Skip to content
Hard

Variance and Standard Deviation

Simple Explanation

Variance and standard deviation measure how far, on average, data values spread out from the mean. Variance, σ², averages the SQUARED distance of each value from the mean; standard deviation, Οƒ, is the square root of variance, bringing the measurement back into the original units.

Why Do We Need It?

Standard deviation is the single most widely-used measure of spread in statistics β€” it uses every data value (unlike the range or IQR), giving the most complete picture of variability.

Formula

Variance and Standard Deviation

Variance: σ² = Ξ£(x βˆ’ xΜ„)Β² / n. Standard deviation: Οƒ = √(σ²)

Measures how far, on average, the data values are spread from the mean β€” variance is the average of the squared deviations; standard deviation is its square root, bringing the units back in line with the original data.

x
β€” each individual data value
xΜ„
β€” the mean of the dataset
n
β€” the number of data values
Οƒ (sigma)
β€” the standard deviation

When to use it: Whenever you need the standard, most widely-used measure of how spread out a dataset is around its mean.

Worked Example

Find the standard deviation of a small dataset

Find the standard deviation of: 2, 4, 6, 8.

    Why Does This Work?

    Squaring each deviation before averaging is necessary because the deviations from the mean always sum to exactly zero (positive and negative deviations cancel) β€” squaring makes every deviation positive first, so the average genuinely measures spread instead of always giving zero. Taking the final square root undoes the earlier squaring, restoring the original units.

    Real-Life Example

    Comparing consistency of two athletes' race times

    Two runners have the same average race time, but one's times vary much more from race to race than the other's.

    The runner with the smaller standard deviation is more consistent β€” standard deviation directly quantifies that consistency, something the mean alone cannot show.

    Practice

    Find the variance of: 1, 3, 5, 7, 9. (Mean = 5.)

    Hard

    Common mistake

    Forgetting to square the deviations before averaging β€” averaging the raw (unsquared) deviations from the mean always gives exactly 0, since they cancel by definition.

    Quick Review

    • σ² = Ξ£(xβˆ’xΜ„)Β²/n (variance); Οƒ = √(σ²) (standard deviation).
    • Squaring prevents positive and negative deviations from cancelling to zero.
    • A smaller standard deviation means data is more tightly clustered around the mean.
    ← Back to StatisticsNext: Cumulative Frequency Tables and the Ogive β†’