Standard Deviation Formula

The Formula

\sigma = \sqrt{\frac{\sum(x - \mu)^2}{n}}

When to use: The typical distance from the average. Low SD = clustered. High SD = spread out.

Quick Example

Data: 5, 5, 5, 5 has SD = 0. Data: 1, 3, 5, 7, 9 has SD \approx 2.83.

Notation

\sigma for population SD, s for sample SD (which divides by n - 1)

What This Formula Means

The standard deviation measures the average distance of data values from the mean, giving a typical spread around the center.

The typical distance from the average. Low SD = clustered. High SD = spread out.

Formal View

\sigma = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_i - \mu)^2} (population); s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2} (sample)

Worked Examples

Example 1

medium
Find the population standard deviation of \{2, 4, 4, 4, 5, 5, 7, 9\}.

Solution

  1. 1
    Compute the mean: \bar{x} = \frac{2+4+4+4+5+5+7+9}{8} = \frac{40}{8} = 5.
  2. 2
    Find each squared deviation: (2-5)^2 = 9, (4-5)^2 = 1 (three times), (5-5)^2 = 0 (twice), (7-5)^2 = 4, (9-5)^2 = 16.
  3. 3
    Sum of squared deviations: 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32.
  4. 4
    Variance: \sigma^2 = \frac{32}{8} = 4.
  5. 5
    Standard deviation: \sigma = \sqrt{4} = 2.

Answer

\sigma = 2
The standard deviation measures how spread out data values are from the mean. A small standard deviation means values cluster near the mean, while a large one indicates greater spread.

Example 2

hard
Find the sample standard deviation of \{10, 12, 23, 23, 16, 23, 21, 16\}.

Common Mistakes

  • Forgetting to square the deviations before summing — using |x - \mu| instead of (x - \mu)^2
  • Dividing by n when the sample formula requires n - 1 (or vice versa)
  • Interpreting SD as a percentage of the mean — SD is in the same units as the data, not a relative measure

Why This Formula Matters

Standard deviation is the most widely used measure of spread in statistics — it appears in confidence intervals, z-scores, normal distributions, and hypothesis tests.

Frequently Asked Questions

What is the Standard Deviation formula?

The standard deviation measures the average distance of data values from the mean, giving a typical spread around the center.

How do you use the Standard Deviation formula?

The typical distance from the average. Low SD = clustered. High SD = spread out.

What do the symbols mean in the Standard Deviation formula?

\sigma for population SD, s for sample SD (which divides by n - 1)

Why is the Standard Deviation formula important in Math?

Standard deviation is the most widely used measure of spread in statistics — it appears in confidence intervals, z-scores, normal distributions, and hypothesis tests.

What do students get wrong about Standard Deviation?

SD uses squared differences, so negative distances become positive.

What should I learn before the Standard Deviation formula?

Before studying the Standard Deviation formula, you should understand: mean, square roots.

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