Statistics⏱ 6 min read

Standard Deviation

σ = √(Σ(x−μ)² / n)

What is the Standard Deviation?

Standard deviation measures how spread out data values are from the mean. A small standard deviation means values cluster tightly around the mean; a large one means they're spread far apart — two classes could have the exact same average test score, but very different standard deviations if one class has everyone clustered near that average and the other has scores ranging from very low to very high.

There are two versions: population standard deviation (dividing by n) is used when the data set represents an entire population. Sample standard deviation (dividing by n−1 instead) is used when the data is only a subset — that adjustment, known as Bessel's correction, corrects for the fact that a sample's own mean is calculated from the same data being measured, which tends to slightly underestimate the true spread of the full population if left uncorrected.

Standard deviation is the building block for the normal distribution's well-known 68-95-99.7 rule, and it's central to how confidence intervals, margins of error, and statistical significance are all calculated — almost every claim in a scientific study or poll that includes an "error bar" or a stated margin of error is really reporting some multiple of a standard deviation.

What Each Variable Means

Standard deviation (sigma)The measure of spread being calculated.

Mean (mu)The average of all data values.

xᵢ

Each data valueEvery individual value in the data set.

CountThe total number of data points. Use (n−1) instead of n for a sample standard deviation.

When to Use It

Quantifying how consistent or variable a data set is
Comparing the spread of two different data sets with similar means
As the basis for the normal distribution's 68-95-99.7 rule

Step-by-Step Example

Problem: Find the standard deviation of {2, 4, 4, 4, 5, 5, 7, 9}.

Find the mean

Add all values and divide by the count.

μ = (2+4+4+4+5+5+7+9)/8 = 40/8 = 5

Subtract the mean from each value, then square

Compute (x−μ)² for every data point.

(2-5)²=9, (4-5)²=1 (×3), (5-5)²=0 (×2), (7-5)²=4, (9-5)²=16

Sum the squared differences

Add all eight squared values.

9+1+1+1+0+0+4+16 = 32

Divide by n, then take the square root

Complete the formula.

σ = √(32/8) = √4

✓

Answer: σ = 2

Common Mistakes

Mistake: Using n instead of (n−1) for a sample's standard deviation.
Fix: If the data is a sample rather than the full population, divide by (n−1), not n — this correction (Bessel's correction) keeps the sample statistic from systematically underestimating the true population spread.
Mistake: Forgetting to square the differences before summing.
Fix: Simply summing (x−μ) without squaring always gives zero, since positive and negative deviations cancel out — squaring first is essential.

Practice Questions

Find the standard deviation of {1, 3, 5, 7, 9}.
Hint: Mean = 5; squared deviations are 16, 4, 0, 4, 16.
Would a data set with all identical values have a standard deviation of 0?

Frequently Asked Questions

What's the difference between variance and standard deviation?

Variance is the average of the squared deviations (σ² in this formula, before the square root). Standard deviation is its square root, which brings the measure back into the same units as the original data.

What does the 68-95-99.7 rule mean?

In a normal distribution, about 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.

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