How do I calculate standard deviation?

Find the mean, subtract it from each value and square the result, add those squared differences, divide by n (population) or n − 1 (sample), then take the square root. This calculator does it automatically and shows every step for your data.

Population or sample standard deviation — which do I use?

Use population (÷ n) when your data is the whole group you care about; use sample (÷ n − 1) when it is a sample meant to represent a larger population. Most statistics courses and studies use the sample version. The calculator shows both.

What is the difference between variance and standard deviation?

Variance is the average of the squared distances from the mean; standard deviation is its square root. Standard deviation is in the same units as your data, so it is easier to interpret, while variance is in squared units.

Is there always a mode?

No. The mode is the most frequently occurring value. If every value appears exactly once there is no mode; if several values tie for the highest frequency, the set is multimodal and all of them are modes. This tool lists them, or shows "none".

Standard Deviation Calculator

Free standard deviation calculator with full descriptive statistics: paste your data for the mean, median, mode, range, variance and standard deviation (population and sample), quartiles, and outliers — with the standard-deviation steps shown. Download as CSV.

Your data (one column per data set) Chart

Statistic	Scores
N	10
Sum	846
Mean	84.6
Median	86.5
Mode	88
Min	73
Max	94
Range	21
Q1	79
Q3	90
IQR	11
Variance (pop.)	42.84
Variance (sample)	47.6
SD (pop.)	6.5452
SD (sample)	6.8993
Outliers	none

Standard deviation, step by step — Scores

Mean = 84.6. Subtract the mean from each value and square it:

x	x − mean	(x − mean)²
82	-2.6	6.76
88	3.4	11.56
76	-8.6	73.96
91	6.4	40.96
85	0.4	0.16
79	-5.6	31.36
94	9.4	88.36
88	3.4	11.56
73	-11.6	134.56
90	5.4	29.16

Sum of squared deviations Σ(x − mean)² = 428.4, with n = 10.

Population: ÷ n = 428.4 ÷ 10 = 42.84 (variance) → √ = 6.5452
Sample: ÷ (n − 1) = 428.4 ÷ 9 = 47.6 (variance) → √ = 6.8993

What descriptive statistics are

Descriptive statistics summarize a set of numbers with a few key figures: measures of center (mean, median, mode) and measures of spread (range, interquartile range, variance, and standard deviation). This calculator computes all of them from data you paste in, and shows the standard-deviation calculation step by step so you can follow — or check — the work.

What standard deviation and variance measure

Standard deviation measures how spread out the values are around the mean — a small standard deviation means the data cluster tightly near the average, a large one means they are widely scattered. It is the square root of the variance, which is the average of the squared distances from the mean. Standard deviation is reported in the same units as the data (variance is in squared units), which is why it is the more commonly quoted figure.

Worked example

Take the eight values 2, 4, 4, 4, 5, 5, 7, 9. The mean is 40 ÷ 8 = 5. Subtract the mean from each value and square the result: (−3)², (−1)², (−1)², (−1)², 0², 0², 2², 4² = 9, 1, 1, 1, 0, 0, 4, 16, which sum to 32. The population variance divides by n: 32 ÷ 8 = 4, so the population standard deviation is √4 = 2. The sample variance divides by n − 1: 32 ÷ 7 ≈ 4.57, so the sample standard deviation is √4.57 ≈ 2.14.

Population vs. sample

There are two versions of variance and standard deviation. The population version divides the sum of squared deviations by n; use it when your data is the entire group you care about. The sample version divides by n − 1 (Bessel’s correction) and gives a slightly larger, less biased estimate of a wider population’s spread from a sample — use it when your data is a sample drawn from something bigger. Most inferential statistics use the sample version, which is why a calculator set to "population" can disagree with your textbook. This tool shows both so you can pick the one your problem calls for.

How to use it

Paste your numbers into the box — one column per data set, straight from a spreadsheet.
Read the summary table: mean, median, mode, range, quartiles, variance and standard deviation (both population and sample), and any outliers.
Follow the step-by-step standard-deviation calculation below the chart to check the work.
Switch the chart between a dot plot and a histogram, and download the summary as CSV or the chart as PNG, SVG, or PDF.

FAQ

How do I calculate standard deviation?: Find the mean, subtract it from each value and square the result, add those squared differences, divide by n (population) or n − 1 (sample), then take the square root. This calculator does it automatically and shows every step for your data.
Population or sample standard deviation — which do I use?: Use population (÷ n) when your data is the whole group you care about; use sample (÷ n − 1) when it is a sample meant to represent a larger population. Most statistics courses and studies use the sample version. The calculator shows both.
What is the difference between variance and standard deviation?: Variance is the average of the squared distances from the mean; standard deviation is its square root. Standard deviation is in the same units as your data, so it is easier to interpret, while variance is in squared units.
Is there always a mode?: No. The mode is the most frequently occurring value. If every value appears exactly once there is no mode; if several values tie for the highest frequency, the set is multimodal and all of them are modes. This tool lists them, or shows "none".