Standard Deviation & Variance Calculator

Measures of Dispersion: Standard Deviation & Variance

In descriptive and inferential statistics, measuring central tendency (mean, median, mode) only reveals half the picture. Two datasets can have identical averages (e.g., 50) while representing vastly different risk profiles: one clustered between 48 and 52, and another scattered between 0 and 100. Standard deviation quantifies the average spread or dispersion of data around the mean.

Mathematical Formulas

1. Mean (Arithmetic Average)

$$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$$

2. Sample Standard Deviation ($s$)

Used when your data is a sample drawn from a larger population: $$s = \sqrt{\frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n - 1}}$$

3. Population Standard Deviation ($\sigma$)

Used when your dataset encompasses all members of the group of interest: $$\sigma = \sqrt{\frac{\sum_{i=1}^N (x_i - \mu)^2}{N}}$$

Step-by-Step Procedure

1. Calculate the Mean ($\bar{x}$): Add all values and divide by the total count $n$.
2. Find the Deviations: Subtract the mean from each individual data point: $(x_i - \bar{x})$.
3. Square Each Deviation: Square each difference to eliminate negative signs: $(x_i - \bar{x})^2$.
4. Sum the Squared Deviations ($SS$): Total all squared deviations: $\sum (x_i - \bar{x})^2$.
5. Divide by Degrees of Freedom: Divide by $(n - 1)$ for sample variance, or by $N$ for population variance.
6. Take the Square Root: The square root yields the standard deviation.