Data Learn — Mean, Median and Spread Basics

What does it mean to summarise a list of numbers?

You replace the whole list with a handful of figures that describe its centre and its spread.

A raw list of numbers is hard to hold in your head. Descriptive statistics compress it: one or two figures for where the values cluster, one or two for how widely they scatter. The compression loses detail on purpose, so the choice of figures matters.

Measures of central tendency are the mean, the median and the mode. Measures of variability are the standard deviation, the variance, the minimum and maximum, kurtosis and skewness. For a first pass you will normally use the mean or median, the min and max, and the standard deviation.

Everything on this page runs on the same eight-number set: 1, 2, 3, 4, 5, 6, 8, 9. Every figure below can be checked by hand in under a minute.

Centre: mean, median or mode — pick one, not all three.
Spread: range for a quick check, standard deviation for a proper one.
Shape: skewness tells you whether the mean and median have drifted apart.

The three measures of central tendency

Mean, median and mode are three different answers to 'where does this list sit?'

The arithmetic mean of n values is their sum divided by n. For the worked set the sum is 38 and n is 8, so the mean is 4.75. The mean uses every value, which makes it sensitive to any extreme one.

The median is the value that separates the higher half of a data sample from the lower half. With an even number of observations there is no single middle value, so the median is defined as the arithmetic mean of the two middle values. For 1, 2, 3, 4, 5, 6, 8, 9 that is (4 + 5) / 2 = 4.5.

The mode is the value that appears most often. The worked set has no repeated value, so it has no mode — a common and legitimate outcome on small lists.

Use the mean when the distribution is roughly symmetric and no value dominates.
Use the median when extreme values carry reduced importance, typically because the distribution is skewed.
Use the mode for categorical or repeating data, such as shoe sizes sold.

The median separates the higher half of a data sample from the lower half — and with an even count it is simply the mean of the two values on either side of the cut.

Procedure dossier

Measures of spread: how far are the values from the centre?

Range gives the width, variance and standard deviation give the typical distance from the mean.

The range is the maximum minus the minimum: 9 − 1 = 8. It is one subtraction and it is driven entirely by the two extreme values.

The variance averages the squared distances from the mean, and the standard deviation is its square root, back in the original units. Squaring first means a value four units from the mean contributes sixteen times as much as one unit away.

Kurtosis and skewness describe the shape of the distribution rather than its width: skewness says which tail is longer, kurtosis says how heavy the tails are. Both are reported alongside the centre and spread in a full summary.

Range: max − min, quick but fragile.
Variance: average squared deviation, in squared units.
Standard deviation: square root of the variance, in original units.
Interquartile range: Q3 − Q1, robust to outliers.

A worked example on the set 1, 2, 3, 4, 5, 6, 8, 9

Sort, count, split, average — four moves give you the centre; two more give you the spread.

Start by sorting: 1, 2, 3, 4, 5, 6, 8, 9. Count n = 8, which is even, so there is no single middle value.

The two middle values are the 4th and 5th: 4 and 5. The median is their arithmetic mean, (4 + 5) / 2 = 4.5. The mean is the sum 38 divided by 8, which is 4.75.

For spread: range = 9 − 1 = 8. The full ordered procedure, with the variance and standard deviation worked out, is on the step-by-step page.

Mean or median: which one should you report?

Add one extreme value and watch the mean jump while the median barely moves.

The median is used as a measure of location when extreme values carry reduced importance, typically because a distribution is skewed. Salary figures, house prices and waiting times are the standard examples: one very large value pulls the mean upward but shifts the median by at most half a rank.

The mean stays the better summary when the distribution is symmetric, because it uses every observation and has nicer mathematical properties for further analysis.

The comparison table on the mean-versus-median page shows both figures before and after a single extreme value is added to the worked set.

Skewed data → report the median, note the mean for reference.
Symmetric data → report the mean, note the standard deviation.
Small n with one odd value → report both and say which you chose.

Key figures at a glance

Key figures for the set 1, 2, 3, 4, 5, 6, 8, 9 (n = 8)
Measure	Value	Unit	How it is found
Mean	4.75	same as data	Sum 38 ÷ 8
Median	4.5	same as data	(4 + 5) ÷ 2
Mode	none	same as data	No value repeats
Minimum	1	same as data	Smallest observation
Maximum	9	same as data	Largest observation
Range	8	same as data	9 − 1
Percentile position of median	50th	percent	2nd quartile, 5th decile

Key figures for the set 1, 2, 3, 4, 5, 6, 8, 9 (n = 8)

The worked set, annotated

Every figure below is derived from 1, 2, 3, 4, 5, 6, 8, 9 and can be recomputed by hand.

The key terms of this guide, drawn to one scale

Where these definitions come from

Standard definitions of mean, median, mode, quartiles, variance, standard deviation, kurtosis and skewness as used in introductory statistics.