Find Standard Deviation with a Checked Worked Example

Define the spread you want to describe

Standard deviation measures the size of deviations around an arithmetic mean using squared differences. It is nonnegative and has the same unit as the observations. Before starting, decide whether the list is the entire finite population you intend to describe or a sample used to estimate a wider population's variation. The arithmetic steps are similar, but the denominator changes. The calculator displays both answers because it cannot infer your sampling purpose from a list of numbers.

Find the mean of a small data set

Take 2, 4, 4, 4, 5, 5, 7 and 9. There are eight observations and their sum is 40, so the mean is 5. Keep each repeated value in the list: the three observations equal to four are three measurements, not one. Enter the values separated by commas in the local form and verify the count and mean. A correct mean provides a useful first check before comparing either standard deviation output.

Square and add the deviations

Subtract the mean five from every observation. The deviations are −3, −1, −1, −1, 0, 0, 2 and 4. Their squares are 9, 1, 1, 1, 0, 0, 4 and 16, with total 32. Squaring prevents positive and negative deviations from cancelling. Taking the ordinary average of the signed deviations would give zero and would not measure spread. The zero deviations here still count as observations even though they contribute nothing to the sum of squares.

Choose population or sample standard deviation

If these eight observations are the complete population of interest, divide 32 by eight to obtain population variance 4 and then take its square root: population SD=2. If they are a sample used to estimate population variance, divide by seven instead. The sample variance is 32/7≈4.571429 and sample SD≈2.13809. The n−1 correction applies to the variance estimator after the mean has been estimated from the same data; it does not make the square-root estimator exactly unbiased in general.

Verify a second list

For 10, 12 and 14, the mean is 12 and the squared deviations total 8. Population variance is 8/3 and population SD≈1.632993. Sample variance is 8/2=4 and sample SD=2. The calculator should show these two distinct standard deviations. This small example makes the denominator difference easy to inspect. With a larger sample the relative difference is smaller, but the correct choice still follows the statistical definition rather than whichever displayed number is more convenient.

Keep spread separate from uncertainty in a mean

The standard deviation describes variation among observations. Under an independent-observation model, the standard error of a sample mean is estimated by s/√n, a different quantity. For the three-value example, s=2 while the estimated standard error is 2/√3≈1.154701. Reporting the latter as though it were the data's SD understates the observed spread. A confidence interval also needs an appropriate critical value and assumptions; neither the SD nor the standard error alone specifies that interval.

Interpret units and transformations

Adding the same constant to every observation leaves standard deviation unchanged because both the values and their mean shift together. Multiplying every observation by a factor a multiplies the SD by |a| and the variance by a². Changing the second list from centimetres to millimetres multiplies its sample SD from 2 to 20 while preserving the relative pattern. These transformation checks are useful for identifying a unit mismatch. Standard deviation cannot meaningfully combine measurements whose units were never made consistent.

Check limitations before making distribution claims

A standard deviation can be calculated without assuming normality. Claims that a fixed percentage of observations lies within one or two SDs require additional distributional context; the familiar normal percentages are not universal. Outliers can strongly increase SD because their deviations are squared. Inspect the source list and explain any exclusions rather than removing inconvenient values. A constant list has SD zero, and sample SD requires at least two observations. The local form requires two even when only a population summary is wanted.

Related guides

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