Standard deviation

Standard deviation calculator

The standard deviation is a statistic that measures the data variability. It is derived from the square root of the distances between each value in the population and the population's mean squared.

Standard deviation formula

standard deviation formula
Variance = σ2

When you use a sample to estimate the population's standard deviation you should use the following formula.

Sample standard deviation formula

standard deviation formula
Sample variance = S2

We use n-1 instead of n, to correct the biased estimation of the variance (partially correct the estimation of the standard deviation) (Bessel's correction).

If you ask a school kid how to measure the variability, he will probably suggest one of the following:
1. Range: Maximum-minimum: Max(𝑥_i)-Min(𝑥i).
2. The average of the absolute differences.

Average =Σi|𝑥i-x̄|
n

Why does the standard deviation formula use squares differences instead of absolute differences?
1. In Mathematics it is more elegant to use second power and square root functions than to use the absolute function, hence easier calculation. (squares is continuously differentiable, easy to derive for a minimum or maximum, while absolute is not).
2. Absolute differences grant equal weight to any difference while square differences grant larger weight to a big difference. This may be good when the outlier is correct, a higher weight emphasizes that the data may spread more, but it may be worse when the outlier is incorrect, See example, and the large weight of the incorrect observation influences more the result.

Example - Low and High Standard deviation

In the following example, 'high' and 'low' are only relative.
MAD - Mean Absolute Deviation, the average absolute distance from the average.

low standard deviation
Data: 6,6.1,5.8,6.9,5.5,6.2,5.9,6.3.
Average absolute distance from the average: 0.29.
Standard deviation: 0.39
low standard deviation chart
High standard deviation
Data: 7,1,6,2,11,3,4,5,2,3,4,5,9,7,5,5,8.
Average absolute distance from the average: 2.03.
Standard deviation: 2.59.
high standard deviation chart

Dartboard example

Following a visual example of the standard deviation calculation.
The red circle in the middle symbolizes the average.
The radius represents the absolute difference from the average.

Values: 4, 1, 9, 10

Average =4 + 1 + 9 + 10=6
4
Radiuses: |4-6|=2, |1-6|=5, |9-6|=3, |10-6|=4.

Press the "run" button to run the demonstration!
Value: 4
Difference (r): 2
s
Value: 4
Difference (r): 2
s
Value: 1
Difference (r): 5
s
Value: 1
Difference (r): 5
s
Value: 9
Difference (r): 3
s
Value: 9
Difference (r): 3
s
Value: 10
Difference (r): 4
s
Value: 10
Difference (r): 4
s
Value: 2.33 or 9.67
Standard deviation (r): 3.67
s
Value: 2.33 or 9.67
Standard deviation (r): 3.67
s
Orange circle: each circle represent one value. (r = |value-average|, area = π r2).
Green circle: has the average area of all the orange circles.
Green circle area=π*22+π*52+π*32+π*42=π*3.672
4
The standard deviation is the radius of the green circle: 3.67.
The π doesn't influence the result, it is only for the visualization. If we would use the square shape instead of the cycle we would get the same result.

Outlier effect - example

In the following example, we took a special case when the standard deviation and the mean absolute deviation are similar.
Even one outlier in a group of 24 influence dramatically both measurements, the MAD and the standard deviation, but it affected much more the standard deviation.

Base
Data: 5.9,6.4,5.9,6.4,5.8,6.2,5.9,6.3,5.9,6.4,5.9,6.4,5.8,6.2,5.9,6.3,5.9,6.4,5.9,6.4,5.8,6.2,5.9,6.3
MAD: 0.23.
Standard deviation (σ): 0.23
low standard deviation chart
Outlier - 8
Data: 5.9,6.4,5.9,6.4,5.8,6.2,5.9,6.3,5.9,6.4,5.9,6.4,5.8,6.2,5.9,6.3,5.9,6.4,5.9,6.4,5.8,6.2,5.9,8
MAD: 0.3 (+30%).
Standard deviation (σ):0.45 (+96%).
high standard deviation chart
Outlier - 12
Data: 5.9,6.4,5.9,6.4,5.8,6.2,5.9,6.3,5.9,6.4,5.9,6.4,5.8,6.2,5.9,6.3,5.9,6.4,5.9,6.4,5.8,6.2,5.9,12
MAD: 0.5 (+117%).
Standard deviation (σ): 1.2 (+420%).
high standard deviation chart