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Mathematics · pattern F, Dataset

Standard deviation, with every intermediate quantity.

Paste your data for the standard deviation, the mean, the sum of squared deviations and the quartiles — everything a marker expects to see written out.

Inputs

Sample or population

Divide by n − 1 for a sample drawn from a wider population; by n only when the data is the entire population.

The formula used

s = √[Σ(x − x̄)² ÷ (n − 1)]

Standard deviation is in the same units as the data, which is why it is the reported measure of spread. The sample form divides by n − 1.

Standard deviation

8.045

n = 10 · sample, ÷ (n − 1)

n
10
Sum
715.000
Mean
71.500
Median
71.000
Mode
no repeated value
Σ(x − x̄)²
582.500
Variance
64.722
Standard deviation
8.045
Standard error
2.5441
Minimum / maximum
59.00 / 85.00
Range
26.000
Q1 / Q3 / IQR
66.50 / 76.25 / 9.75

Your data — accent bars sit beyond one deviation

59.0085.00

Sample or population matters most at small n.

On ten observations, dividing by nine rather than ten raises the standard deviation by about five per cent; on a thousand the difference is negligible. Use the sample form unless the data genuinely is the whole population.

Questions about standard deviation

What does one standard deviation cover?
About 68% of a normal distribution, 95% within two and 99.7% within three. Those figures do not hold for non-normal data.
Why not use mean absolute deviation?
It is valid and less tractable. Squaring gives the additive property that statistical theory relies on.
How does it relate to standard error?
Standard error is the standard deviation divided by √n, and describes the precision of the mean rather than the spread of the data.

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