More probability density is found as one gets closer to the expected (mean) value in a normal distribution. Statistics used in standardized testing assessment are shown. The scales include standard deviations, cumulative percentages, percentile equivalents, Z-scores, T-scores, standard nines, and percentages in standard nines.
Example of two samples with the same mean and different standard deviations. Red sample has mean 100 and SD 10; blue sample has mean 100 and SD 50. Each sample has 1000 values drawn at random from a gaussian distribution with the specified parameters.
Normal distribution curve that illustrates standard deviations. Each band has 1 standard deviation, and the labels indicate the approximate proportion of area.
Remember that the probability of observing a result given that some hypothesis is true is not equivalent to the probability that a hypothesis is true given that some result has been observed. Using the p-value as a “score” is committing the transposed conditional fallacy.