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.

Where Mean and SD Is Used

Summarising data

Reporting the centre and spread of measurements.

Comparing groups

Seeing which data set is more variable.

Spotting unusual values

Using z-scores to find points far from the mean.

Coursework

Checking mean and standard deviation by hand.

Common Mean and SD Mistakes to Avoid

Use the sample SD when your data are a sample from a larger group.
Use the population SD only when you have every member of the group.
Outliers inflate the SD, so check the dot plot.

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?

Add the values and divide by n for the mean, then take the square root of the average squared deviation for the SD.

Should I use n or n − 1?

Use n − 1 for a sample and n for a complete population.

What does a large SD mean?

The values are spread widely around the mean.

Can the SD be negative?

No. It is a square root and is zero only when all values are equal.

Mean and SD Calculator

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