# What is the difference between empirical variance and variance?

As far as I know variance is calculated as

$$\text{variance} = \frac{(x-\text{mean})^2}{n}$$

while

$$\text{Empirical Variance} = \frac{(x-\text{mean})^2}{n(n-1)}$$

Is it correct? Or is there some other definition? Kindly explain with example or any refence for reading on this topic

-

I have used Latex to alter the presentation of your question.  If this is not what you intended, let me know – Henry Jun 1 '11 at 9:34

## 1 Answer

In your expression for the variance, you need to take a sum (or integral) across the population

$$\text{variance} = \frac{\sum_i(x_i-\text{mean})^2}{n}$$

If your data is a sample from the population then this expression will give you a biased estimate of the population variance. An unbiased estimate would be as follows (note the change in the denominator from your expression), often called the sample variance

$$\text{Sample variance} = \frac{\sum_i(x_i-\text{mean})^2}{n-1}$$

If on the other hand you were trying to estimate the variance of the sample mean, then you vould have a smaller number, closer to your expression. The square root of this is called the standard error of the mean and a reasonable estimate is

$$\text{Standard error} = \sqrt{\frac{\sum_i (x_i-\text{mean})^2}{n(n-1)}}$$

-

2
See en.wikipedia.org/wiki/Bias_of_an_estimator#Sample_variance for an explanation why the variance $1/n \sum_{i} (x_{i} - \bar{x})^2$ is a biased estimator, and vdov.net/~acosta/content/mle-normal for an explanation why it is the maximum-likelihood estimator for normal variables. – caracal Jun 1 '11 at 13:48
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 5, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9117439389228821, "perplexity_flag": "head"}