Standard Deviation Formula

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

The Formula

ฯƒ=โˆ‘(xโˆ’ฮผ)2n\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= 0. Data: 1, 3, 5, 7, 9 has SD โ‰ˆ2.83\approx 2.83.

Notation

ฯƒ\sigma for population SD, ss for sample SD (which divides by nโˆ’1n - 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

ฯƒ=1nโˆ‘i=1n(xiโˆ’ฮผ)2\sigma = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_i - \mu)^2} (population); s=1nโˆ’1โˆ‘i=1n(xiโˆ’xห‰)2s = \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}\{2, 4, 4, 4, 5, 5, 7, 9\}.

Answer

ฯƒ=2\sigma = 2

First step

1
Compute the mean: xห‰=2+4+4+4+5+5+7+98=408=5\bar{x} = \frac{2+4+4+4+5+5+7+9}{8} = \frac{40}{8} = 5.

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Example 2

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

Example 3

medium
Find the population standard deviation of {3,7,7,19}\{3, 7, 7, 19\}.

Common Mistakes

  • Forgetting the square root โ€” that gives variance, not standard deviation.
  • Dropping the squaring of deviations โ€” positive and negative distances would cancel; square them first.
  • Dividing a sample by nn when you should divide by nโˆ’1n-1 โ€” sample SD (ss) uses nโˆ’1n-1, population SD (ฯƒ\sigma) uses nn.

Why This Formula Matters

Standard deviation is the workhorse of spread: it sets the scale for z-scores, defines the width of the normal curve, and lets you compare a value's unusualness across different data sets. Because it is in the original units (dollars, inches), it is the spread number people actually interpret. Recognizing it by "Am I measuring how far values typically fall from the mean, in the data's own units?" โ€” rather than by familiar numbers โ€” is what lets a student tell it apart from variance and range and mean absolute deviation (mad) in a mixed problem set.

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?

ฯƒ\sigma for population SD, ss for sample SD (which divides by nโˆ’1n - 1)

Why is the Standard Deviation formula important in Math?

Standard deviation is the workhorse of spread: it sets the scale for z-scores, defines the width of the normal curve, and lets you compare a value's unusualness across different data sets. Because it is in the original units (dollars, inches), it is the spread number people actually interpret. Recognizing it by "Am I measuring how far values typically fall from the mean, in the data's own units?" โ€” rather than by familiar numbers โ€” is what lets a student tell it apart from variance and range and mean absolute deviation (mad) in a mixed problem set.

What do students get wrong about Standard Deviation?

The procedure for standard deviation is the easy part; the trap is forgetting the square root. Asking "Am I measuring how far values typically fall from the mean, in the data's own units?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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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