Variance, Standard Deviation and Standard Error

This is an short explanation of Variance, Standard Deviation and Standard Error.

The standard deviation of the mean (SD) is the most commonly used measure of the spread of values in a distribution. SD is calculated as the square root of the variance (the average squared deviation from the mean).

Variance in a population is

\[\sigma^2 = \frac{\sum_{i=1}^N(y_i - \mu)^2}{N}\]

where:

• \(y_i\): individual value from population
  
• \(\mu\): mean of all \(y_i\)
  
• \(N\): total individual

Variance is usually estimated from a sample drawn from a population. The unbiased estimate of population variance calculated from a sample is

\[s^2 = \frac{\sum_{i=1}^n(y_i - \bar{y})^2}{n-1}\]

where:

• \(y_i\): individual value from the sample
  
• \(\bar{y}\): mean of all \(y_i\) in the sample
  
• \(n\): sample size
  
• \(n-1\): degree of freedom

The spread of a distribution is also referred to as dispersion and variability. All three terms mean the extent to which values in a distribution differ from one another.

The standard error of the mean (SEM) is the expected value of the standard deviation of means of several samples, this is estimated from a single sample as:The standard error of the mean is the expected value of the standard deviation of means of several samples, this is estimated from a single sample as:

\[ SEM = \frac{s}{\sqrt{n}}\]

where

• \(s\): standard deviation
  
• \(n\): sample size

In conclusion, the standard deviation (SD) measures the amount of variability, or dispersion, from the individual data values to the mean, while the standard error of the mean (SEM) measures how far the sample mean (average) of the data is likely to be from the true population mean. The SEM is always smaller than the SD.