How to Calculate Standard Deviation: Sample and Population

 

Standard deviation describes how spread out a set of numbers is around its arithmetic mean. A small value indicates that observations are relatively close to the mean in their measurement units. A larger value indicates more spread, but whether that spread is important depends on the context and scale of the data.

The Standard Deviation Calculator calculates sample or population standard deviation, variance and mean from a list. It also shows the count, sum of squared deviations, divisor, minimum and maximum so you can follow the calculation.

Choose sample or population

Use population standard deviation when the list contains the complete group you want to describe. Divide the sum of squared deviations by the number of observations, n. A single population observation has zero variance and zero standard deviation.

Use sample standard deviation when observations are used to estimate variability in a broader population under the usual sampling assumptions. The sample variance divides by n - 1 rather than n. This correction makes sample variance unbiased under standard independent sampling assumptions with finite variance; its square root is not exactly unbiased for population standard deviation.

Sample standard deviation requires at least two observations. The choice depends on what the list represents, not simply whether the list is large or small.

Calculate the mean and squared deviations

Start with these eight values: 2 4 4 4 5 5 7 9. Their sum is 40, so the mean is 40 / 8 = 5. Subtract 5 from each observation to obtain -3, -1, -1, -1, 0, 0, 2 and 4.

Square those differences to obtain 9, 1, 1, 1, 0, 0, 4 and 16. Their sum is 32. Squaring prevents positive and negative differences from canceling, and gives larger deviations greater influence.

Find population variance and standard deviation

For the complete population, divide 32 by 8. Population variance is 4. Take the square root to obtain population standard deviation: sqrt(4) = 2.

In the calculator, choose Calculation type as Population (divide by n). Enter Numbers as 2 4 4 4 5 5 7 9, then select Calculate standard deviation. Check that Count (n) is 8, Mean is 5 and Sum of squared deviations is 32.

Calculate sample standard deviation

Change Calculation type to Sample (divide by n - 1) and calculate again. The divisor becomes 7. Sample variance is 32 / 7 = 4.571429, rounded. Its square root gives sample standard deviation of approximately 2.138090.

The observations, mean and sum of squared deviations remain unchanged. Only the divisor changes. For the same nonconstant list, sample standard deviation is larger than population standard deviation because it uses the smaller divisor.

Interpret units and unusual observations

Standard deviation uses the original measurement units. If observations are lengths in centimeters, standard deviation is in centimeters, while variance is in square centimeters. Adding a constant to every observation shifts the mean but leaves the spread unchanged. Multiplying every observation by three multiplies standard deviation by three and variance by nine.

Outliers can substantially increase standard deviation. Check unusual entries against your source before deciding whether they are errors. These descriptive calculations do not require normality, but interpreting 68 percent of values as lying within one standard deviation does require an approximately normal distribution.

Enter and check the list

Separate observations with spaces, commas, semicolons or line breaks. Use a period for decimals and avoid thousands separators: 1,000 is interpreted as two observations. Each repeated number counts separately. The calculator accepts up to 10000 observations but does not accept frequency tables or weights.

Results are rounded for display and use floating-point arithmetic. Very small differences relative to large values can lose precision. For a standardized value under a normal model, use the Z-Score Calculator.