Free Math tool · Step by step

Standard Deviation Calculator

Calculate sample or population standard deviation, variance, mean, median, mode and quartiles from a list of numbers, with the full deviation table. Free for AP Statistics, IB and A-level.

Sample standard deviation

2.44949

Variance 6 · Mean 14.5 · n = 8

Count (n)
8
Sum
116
Mean
14.5
Median
14.5
Mode
none
Min / Max
11 / 18
Range
7
Q1 / Q3
12.5 / 16.5
Population SD (σ)
2.29129
Sample SD (s)
2.44949
Population variance
5.25
Sample variance
6
xx − mean(x − mean)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
Sum of squares42
  1. 1

    Mean

    xˉ=∑xn=1168=14.5\bar{x} = \frac{\sum x}{n} = \frac{116}{8} = 14.5
  2. 2

    Square each deviation from the mean and add

    ∑(x−xˉ)2=(12−14.5)2+(15−14.5)2+(11−14.5)2+(18−14.5)2+(14−14.5)2+(16−14.5)2+(13−14.5)2+(17−14.5)2=42\sum (x-\bar{x})^2 = (12-14.5)^2 + (15-14.5)^2 + (11-14.5)^2 + (18-14.5)^2 + (14-14.5)^2 + (16-14.5)^2 + (13-14.5)^2 + (17-14.5)^2 = 42
  3. 3

    Sample variance: divide by n − 1

    s2=427=6s^2 = \frac{42}{7} = 6
  4. 4

    Standard deviation = square root of variance

    s=6=2.44949s = \sqrt{6} = 2.44949

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FAQ

Sample or population — which one do I use?

Use population (divide by n) when your data is the whole group. Use sample (divide by n − 1) when the data is a sample from a bigger group — the usual case in AP Statistics.

How is standard deviation calculated?

Find the mean, square each value’s distance from the mean, add them, divide by n (or n − 1) to get the variance, then take the square root.

How are the quartiles found?

Q1 and Q3 are the medians of the lower and upper halves (the median itself is left out when n is odd) — the method used in most school courses.

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