytical Spectrometry Services (MASS) Facility, University of Adelaide. An initial group of 378 samples were analysed for stable isotopes at both MASS and the Stable Isotope Facility at the Waite Campus of CSIRO in 2019. A subsequent group of 162 plant samples were analysed in 2020 at MASS.

### Stable carbon isotope analysis at CSIRO

2 to 2.5 mg of powdered plant samples were weighed into tin cups and analysed for δ13C using a continuous flow isotope ratio mass spectrometer (IRMS Delta V, ThermoBremen, Germany) equipped with an elemental analyser (Flash EA, Thermo, Bremen, Germany). Stable isotope ratios were expressed in δ notation as deviations from a standard in parts per mil (‰):

$${\delta }^{13}C=[({R}_{{\rm{s}}{\rm{a}}}/{R}_{{\rm{r}}{\rm{e}}{\rm{f}}})-1]\times 1000$$
(1)

where Rsa is the ratio of abundances of 13C/12C in the sample, and Rref is this ratio in the reference gas47. δ13C was reported relative to the standard Vienna Pee Dee Belemnite (VPDB). See the “Technical Validation” section for normalisation methods and precision estimates.

### Stable carbon isotope analysis at MASS, University of Adelaide

Like the procedures at CSIRO, 2 to 2.5 mg of powdered plant samples were weighed into tin cups and analysed for δ13C using a continuous flow isotope ratio mass spectrometer (Nu Horizon, Wrexham, UK) equipped with an elemental analyser (EA3000, EuroVector, Pavia, Italy). Stable isotope ratios were expressed in δ notation as deviations from a standard in parts per mil (‰) using Eq. 1. δ13C was reported relative to the standard Vienna Pee Dee Belemnite (VPDB). See the “Technical Validation” section for normalisation methods and precision estimates. Once all stable isotope analysis was complete, a final dataset was compiled that listed the photosynthetic pathway of 2429 plant species detected in TERN plots47.

## Data Records

All data records are stored in the TERN Geospatial Catalogue repository and can be found via the TERN Data Discovery Portal47. Data has been released under a CC‐BY Creative Commons license (https://creativecommons.org/licenses/by/4.0/), which allows reuse with attribution. Any work or publications using these data should cite this descriptor and, if applicable, the original sources (Table 1). The data set is comprised of two data tables and one data descriptor file that defines the values in the two data tables (Table 2). All tables and files are in MS Excel (.xlsx). The first table contains a list of each species and its photosynthetic pathway. It specifies the method used to determine the photosynthetic pathway (i.e. peer-reviewed literature, inferred from lineage, or δ13C analysis), as well as the peer-reviewed source or δ13C value of the tested specimen, as applicable. The plot number, location, and date that specimens were collected, the facility where the stable isotope analysis was conducted, and any replicate δ13C values are also provided. Details on commonly used species name synonyms are also listed (see Usage Notes for details). Any discrepancies in photosynthetic pathway assignments between sources, or notes about the need for further testing to confirm tentative assignments, are also recorded for each species. The second table includes a list of all the peer-reviewed sources used to create this dataset. Updates to the dataset will be managed through the TERN Geospatial Catalogue by creating a new version of the dataset. As TERN continues to expand its plot network, we will aim to include new species on an annual basis. We will also re-evaluate species taxonomy and photosynthetic pathways as new information becomes available.

## Technical Validation

TERN Ecosystem Surveillance plot surveys have been performed by different individuals and teams, which has the potential to introduce errors in plant identification in the field by ground observers. For this reason, all collections are given a temporary field name identification and assigned a permanent primary genetic barcode that is associated with a physical plant sample. Each data point and sample are tracked and recorded using the primary genetic barcode, which ensures each data point in the transect is correctly associated with a physical sample for later identification. TERN data is not published until the temporary field names are confirmed or corrected by expert local taxonomists at regional herbaria. Prior to publication of plot plant data, each species is cross-referenced against the Australian Plant Census (https://www.anbg.gov.au/chah/apc/) to confirm correct nomenclature. The whole database is also routinely compared to the Plant Census to detect changes in taxonomy over time.

Photosynthetic pathway assignments obtained from published sources have already been subject to scientific scrutiny and are well-validated. The assumption that all species within a given genus possess the same photosynthetic pathway is realistic in most circumstances3. However, our own work and the work of others has identified multiple exceptions. C4 and CAM photosynthesis have independently evolved multiple times across dozens of lineages48,49, which introduces the potential for misclassifications. To minimise this potential source of error, all species within a given family that are known to include C4 species were targeted for δ13C analysis. We targeted species in the families Aizoaceae, Asteraceae, Boraginaceae, Caryophyllaceae, Chenopodiaceae, Euphorbiaceae, Poaceae, Portulacaceae, and Zygophyllaceae. We recognize that Chenopodiaceae is now a subfamily of Amaranthaceae; however, chenopods have traditionally been examined as a separate family in past C4 analysis50,51,52. Therefore, to enable consistent comparisons with previous work and datasets we distinguished Chenopodiaceae independent of Amaranthaceae. As previously discussed, CAM or C3-CAM photosynthesis is particularly difficult to identify using δ13C, therefore any CAM or C3-CAM designations based on δ13C values should be considered tentative and warrant further investigation. Special mention should also be made of the genus Portucula (Portulacaceae). Traditionally considered a C4 genus, recent evidence has found some Portucula species have CAM potential53,54. Until species-specific information becomes available, most Portucula species in the dataset have been assigned to the C4 pathway, but the possibility of C4-CAM should be considered.

Stable isotope analysis was performed at two different laboratories over multiple years, therefore technical validation needs to be considered. Each laboratory measured plant δ13C using well-established analytical techniques. All samples where corrected for instrument drift and normalized according to reference values55 using a combination of certified and in-house calibrated standards (Table 3). For the stable isotope analysis conducted at CSIRO in 2019, all samples were normalized using a multipoint linear regression, where the slope and intercept are used to correct the isotope data on the δ13CVPDB scale56. Using the multipoint normalization procedure, measured δ values for the analysed standards are plotted on the x-axis, and the “true” accepted δ values expressed on the δ13CVPDB scale are plotting on the y-axis. These points create a regression line (Eq. 2) that covers the range of δ values:

$${\delta }_{Spl}^{T}=a\times {\delta }_{Spl}^{M}+b$$
(2)

Where a is the slope and b is the intercept. To normalize data, the measured δ value of the sample (δMSpl) is multiplied by the slope and the value of the intercept is added. Stable carbon isotope values had uncertainties of ≤0.77‰ δ13C based on repeat analysis of all the standards (n = 141). The mean and standard deviation of the absolute difference between replicate samples (10% of all samples) was 0.20 ± 0.34‰ δ13C.

MASS standards were calibrated using a two-point correction57:

$${{\rm{\delta }}}_{{\rm{sa,c}}}={{\rm{\delta }}}_{{\rm{std1}}}+\left[\left({{\rm{\delta }}}_{{\rm{sa,i}}-{{\rm{\delta }}}_{{\rm{std1,m}}}}\right)\right]/\left({{\rm{\delta }}}_{{\rm{std2,m}}-{{\rm{\delta }}}_{{\rm{std1,m}}}}\right)$$
(3)

where δsa,c is the corrected value of the measurement, δstd1,m and δstd2,m are the measured values of the standards, and δstd1 and δstd are the known values of the standards. For the isotope analysis conducted at MASS in 2019, isotope values had uncertainties of ≤0.31‰ δ13C based on repeat analysis of all the standards (n = 30). For the isotope analysis conducted at MASS in 2020, isotope values had uncertainties of ≤0.09‰ δ13C based on repeat analysis of all the standards (n = 75). The mean and standard deviation of the absolute difference between replicate samples (10% of all samples) in 2020 was 0.24 ± 0.48‰ δ13C. Given the broad but unique range of isotope values exhibited by C3 and C4 species, small deviations in values between laboratories are not likely to affect photosynthetic pathway assignment.

## Usage Notes

All photosynthetic pathways assignments in this dataset are available in the public plant trait database ‘Austraits’, which aggregates trait values for Australian plants. Site descriptions and complete species and specimen lists can be freely accessed for all TERN plots via the TERN ausplotsR package (available via CRAN and with the latest development version and patches at https://github.com/ternaustralia/ausplotsR)34,35, or the TERN Data Discovery Portal (https://portal.tern.org.au/). As previously described, ausplotsR allows users to directly access all TERN plot-based data on vegetation and soils across Australia34,35. It also provides functions that calculate and visualise species presence, richness and cover (%) at all TERN plots. The photosynthetic pathway dataset presented here was designed to be easily combined with TERN ausplotsR species distribution data to investigate national distribution patterns of different photosynthetic pathways. As an example, we have provided sample code for the R statistical environment to demonstrate how the TERN photosynthetic pathway dataset presented here and % species cover calculated at TERN plots can be combined to calculate C4 plant cover (relative to C3) across Australia, and relate relative C4 cover values to changes in climate and local factors. As detailed in Supplementary File 1, simple functions in ausplotsR can quickly calculate % species cover at each TERN plot, and then each species in each plot can be assigned its correct photosynthetic pathway using the TERN photosynthetic pathway dataset. This enables the calculation of relative C4 plant cover at each plot. Relative C4 cover can then be regressed against climate and local parameters by using TERN plot coordinates to extract site-specific environmental data from other national climate58 and soil59 rasters.

Additional TERN data infrastructure can be found via the TERN Data Discovery Portal. For more information and tutorials on how to access TERN data, visit www.tern.org.

As previously discussed, scientific names for species in the TERN database are provided by state herbaria and are the most commonly used names in a given state. However, valid scientific names may vary between states due to differences in nomenclature (although this is rare). TERN Ecosystem Surveillance uses the scientific names as provided by the local herbaria as the point of truth in all its analysis and datasets. To enable the integration of this dataset with other data records, where there are known nomenclature issues between jurisdictions, we have notated alternative synonyms in the species name comments field of Table 1 in the dataset. When using this dataset, users should take care to select the most relevant synonym for their work.

## Code availability

No custom code was used in this analysis. Examples of how to combine this photosynthetic pathway dataset with other TERN data infrastructure in the R statistical environment has been provided in the supplementary material (Supplementary File 1).

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## Acknowledgements

We acknowledge the TERN Ecosystem Surveillance field team for their work collecting the voucher specimens used in the δ13C analysis. We also acknowledge the support of TERN by the Australian government through the National Collaborative Research Infrastructure Strategy. Additional financial support for this project was provided by the AMP Foundation and the AMP Tomorrow Fund awarded to S.M., the Australian Research Council Future Fellowship (FT110100100793) awarded to F.A.M., and Australian Government Research Training Program and University of Adelaide Faculty of Sciences Divisional scholarships awarded to J.W.A.

## Author information

Authors

### Contributions

S.M., F.A.M. and J.A. originally formulated the idea, S.M., F.A.M., J.A., E.L., G.G. and B.S. designed the study and developed the methodology. S.M., N.W., F.A.M., J.A., E.L., T.H., S.S., S.C.R. and R.A. collected plant samples, performed the experiments, and analysed the data. S.M. wrote the manuscript; all other authors provided editorial advice.

### Corresponding author

Correspondence to Samantha E. M. Munroe.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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The Creative Commons Public Domain Dedication waiver http://creativecommons.org/publicdomain/zero/1.0/ applies to the metadata files associated with this article.

Reprints and Permissions

Munroe, S.E.M., McInerney, F.A., Andrae, J. et al. The photosynthetic pathways of plant species surveyed in Australia’s national terrestrial monitoring network. Sci Data 8, 97 (2021). https://doi.org/10.1038/s41597-021-00877-z


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Article Text

The inverse relationship between national food security and annual cholera incidence: a 30-country analysis
1. Aaron Richterman1,2,
2. Andrew S Azman3,
3. Georgery Constant4,
4. Louise C Ivers2,5
1. 1 Division of Infectious Disease, Brigham and Women's Hospital, Boston, Massachusetts, USA
2. 2 Center for Global Health, Massachusetts General Hospital, Boston, Massachusetts, USA
3. 3 Department of Epidemiology, Johns Hopkins University Bloomberg School of Public Health, Baltimore, Maryland, USA
4. 4 Partners in Health / Zanmi Lasante, Hinche, Haiti
5. 5 Department of Global Health and Social Medicine, Harvard Medical School, Boston, Massachusetts, USA
1. Correspondence to Dr Aaron Richterman; arichterman{at}partners.org

Abstract

Introduction Individual and household-level evidence suggests a relationship between food insecurity and cholera risk. The relationship between national food security and the size of cholera outbreaks is unknown.

Methods We analysed the relationship between national food security and annual cholera incidence rate from 2012 to 2015 across 30 countries. We used components of the Global Food Security Index (GFSI) as measures of food security. We included countries with available GFSI reporting cases of cholera during the study period, excluding high-income countries. We developed multivariable zero-inflated negative binomial models with annual cholera incidence rate as the outcome, GFSI components as the exposure of interest, fixed effects for country and year, and time-varying effects related to water, sanitation, and hygiene, oral cholera vaccine deployment, healthcare expenditure, conflict and extreme weather.

Results The 30 countries reported 550 106 total cases of cholera from 2012 to 2015, with a median annual incidence rate of 3.1 cases per 100 000 people (IQR 0.3–9.9). We found independent inverse relationships between cholera and Overall GFSI (incidence rate ratio (IRR) 0.57, 95% CI 0.43 to 0.78), GFSI-Availability (IRR 0.81, 95% CI 0.70 to 0.95) and GFSI-Affordability (IRR 0.76, 95% CI 0.62 to 0.92).

Conclusions We identified a strong inverse relationship between national food security and annual incidence rate of cholera. In the context of prior evidence at the individual and household levels, this suggests that there is a linkage between food insecurity and cholera at the national level that should be further considered in assessing cholera risk in vulnerable regions and in designing cholera control interventions.

• food security
• cholera

This is an open access article distributed in accordance with the Creative Commons Attribution 4.0 Unported (CC BY 4.0) license, which permits others to copy, redistribute, remix, transform and build upon this work for any purpose, provided the original work is properly cited, a link to the licence is given, and indication of whether changes were made. See: https://creativecommons.org/licenses/by/4.0/.

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Key questions

• In two prior studies at the individual and household levels in Haiti, we identified an independent inverse relationship between food security and risk of cholera and risk of death from cholera.

• The directionality and mechanism of the relationship between food security and cholera, as well as the impact of interventions or policies targeting food insecurity on cholera risk, are unknown.

• Food security is not explicitly considered in any existing programmatic guidance for cholera control, and the impact of cholera on food security (and its subsequent consequences for communities) is rarely discussed in the literature and in policy debates.

What are the new findings?

• In this study of 30 countries from 2012 to 2015, we identified a strong inverse relationship between national food security, in particular food availability and affordability, and annual incidence rate of cholera.

What do the new findings imply?

• Together with prior evidence at the individual and household levels, this study suggests that there is a linkage between food insecurity and cholera at the national level that should be further considered in cholera risk assessments and when designing cholera control interventions.

Introduction

There has been no substantive decrease in the annual number of cholera cases reported over the last three decades, and the 172 454 cases reported in 2015 likely represent a fraction of the total estimated incidence of 1.3–4 million annual cases worldwide.1–3 This is despite the identification of numerous modifiable individual and household risk factors for cholera,4 including those related to water, sanitation and hygiene (collectively termed WASH), which continue to be emphasised in guidelines for cholera control despite an unfortunate lack of evidence about which specific household or individual WASH interventions are effective in any given context.5 6 Oral cholera vaccines (OCVs) show great promise as one tool for cholera prevention but have not yet been broadly implemented.7 This is in part because the global supply of OCV, while increasing, currently remains too limited for integration into routine health system activities.8 In the meantime, additional evidence-based interventions are needed to meet the goal set by the Global Task Force on Cholera Control for the elimination of cholera from 20 countries with a 90% reduction in global cholera deaths by 2030.9

In two recent studies in Haiti, where cholera is now endemic after being introduced in 2010, we found an independent association between household food insecurity and both reported history of cholera and reported death from cholera.10 11 Food insecurity is defined as a lack of stable access to food in adequate quantity or quality.12 Food insecurity is closely related to poverty,13 but has been linked to a variety of adverse health outcomes (including mortality) independent of wealth and other measured social determinants of health.10–12 14 The directionality of the relationship between food insecurity and cholera is unknown and may be bidirectional, as it is with other diseases.12 15 Plausible pathways from food insecurity to cholera-related risk include malnutrition, causing impaired immunity and gut barrier function; behavioural changes during periods of hunger or food scarcity leading to a higher likelihood of drinking unsafe water or eating unsafe food; and worse mental health, affecting an individual, household or community’s ability to respond to illness.11 15 To our knowledge, there have been no studies of cholera and food security outside of Haiti. The impact of interventions or policies targeting food insecurity on cholera risk is unknown.

A stark example of the intersection between food insecurity and cholera is in Yemen, which is currently experiencing both the greatest food security crisis in the world and a major cholera outbreak.16 17 While war is a common factor in both of these emergencies,18 understanding a possible relationship between food security and cholera in this and other settings may inform new potential avenues for cholera risk reduction and/or the alleviation of the consequences of cholera on communities.

Food security can be considered across multiple dimensions, including availability, access, affordability, utilisation, and quality and safety—and levels—national, subnational, household and individual.12 While food security and cholera incidence are often heterogeneous within a country,19 a multicountry analysis of the association between national level cholera and food security indicators may reveal broad trends that can help provide generalisable insight for policymakers. This kind of analysis may both suggest new approaches for cholera control and allow for a greater understanding of the possible full health and economic benefits of food security interventions. We thus sought to estimate the relationship between multiple dimensions of national food security and the annual incidence rate of cholera in this study of 30 countries from 2012 to 2015.

Methods

Data

As our measure of national food security, we used the Global Food Security Index (GFSI), a composite measure of national food security that is reported annually, incorporates 28 unique indicators and allows for interdimensional and cross-national comparisons over time.20 21 The indicators come from a number of international organisations, including the United Nations, International Monetary Fund, Food and Agricultural Organisation, the World Bank, the WHO and others. The GFSI is available for 113 countries, selected based on regional diversity, economic importance and size of population.20 The GFSI is reported as a normalised overall score from 0 to 100 (with a higher number indicating greater food security and no specific threshold for food security). The GFSI can be broken up into three dimensions of food security, each also reported as a score from 0 to 100: GFSI-Availability, GFSI-Affordability and GFSI-Quality and Safety.

We identified the total number of suspected cholera cases reported per country and year to the WHO using the Global Health Observatory Database.3 If a country did not report whether or not it had cases of cholera in a given year, we treated it as having zero cases and treated it as missing data in sensitivity analyses. We calculated the annual incidence rate of cholera using the midyear population size of each country.22

We obtained additional time-varying covariates for each country and year that were likely to be related to changes in the cholera rate or food security over time: gross national income (GNI) per capita in 2011 Purchasing Power Parity (PPP);22 health expenditure per capita in current PPP;22 percentage of the population within each country that had access to basic drinking water services (defined as drinking water from an improved source within 30 min or less), access to basic sanitation (defined as the use of improved facilities not shared with other households), and access to basic handwashing facilities;23 whether OCV was deployed within the country during a given year;8 whether a country experienced armed conflict in a given year;24 25 and the Global Climate Risk Index, an annually reported composite estimate of the direct impacts of extreme weather events in a country.26

Analysis

We focused our analysis on 2012–2015 because all covariates of interest were available during these years. We included countries with available GFSI that reported cases of cholera during that time period, excluding high-income countries (as defined by the World Bank’s 2017 classification) because the majority of the cholera cases in these countries were imported.27

We modelled the relationship between GFSI components and the annual cholera incidence rate using a zero-inflated negative binomial (NB) regression (online supplementary appendix, online supplementary figure 2).28 29 The NB portion of our regression model had the following generic form:

Supplementary data

where C is the number of cholera cases for a country j experiencing an outbreak in year t; θ is the dispersion parameter; αt is the year-specific intercept; F is the food security metric (Overall GFSI, GFSI-Availability, GFSI-Affordability or GFSI-Quality and Safety) for a country j in year t; X is a vector of time-varying controls previously associated with food security and/or cholera rate; S is a country fixed effect; and O is the natural log of population size for country j in year t, an offset variable. Country and year fixed effects control for unmeasured time-invariant differences across countries (eg, environmental, socioeconomic, infrastructural, health systems factors).

Our key effect estimate of interest is β, which estimates the association between national food security and annual cholera incidence rate. Holding other variables equal, a unit increase in F would be expected to be associated with a multiplicative change of eβ for C . We calculated three effect estimates for each food security metric, reported as incidence rate ratio (IRR) with 95% CI: (1) unadjusted, (2) adjusted for country and year fixed effects and (3) adjusted for fixed effects and additional time-varying covariates (online supplementary file 1).

We performed three sensitivity analyses to assess the robustness of our results (online supplementary appendix). First, we used a simpler model using mean annual incidence of cholera over the included years as the outcome. Second, we treated country-years during which a country did not report whether or not it had cases of cholera as missing data rather than as having zero cases. Finally, we used an alternate indicator of food insecurity, the prevalence of undernourishment.30

We performed statistical analysis using SAS V.9.4 and R V.3.5.2 using the ggplot2 package.31

Patient and public involvement

Patients and the public were not involved in the design, conduct and reporting of this research.

Results

Thirty low-income or middle-income countries with available GFSI reported cases of cholera from 2012 to 2015, for a total of 120 country-years (table 1). Fifteen of these countries are low income, ten are lower middle income and five are upper middle income.27

Table 1

Characteristics of countries with available Global Food Security Index (GFSI) reporting cases of cholera from 2012 to 2015

Three countries reported cholera during 1 year, 4 countries reported cholera during 2 years, 7 countries reported cholera during 3 years and 16 countries reported cholera during all 4 years within the study period. There were 550 106 total cases of cholera reported, with a median of 3333 cases per country (IQR 571–10 999) and a median annual incidence rate of 3.1 cases per 100 000 people (IQR 0.3–9.9). The median GNI per capita was $2574 (IQR 1599–5526) at the level of a lower middle-income country. There was a median coverage with basic sanitation services of 35.5% (IQR 21–60), basic water services of 65.5% (IQR 56–87) and basic handwashing facilities of 13% (IQR 9–24). The median health expenditure per capita was$143.5 (IQR 94–220). Seven countries experienced armed conflict during every year in the study period, 9 experienced conflict during some years and 14 did not experience conflict during any year.24 25 Five countries had OCV deployed during at least 1 year with target populations ranging from <0.1% to 1.5% of the national population.8

Included countries were, in general, food insecure—among all countries with available GFSI, they had a median rank of 90 (IQR 77–103) out of 113 countries for Overall GFSI during the study period and included eleven of the 13 most food insecure countries. The median annual change in food security was greatest for Overall GFSI (1, IQR 0.4–1.9) and GFSI-Availability (1.8, IQR 0.6–3.6) and less for GFSI-Affordability (0.4, IQR 0.1–1.5) and GFSI-Quality and Safety (0.7, IQR 0.3–1.4) (figure 1; greater detail in online supplementary figure 1).

Supplementary data

Figure 1

Trends in Global Food Security Index (GFSI) components by country for countries with available GFSI that reported cases of cholera between 2012 and 2015.

Unadjusted models showed a consistent inverse relationship between the incidence rate of cholera and food security, including a 12% reduction in incidence rate from a one point increase in Overall GFSI (IRR 0.88, 95% CI 0.84 to 0.91), a 14% reduction in incidence from a one point increase in GFSI-Availability (IRR 0.86, 95% CI 0.82 to 0.89), a 9% reduction in incidence from a one point increase in GFSI-Affordability (IRR 0.91, 95% CI 0.87 to 0.96) and a 10% reduction in incidence from a one point increase in GFSI-Quality and Safety (IRR 0.90, 95% CI 0.88 to 0.93) (table 2, figure 2). After adjusting for country and year in the model (as fixed effects), our estimates of the relationship between cholera incidence and three of the four dimensions of food security were stronger, with an estimated 41% reduction in incidence per point increase in Overall GFSI (IRR 0.59, 95% CI 0.44 to 0.78), an 18% reduction in incidence per point increase in GFSI-Availability (IRR 0.82, 95% CI 0.71 to 0.95) and a 24% reduction in incidence per point increase in GFSI-Affordability (IRR 0.76, 95% CI 0.63 to 0.93), while there was a complete attenuation of the effect estimate for GFSI-Quality and Safety (IRR 1.22, 95% CI 0.92 to 1.61). The effect estimates were largely unchanged after including the other time-varying covariates in the multivariable models. None of the other time-varying covariates were associated with the annual incidence rate of cholera in the multivariable models (online supplementary table 1).

Figure 2

Scatter plot of the annual cholera incidence rate per 100 000 people against Global Food Security Index components. each dot represents a country-year, the lines represent linear model lines of best fit, and the envelopes represents 95% CI for the lines.

Table 2

The relationship between food security (GFSI-Overall, GFSI-Affordability, GFSI-Availability and GFSI-Quality and Safety) and the annual incidence rate of cholera using zero-inflated negative binomial regression models

Results from sensitivity analyses were consistent with those from our primary analysis (online supplementary appendix). In simpler multivariable models using the mean annual incidence rate of cholera over the time period as the outcome, all GFSI components showed an inverse relationship with cholera incidence (online supplementary table 2). Treating country-years during which a country did not report whether or not there were cholera cases as missing values rather than as zero-case country-years did not substantially change IRRs or 95% CI from our primary analysis (online supplementary table 4). Use of an alternate potential endpoint of food security yielded consistent results with our primary analysis, whereby a 1% increase in prevalence of undernourishment was associated with a 10% increase in cholera incidence (IRR 1.10, 95% CI 1.00 to 1.20) (online supplementary table 6, online supplementary figure 3, online supplementary figure 4).

Discussion

In this analysis of 30 low-income or middle-income countries reporting cases of cholera from 2012 to 2015, we found that Overall GFSI, GFSI-Availability and GFSI-Affordability were independently and inversely associated with the annual incidence rate of cholera. Our model estimates the relationship between food security and rate of cholera by controlling for unmeasured time-invariant country and year effects, as well as a variety of measured time-varying effects including national health expenditure, WASH indicators, conflict, OCV deployment and impacts of extreme weather. These findings suggest that national food insecurity may impact the size of cholera outbreaks, or that, alternatively, more extensive cholera outbreaks have an adverse effect on national food security. For both of these reasons, both short-term and long-term interventions and policies targeting food insecurity, and in particular food availability and affordability, should be evaluated as additional strategies to be used as part of cholera response. Food security is not currently considered in any programmatic guidance for cholera control, and the impact of cholera on food security (and its subsequent consequences for communities) is not discussed.5 9

Both GFSI-Availability and GFSI-Affordability contain a number of indicators that are feasible targets for intervention or policy change, and specific contexts within countries will dictate which areas have the greatest room for improvement. In descending order of weight, GFSI-Availability is comprised of the following: sufficiency of food supply based on domestic supply, imports and chronic food aid; public expenditure on agricultural research and development; agricultural infrastructure; volatility of agricultural production; food loss; political stability risk; corruption; and urban absorption capacity (the capacity of a country to absorb the stress placed on it by urban growth and still ensure food security). GFSI-Affordability contains food consumption as a share of household expenditure; gross domestic product per capita; proportion of population under global poverty line; agricultural import tariffs; presence of food safety net programme; and access to financing for farmers. The median country in our analysis was at the 80th percentile for food security, suggesting that there is substantial opportunity to positively impact food security in these settings.

Our findings are consistent with individual and household-level analyses that we have previously reported from Haiti. In a multivariable analysis of 2320 people living in HIV-affected households, we found that severe household food insecurity was associated with a greater than threefold increase in an individual’s odds of reporting a history of cholera since the onset of the cholera epidemic in Haiti.10 In a subsequent study of the general population in Haiti using data from the 13 181 households in the 2012 Demographic and Health Survey, we similarly found that both moderate and severe household food insecurity were associated with a household reporting at least one member with history of cholera, independent of household wealth and other measured confounders.11 We also found that severe household food insecurity was associated with nearly double the odds of a household reporting that a member had died from cholera.11 To assess for the presence of food insecurity, these studies used the Household Hunger Scale, a cross-cultural experience-based scale focused on household food access.32 Food access at the household level is thought to flow downstream, at least in part, from food availability and affordability at the national level,12 and thus these prior analyses are consistent with the current study.

This analysis did not allow us to determine the specific mechanisms by which food security at the national level might impact the number of cases of cholera in a given year (or vice versa), but there are a number of possibilities which can be thought of within the context of a social ecological model of cholera (figure 3). At the government and organisational levels, poor food availability may be a partial cause and/or result of systemic susceptibility to an extensive cholera outbreak along with an ineffective response. Governments and national and international organisations responding to both cholera and food insecurity crises are likely to operate under strain, impacting programmatic success in cases of limited resources. Lack of national food availability can contribute to worse food affordability and flow downstream to inadequate food access at the community level, manifesting in, for example, shortages in local markets or communal granaries that may impact (or be impacted by) regional cholera response coordination. At the household level, poor food access may be related to the ability to access healthcare, intra-household transmission dynamics or the care of ill household members.

Figure 3

Social ecological model of cholera and possible relationships with food security. WASH, water,sanitation and hygiene.

At the individual level, people experiencing food insecurity make pressured choices, and may thus have a greater likelihood of engaging in higher risk behaviours like drinking from unsafe water sources or eating unsafe food. Food insecurity is associated with worse mental health in a dose–response fashion.12 33 34 Poorer mental health may, in turn, limit a person’s ability to respond to cholera by accessing healthcare, engaging in preventative measures or in other ways. Insufficient household food access can result in decreased individual food utilisation, which may cause clinically significant malnutrition. Malnutrition correlates with impaired gut and immune function and may plausibly increase an individual’s susceptibility to cholera infection, particularly for young children.35

While it is unknown which specific interventions or policies targeting food insecurity might best decrease cholera incidence and/or alleviate the impact of cholera, there are a range of possibilities, both short and long term, that could be considered across all levels within the social ecological model. Some examples include the expansion of food safety net programme, temporary changes in agricultural import tariffs during an epidemic, investment in communal grain storage programme and infrastructure designed to improve agricultural climate resiliency.

We note that cholera data reported to the WHO are thought to reflect under-reporting in some countries (eg, some high-burden countries in South Asia consistently do not report or report only confirmed cases) and over-reporting in others (when acute watery diarrhoea is used as a proxy measure for cholera with no regular laboratory confirmation). Our results may be biassed if the proportion of reported cases to actual cases varied over time within included countries. We believe this bias, if present, is likely to be small because all included countries had experienced cholera prior to 2012 and we are not aware of major changes in their reporting apparatuses during the study period.

This study has some additional limitations. Average changes in cholera incidence over multiple years are more reflective of overall changes in a country’s cholera burden of disease compared with year-to-year trends, which are more influenced by local changes in immunity. Because of this, the relationships identified in this analysis are probably more reflective of longer term trends than true year-to-year changes. The presence of unmeasured time-varying variables associated with both food security and cholera may bias our findings. While our analysis included diverse countries from a variety of settings, some countries that have experienced food insecurity and contribute to the global cholera burden (eg, Iraq, Somalia and Yemen) were not included because GFSI was not available or because they did not report cases of cholera in more than 1 year during the study period. We relied on aggregate country-level data, and analyses which include local data would provide greater nuance in exploring the relationship between food security (at all levels) and cholera.19

Conclusion

In conclusion, we identified an inverse relationship between multiple dimensions of national food security and the reported annual incidence rate of cholera across 30 countries, consistent with prior literature at the individual and household levels. There are multiple potential pathways through which food security may be linked to cholera, and this association should be further considered in assessing cholera risk in vulnerable regions and in designing cholera control interventions.

Acknowledgments

We thank Professor Mauricio Avendano Pabon for methodological advice.

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Footnotes

• Handling editor Alberto L Garcia-Basteiro

• Contributors AR and LCI conceptualised the study. AR performed the primary quantitative analysis with methodological guidance by ASA. AR wrote the first draft of the manuscript with critical input on revisions by ASA, GC and LCI.

• Funding This study was supported by NIAID R01AI099243 (LCI), The Bill and Melinda Gates Foundation OPP1148213 (LCI, AR).

• Competing interests None declared.

• Patient consent for publication Not required.

• Provenance and peer review Not commissioned; externally peer reviewed.

• Data availability statement Datasets are freely available for download from the Harvard Dataverse (https://doi.org/10.7910/DVN/CIBFJN).

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# Sample Mean vs. Population Mean

I did not mention this is my last post that outlined some basic statistical functions related to central tendency, but when studying stats, it is important to understand that you will be dealing with two kinds of means: a sample mean, and a population mean.  Conceptually, they both do the same kind of thing, though their meanings are slightly different.  It is a very good idea to know when to use sample mean vs population mean, and I will try to go over these concepts and their uses in this post.

Mathematically, I already explained how you determine a mean value.  It is what you have likely always known as an average value, and you can very easily find it using the mean formula.  Without actually writing out the equation, you already know that the mean is the sum of all your values, divided by the total number of your values.  This is straightforward and nothing new.  Here is where I am going to make a distinction that you need to be aware of.

In statistics, you deal with populations.  Populations are complete groups of people, of things, of measurements.  As an example, you likely know of the population of the planet Earth.  That refers to all of the people on the planet.  Or, you could have a population of bald eagles in a nesting ground, or a population of Ferrari sports car manufactured in 2011.  Populations refer to the whole group of whatever you are talking about.  However, in many cases, you don't have access to data about the entire population.  You only have access to a subset of that population... a sample of the population.  So, a sample can be considered to be a small part of the population, but is representative of that population as a whole.

A sample could also be looked at as only an estimate of the larger population.  They are frequently sufficient enough to work with, since having data for an entire population could involve a very complicated and long set of data, and the closer your sample size is to your population size, the more accurate this estimate becomes.  This is why people tend to question things that are only based upon a few observations... error is higher when sample size is smaller.  More observations means less error.

So then, with those definitions in mind, you should hopefully be able to understand what is meant by population mean and sample mean.  Literally, a population mean is the average of the entire population, whereas the sample mean is the average of a sample (which represents a larger population).  Of course, since this is mathematics, we have different ways to write the notation for these two statistics concepts.

When we are talking about a population mean, where we have data about all of the subjects or measurements of a given population, we represent that data by the Greek letter mu, which looks like a fancy lower-case u:

This is calculated by summing all of the values in the entire population, and then dividing by the total number of values in that population, which is denoted by a capital N for a population.

On the other hand, when we are dealing with a sample mean (a subset that is representative of a whole population), we denote this function by the aforementioned symbol, x-bar:

As before, we find this by summing all of the values in your set, and then dividing by the total number of values in your set, in this case, the number being denoted by a lower-case n for a sample.

As I mentioned, calculating these values means essentially doing the same thing.  However, in stats, it is wise to pay attention to the group that you are analyzing.  Making a mistake at this point could lead to much larger errors in any further statistical analysis.  Keep in mind that a sample mean is an approximation of a population mean, and that approximation becomes more accurate as the size of your sample (n values) approaches the size of your whole population (N values).


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# Math Help - Definition of Sum

1. ## Definition of Sum

I have been trying to find an acceptable definition of the "sum" of two natural numbers. Wikipedia has come close to what I am looking for. They begin by saying:

Let n' be the successor of n, that is the number following n in the natural numbers, so 0'=1, 1'=2. Define a+0=a.

I get all the above. They then continue by saying:

Define the general sum recursively by a + (b') = (a+b )'. Hence 1+1=1+0' = (1+ 0)'=1'=2.

I see the truth in everything that they are saying in the immediately preceding. However, I fail to see how that gives us a general definition for the sum of two natural numbers. I was really expecting something more along the lines of, "c is the sum of the natural numbers a and b if and only if blah blah blah." Anyway, can someone please tell me how I interpret the above as the definition of two natural numbers?

Thanks for any input.

... doug

2. ## Re: Definition of Sum

Well first consider how the natural numbers were constructed:

1 = $\{\emptyset\}$
2 = $\{\emptyset, \{\emptyset\}\}$
3 = $\{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}$

So each element of the new number is the set of one of the numbers that came before it. We also include the empty set as an element of each new number as well. (It's also how we get ordering, as seen 2 is a subset of 3)

So C is the sum of A,B if and only if the cardinality of C is equal to the trivial cardinality (equivalent elements are treated separately) of $A \cup B$.

Just so you know I just came up with this... so if its flawed or just dumb... don't hate too hard

3. ## Re: Definition of Sum

The recursive definition of Wikipedia and the set-theoretic definitions are two different approaches, One is not needed for the other.

Recursive functions have definitions like

f(0) = m
f(n') = G(f(n))

for some already known function G. Since you know f(0), you can calculate G(f(0)) and so f(0'), i.e., f(1). Then you can calculate G(f(1)), which is f(1'), or f(2), and so on. This is calculating bottom up. In practice, you need to compute, e.g., f(0'''), which you reduce to G(f(0'')), G(G(f(0')) and so on (this computation is called top-down).

Addition is a function of two arguments defined by recursion on the second argument (it could as well be defined by recursion on the first one). We have

a + 0 = a
a + b' = (a + b)'

E.g., 2 + 3 = 2 + 0''' = (2 + 0'')' = (2 + 0')'' = (2 + 0)''' = 2''' = 5.

4. ## Re: Definition of Sum

Does it bother anyone else that we are here talking about a successor element within the context of set? My understanding was that the order of elements within a set is irrelevant, e.g., { 3, 2, 1 } = { 1, 2, 3 }.

... doug

5. ## Re: Definition of Sum

Originally Posted by ddjolley
Does it bother anyone else that we are here talking about a successor element within the context of set? My understanding was that the order of elements within a set is irrelevant, e.g., { 3, 2, 1 } = { 1, 2, 3 }.

... doug
The successor function is a map from a set to itself and is not concerned by the order in which you list the elements of the set in the notation you use above.

CB


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## Factor quadratics with positive coefficients

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Suppose your height above sea level in feet when traveling over a hill can be represented by the expression , where is the horizontal distance traveled. If you wanted to factor this expression, could you do it? What would be the steps that you would follow?

Previously, we factored common monomials, so you already know how to factor quadratic polynomials where . Now, we will learn how to factor quadratic polynomials for different values of , and .

Quadratic polynomials are polynomials of degree 2. The standard form of a quadratic polynomial is , where , and are real numbers.

A quadratic of the form factors to the product of two binomials:  where  and .

There are some important rules to know when factoring quadratics:

• If and are positive then both and  are positive.
• Example: factors as .
• If is negative and is positive then both and  are negative.
• Example: factors as .
• If is negative then either is positive and  is negative or vice-versa.
• Example: factors as .
• Example: factors as .
• If , factor a common factor of –1 from each term in the trinomial and then factor as usual. The answer will have the form .
• Example: factors as .

#### Let's factor the following polynomials:

We are looking for an answer that is the product of the two binomials in parentheses: .

To fill in the blanks, we want two numbers and that multiply to 6 and add to 5. A good strategy is to list the possible ways we can multiply two numbers to give us 6 and then see which of these pairs of numbers add to 5.

Since both  and  are positive, then both spaces will be filled by positive numbers.

The number six can be written as the product of 1 and 6 or 2 and 3. Testing the product and sums we get:

So the answer is  because 2 and 3 multiply to 6 and sum to 5.

We can check to see if this is correct by multiplying .

is multiplied by and .

2 is multiplied by and .

Combine the like terms: .

We are looking for an answer that is the product of the two binomials in parentheses: .

Since  is negative and  is positive, both of the blanks are going to be negative. We only want the negative factors of 8.

The number 8 can be written as the product of -1 and -8 or -2 and -4. Testing the product and sums, we get:

-2 and -4 are the factors that multiply to 8 and add to -6. This is the correct choice.

We are looking for an answer that is the product of the two binomials in parentheses: .

Since  is positive and  is negative, one of the blanks will be negative and one will be positive. The number –15 can be written as the product of -1 and 15, 1 and -15, 3 and -5, or -3 and 5. Testing the product and sums, we get:

-3 and 5 are the factors that multiply to -15 and add to 2. The answer is .

### Examples

#### Example 1

Earlier, you were told that your height above sea level when traveling over a hill can be represented by the expression , where  is the horizontal distance traveled. What is the factored form of this expression?

First, factor the common factor of -1 from each term in the trinomial. Factoring -1 changes the signs of each term in the expression.

We are looking for an answer that is the product of the two binomials in parentheses: .

Now our job is to factor . Since  is negative and  is negative, one of the blanks is positive and one is negative.

The number –63 can be written as the product of -1 and 63, 1 and -63, -3 and 21, 3 and -21, -7 and 9, or 7 and -9. Testing the products and sums, we get:

3 and -21 are the factors that multiply to -63 and sum to -18.The answer is .

#### Example 2

Factor .

Like in Example 1, first factor the common factor of –1 from each term in the trinomial.

We are looking for an answer that is the product of the two binomials in parentheses: .

Now our job is to factor .

The number –6 can be written as the product of the following numbers.

-3 and 2 are the factors that multiply to -6 and add to -1. The answer is .

### Review

Mixed Review

1. Evaluate when .
2. Simplify .
3. Graph the following on a number line: .
4. What is the multiplicative inverse of ?

#### Quick Quiz

1. Name the following polynomial. State its degree and leading coefficient: .
2. Simplify .
3. A rectangular solid has dimensions by by . Find its volume.
4. Simplify .
5. Find the solutions to .
6. Multiply .

To see the Review answers, open this PDF file and look for section 9.8.

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English Spanish

TermDefinition
Quadratic Polynomials A quadratic polynomial is a polynomial of the 2nd degree, in other words, a polynomial with an $x^2$ term.
standard form of quadratic polynomials The standard form of a quadratic polynomial is $ax^2+bx+c$, where $a,\ b$, and$c$ are real numbers.
Greatest Common Factor The greatest common factor of two numbers is the greatest number that both of the original numbers can be divided by evenly.
linear factors Linear factors are expressions of the form $ax+b$ where $a$ and $b$ are real numbers.
Polynomial A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.
Quadratic Formula The quadratic formula states that for any quadratic equation in the form $ax^2+bx+c=0$, $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$.
Trinomial A trinomial is a mathematical expression with three terms.


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# when g of calcium metal is reacted manufacture

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