Variance and Standard Deviation
Understanding how spread out data is from the mean — the foundation of statistical analysis.
Why Do We Need Measures of Spread?
Consider two students who both have a test average of 80%. One scored 78, 80, 82, 80 while the other scored 50, 100, 70, 100. Their means are the same, but their data tells very different stories. Variance and standard deviation quantify this difference — they measure how far data values tend to be from the mean.
Population Variance & Standard Deviation
When you have data for an entire population (every member of the group):
Population Variance:
$$ \sigma^2 = \frac{\sum_{i=1}^{N}(x_i - \mu)^2}{N} $$
Population Standard Deviation:
$$ \sigma = \sqrt{\frac{\sum_{i=1}^{N}(x_i - \mu)^2}{N}} $$
- $ \mu $ = population mean
- $ N $ = population size
- $ x_i $ = each data value
Sample Variance & Standard Deviation
When you have data from a sample (a subset of the population):
Sample Variance:
$$ s^2 = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n - 1} $$
Sample Standard Deviation:
$$ s = \sqrt{\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n - 1}} $$
- $ \bar{x} $ = sample mean
- $ n $ = sample size
- $ n - 1 $ = degrees of freedom (Bessel's correction)