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Mean and standard deviation
Intermediate
Statistics describing the centre and spread of repeated measurements.
σ = √(Σ(x−x̄)²/(n−1))The mean x̄ is the best estimate of the true value from repeated readings. The standard deviation σ measures the typical scatter of individual readings, and σ/√n, the standard error, measures how well the mean is known. Report results as x̄ ± standard error.
Readings 9.8, 9.9, 10.1, 10.2: mean 10.0, deviations −0.2, −0.1, 0.1, 0.2, σ ≈ 0.18.
Standard deviation does not shrink with more data; the standard error of the mean does.
In more detail
The mean is the average of repeated readings, and the standard deviation describes how much they spread around that mean. The standard error of the mean, the standard deviation divided by the square root of n, shows how precisely the average is known.
Real-world uses
- Physics experiments report the mean and its standard error from repeated timings.
- Quality control uses the standard deviation to set process limits on production lines.
- A quality control team reports the mean diameter of 50 bolts along with the standard deviation to show how consistent the production line is.
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Frequently asked questions
What is the difference between standard deviation and standard error?
Standard deviation describes the spread of single readings. Standard error describes the uncertainty in the mean and shrinks as you take more readings.
Is a low standard deviation always good?
It means the readings are consistent, but a consistent instrument can still have a systematic error.
How is the standard error of the mean related to the standard deviation?
The standard error equals the standard deviation divided by the square root of the number of readings, so it shrinks as you take more data.
Related topics in Measurement & Math Tools
Error in a sum or difference · Error in product or quotient · Error in powers · Dimensions of a quantity · Dimensional homogeneity · Checking equations by dimensions