Mean and SD Calculator: Average, Sample and Population SD

Each value, its deviation and z-score
x	x − x̄	(x − x̄)²	z = (x − x̄)/SD
12	-3	9	-1.162
15	0	0	0
11	-4	16	-1.549
18	3	9	1.162
14	-1	1	-0.3873
16	1	1	0.3873
13	-2	4	-0.7746
19	4	16	1.549
15	0	0	0
17	2	4	0.7746

MeanΣx / n

Sample SD√(Σ(x−x̄)²/(n−1))

Population SD√(Σ(x−μ)²/N)

z-score(x − x̄)/SD

What Is Mean and SD?

The mean is the average of a data set and the standard deviation (SD) measures how far values typically lie from it. Because the SD depends on whether your data are a sample or a complete population, this calculator shows both, divided by n − 1 and by n.

Besides the mean and SD it returns the variances, the coefficient of variation, the ranges mean ± 1 SD and mean ± 2 SD with the count of values inside each, and the z-score of every value in a working table. A dot plot and histogram show the shape of the data.

Below you will find the formulas, a variables table, a worked example and answers about the difference between sample and population SD.

How to Use the Mean and SD Calculator

Paste or type your numbers, separated by commas, spaces or new lines.
Choose whether the headline SD is the sample or the population version.
Read the mean, both standard deviations and the variances.
Check the ±1 and ±2 SD ranges and the z-score of each value.

Mean and SD Formula

Mean

\bar x=\frac{\sum x_i}{n}

Sample standard deviation

s=\sqrt{\frac{\sum (x_i-\bar x)^2}{n-1}}

Population standard deviation

\sigma=\sqrt{\frac{\sum (x_i-\mu)^2}{N}}

Mean and SD Variables Explained

Symbol	Meaning	Notes
x̄	Mean	Σx ÷ n
s	Sample standard deviation	divides by n − 1
σ	Population standard deviation	divides by N
z	z-score of a value	(x − x̄) ÷ SD

Mean and SD Worked Example

For 2, 4, 4, 4, 5, 5, 7, 9: the mean is 40 ÷ 8 = 5. The squared deviations sum to 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32. The population SD is √(32 ÷ 8) = 2 and the sample SD is √(32 ÷ 7) = 2.138.

With the example values loaded in the calculator above, the steps are:

Mean x̄ = Σx ÷ n = 150 ÷ 10 = 15.
Sum of squared deviations Σ(x − x̄)² = 60.
Sample SD s = √(60 ÷ 9) = 2.5819889.
Population SD σ = √(60 ÷ 10) = 2.4494897.

Result: Mean x̄ = 15; Sample SD s = 2.5819889.

How to Read the Mean and SD

The mean is the centre and the SD is the typical distance from it. For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the sum of squared deviations is 32, the population SD is √(32 ÷ 8) = 2 and the sample SD is √(32 ÷ 7) = 2.138.

For roughly bell-shaped data, about 68% of values fall within one SD of the mean and about 95% within two. The table lists each value, its deviation and its z-score, so you can see which points are far from the mean.

Where Mean and SD Is Used

Describing data

Report a mean and SD for a sample.

Spotting extremes

Check the z-score of each value.

Comparing spread

Use the coefficient of variation for data on different scales.

Homework

Check the squared deviations by hand.

Common Mean and SD Mistakes to Avoid

Use n − 1 for a sample and n for a population.
The 68–95 pattern applies only to roughly normal data.
A large SD relative to the mean means the data are widely spread.
Report the mean and SD with sensible rounding.

Mean and SD Terms and Definitions

SD: Standard deviation: a typical distance of values from the mean.
z-score: How many SDs a value is above or below the mean.

Mean and SD FAQs

How do I calculate the mean and SD?

Find the mean, subtract it from each value, square the results, add them, divide by n − 1 for a sample, and take the square root.

Should I use n or n − 1?

Use n − 1 for a sample from a larger group, and n only when you have the entire population.

What does a large SD mean?

The values are spread widely around the mean. Compare it with the mean through the coefficient of variation to judge how large it really is.

Can the SD be negative?

No. It is a square root of an average of squares, so it is zero or positive. It is zero only when all values are equal.

Mean and SD Calculator

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