13 Research methods, statistics, critical review and evidence-based practice13.8 Descriptive statistics

13.8.5 Dispersion and standard error

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Written by Resident Doctor Committee · Published

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Where the standard error comes from

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The means of 20 samples of 25 observations, drawn from a population with a mean of 100 and a standard deviation of 15. They scatter with a standard deviation of 3.00 around a standard error of 3.00.Population mean60708090100110120130140Sample mean

Sample means drawn so farThe sampling distribution in theory

Each sample is this many observations; its mean is one bar.

Samples drawn
20
Standard error
3.0015 over the root of 25
Spread of the means so far
3.00
Mean of the means so far
100.05

The population never changes. Only the sample size does, and the standard error falls with the square root of it: to halve the error you need four times the patients.

Draw repeated samples of 25 from a population with a standard deviation of 15. The sample means pile up with a spread of about 3, the standard error. Raise the sample size and the means close in, while the spread of the data themselves stays at 15.

New in the 2027 syllabus

The College added this topic for exams from January 2027.

Measures of spread

Dispersion means how spread out the values are.

Measure

What it is

Units

Range

Largest value minus smallest

Original units

Interquartile range

Upper quartile minus lower quartile

Original units

Variance

Average squared distance from the mean, using the right denominator

Squared units

Standard deviation

Square root of the variance

Original units

Standard error of the mean

Estimated spread of sample means across repeated samples

Original units

  • Sample variance divides the sum of squared deviations by n minus 1. Use it to estimate the population variance from independent observations.

  • Population variance divides by the population size. Use it when you are describing every value in the population.

Keep the two ideas apart. The standard deviation (SD) describes how much individual observations vary. The standard error describes how uncertain an estimate is.

Worked example

Four hypothetical measurements have a mean of 5 units.

Value

Value minus mean

Squared deviation

2

-3

9

4

-1

1

6

1

1

8

3

9

  1. Work out the mean: (2 + 4 + 6 + 8) / 4 = 5.

  2. The range is 8 minus 2 = 6 units. The squared deviations add up to 20.

  3. Sample variance = 20 / (4 minus 1) = 6.67 squared units.

  4. Sample SD = square root of 6.67 = 2.58 units.

  5. Standard error of the mean = 2.58 / square root of 4 = 1.29 units.

Standard error and sample size

  • The usual standard error formula assumes independent sampling. Clustered observations, or observations correlated over time, need an analysis that allows for this dependence.

  • If the underlying variability stays the same, quadrupling n halves the standard error. It does not make individual observations any less variable.

  • The standard error alone is not an interval. A confidence interval combines the estimate, its standard error and a suitable critical value.

These are distributions of sample means for normal observations with population standard deviation 1; increasing n from 16 to 64 halves the standard error. Biggerj1, CC BY 4.0, via Wikimedia Commons

Outliers

An outlier is an observation that lies unusually far from the others. It may be an error, come from a different population or be a genuine extreme value.

  • The range, variance and SD are sensitive to extreme values.

  • The interquartile range looks only at the middle half of the data.

When you find an outlier, check where it came from and how much it changes the results. Do not delete it automatically. Doing so can remove real information and introduce bias.

Revision material for the MRCPsych examinations. Not clinical guidance, and not for use in patient care.