Standard Deviation Calculator

Standard deviation measures how far values typically sit from their mean. Paste a data set to get both the sample (s) and population (σ) standard deviation, the variance and a full deviation table you can copy into homework.

8 values read.

Sample SD (s)
2.4495
Population SD (σ)
2.2913
Variance s²
6
Mean
14.5
Count n
8
Sum of squares
42
±1 SD1112131415161718
▲ mean 14.5 · ◆ median 14.5 · shaded band = mean ± 1 SD (12.0505 to 16.9495)

Step-by-step working

  1. Count the valuesn = 8
  2. Add the values and divide by the count to get the meanx̄ = (12 + 15 + 11 + 18 + 14 + 16 + 13 + 17) ÷ 8 = 116 ÷ 8 = 14.5
  3. Subtract the mean from each value, then square the result (see the table)(x − x̄)² for each x, e.g. (12 − 14.5)² = 6.25
  4. Add up the squared deviations (the sum of squares)SS = Σ(x − x̄)² = 42
  5. Divide by n − 1 for the sample variance (Bessel’s correction)s² = 42 ÷ 7 = 6
  6. Take the square root to get the standard deviations = √6 = 2.44949
Deviation table
xx − x̄(x − x̄)²
12−2.56.25
150.50.25
11−3.512.25
183.512.25
14−0.50.25
161.52.25
13−1.52.25
172.56.25
Σx = 116≈ 0SS = 42

Formula s = √[ Σ(x − x̄)² ÷ (n − 1) ] σ = √[ Σ(x − μ)² ÷ N ]

Ask the tutor about your result

An AI tutor reads your inputs and results above and explains them in plain English. The numbers come from the calculator; the explanation is AI-written, so check it against your notes.

See why standard deviation works

Drag any dot (or focus it and use ← →). Each square’s side is that value’s distance from the mean, so its area is the squared deviation. Standard deviation is the side of the “average” square.

012345678910mean 5.17

Σ(x − x̄)² = 10.03 + 1.36 + 1.36 + 0.03 + 3.36 + 14.69 = 30.83
s = √(30.83 ÷ 5) = 2.48

Why n − 1? A sample’s values sit closer to their own mean than to the true population mean, so squared deviations from x̄ run slightly small. Dividing by one less (Bessel’s correction) makes the variance unbiased. Notice the difference shrinks as n grows.

How to calculate standard deviation by hand

  1. Find the mean: add the values and divide by how many there are.
  2. Subtract the mean from each value to get its deviation.
  3. Square each deviation (this makes them all positive).
  4. Add the squares. This is the sum of squares, SS.
  5. Divide SS by n − 1 for a sample or by N for a whole population. That is the variance.
  6. Take the square root.

Example: for 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the squared deviations add to 32, so σ = √(32 ÷ 8) = 2 and s = √(32 ÷ 7) ≈ 2.138.

Sample or population?

Use the sample formula (n − 1) when your numbers are a subset used to estimate a larger group: a class survey, lab replicates, 30 bags from a production line. Use the population formula (N) only when you have every member of the group you care about, such as the scores of every student in one class when you only want to describe that class. When unsure, homework almost always wants the sample version. On a TI-84, Sx is sample and σx is population; in Excel, STDEV.S and STDEV.P.

What the number tells you

Standard deviation is in the same units as the data. For roughly bell-shaped data, about 68% of values fall within one SD of the mean and about 95% within two. A small SD means the values cluster tightly; a large one means they are spread out. To compare spread between data sets with different units or very different means, use the coefficient of variation.

Frequently asked questions

Why divide by n − 1 instead of n?

Deviations are measured from the sample mean, which is always the best-fitting centre for that sample, so they come out slightly too small compared with deviations from the true population mean. Dividing by n − 1 (Bessel’s correction) makes the sample variance an unbiased estimate of the population variance.

Can standard deviation be negative?

No. It is the square root of an average of squared numbers, so it is always zero or positive. It is zero only when every value is identical.

What is the difference between standard deviation and variance?

Variance is the average squared deviation; standard deviation is its square root. Standard deviation is easier to interpret because it is in the original units rather than squared units.

How do I find standard deviation on a TI-84?

Enter the data in L1 (STAT → Edit), then STAT → CALC → 1-Var Stats. Sx is the sample standard deviation and σx is the population standard deviation.