What is the difference between the mean and the median?

The mean shares the total out evenly across every observation, so each number affects it in proportion to its size. The median is the middle value of the sorted data — it cares only about order, not about how extreme an extreme value is. That is why one very large number shifts the mean a long way and barely moves the median.

How do you find the median of an even sample?

Sort the data, take the two middle values and average them. For 3, 5, 8, 10 the middles are 5 and 8, so the median is (5 + 8)/2 = 6.5. The result need not belong to the data set — and very often it does not.

When does the mean mislead?

Whenever the data has extreme values or is clearly skewed — salaries, house prices, waiting times. One large number lifts the mean until most of the data sits below it, so the "average" result stops describing anybody. In sets like that the median is the honest measure.

Can a data set have no mode?

Yes. If every value occurs exactly once, none is the most frequent and there is no mode. The opposite happens too: when two values share the highest frequency the set has two modes and is called bimodal. The mode is the only one of the three measures that need not exist and need not be unique.

How do you find a missing value when the mean is known?

Use the fact that the total equals n · x̄. Work out what the whole total has to be, subtract the values you know, and what is left is the number you are after. This is the commonest way a mean shows up in a "backwards" problem.

What is a weighted mean?

A mean in which every value carries its own weight — it counts as many times as the weight says. That is how a grade average works when the exam counts three times and a quiz once. The ordinary arithmetic mean is the special case where all the weights are equal.

Does the median always belong to the data set?

Only when the sample size is odd. Then the median is one specific middle observation. With an even sample it is the mean of the two middle values, so it may be a number that never occurs in the data — a result ending in .5, for instance.

Which measure should I use?

The mean when the data is reasonably symmetric and free of extremes, because it uses every observation. The median when a few very large or very small numbers sit in the set. The mode when the data are categories rather than numbers — the commonest shoe size, the commonest grade. The safest answer is to give two of them and point out the gap if there is one.

Basic level

Mean, median and mode

Three answers to one question: which single number best describes a whole data set. The mean shares the total out evenly, the median stands exactly in the middle, and the mode points at the most frequent value — and which one you pick can change the conclusion.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

DivisionDivision is the inverse of multiplication — divide the dividend by the divisor to get the quotient. Learn the names, the link to multiplication, division with a remainder and why you must never divide by zero.
DecimalsA decimal writes part of a whole with a decimal point instead of a fraction bar. Learn place value, converting a fraction to a decimal, adding, multiplying and rounding.

All formulas

Arithmetic mean
$\bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n}$
the sum of all the data divided by how many there are
Total from the mean
$x_1 + x_2 + \dots + x_n = n \cdot \bar{x}$
this is how a missing value is recovered when the mean is known
Median — an odd sample
$\mathrm{Me} = x_{\left(\frac{n+1}{2}\right)} \quad (n \ \text{odd})$
once sorted, it is exactly the middle number
Median — an even sample
$\mathrm{Me} = \frac{x_{\left(\frac{n}{2}\right)} + x_{\left(\frac{n}{2}+1\right)}}{2} \quad (n \ \text{even})$
the mean of the two middle values
Mode
$\mathrm{Mo} = \text{the value with the highest frequency}$
the number that occurs most often — there may be none at all
Weighted mean
$\bar{x}_w = \frac{w_1 x_1 + w_2 x_2 + \dots + w_k x_k}{w_1 + w_2 + \dots + w_k}$
every value counts as many times as its weight says

A data set is usually a long list of numbers, and the question is always the same: in one number — what does it look like? Descriptive statistics gives three answers, and each measures something different. All three are worth knowing, because the gap between them is often the whole point of a problem.

The arithmetic mean

The mean is the sum of all the data divided by how many there are:

\bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n}

The symbol $\bar{x}$ is read "x bar". The idea is one of sharing out: if everyone had the same, everyone would have exactly the mean.

Find the mean of 12, 7, 9, 8, 14.

A second, equally useful formula follows straight from the first — multiply both sides by $n$ :

x_1 + x_2 + \dots + x_n = n \cdot \bar{x}

Knowing the mean means knowing the total. That is all it takes to recover a missing number.

The mean of five numbers is 8. Four of them are 6, 9, 10 and 5. What is the fifth?

A frequency diagram

Instead of writing every observation out, you can show how many times each value occurs. Here is the spread of test grades in a class of 24:

The height of a bar is the number of students with that grade. The vertical guide stands where the mean falls.

The mean of such a diagram is found by multiplying every grade by its frequency:

\bar{x} = \frac{1 \cdot 1 + 2 \cdot 3 + 3 \cdot 6 + 4 \cdot 8 + 5 \cdot 4 + 6 \cdot 2}{24} = \frac{89}{24} \approx 3.71

That is already a weighted mean, with the frequencies as the weights:

\bar{x}_w = \frac{w_1 x_1 + w_2 x_2 + \dots + w_k x_k}{w_1 + w_2 + \dots + w_k}

The ordinary arithmetic mean is its special case — the one where every weight is the same.

Notice one thing: $3.71$ is not a real grade at all. A mean need not belong to the data set, and usually does not.

The median

The median is the middle value of the data once it is sorted. The sorting is part of the recipe, not decoration — without it the "middle" number means nothing.

For an odd sample size there is a single middle:

\mathrm{Me} = x_{\left(\frac{n+1}{2}\right)}

For an even one there are two, so we average them:

\mathrm{Me} = \frac{x_{\left(\frac{n}{2}\right)} + x_{\left(\frac{n}{2}+1\right)}}{2}

Find the median of 7, 2, 9, 4, 5 and of 10, 3, 6, 8.

The mode

The mode is the value that occurs most often. On a frequency diagram you can see it immediately — it is the tallest bar. In the class drawn above the mode is the grade $4$ : eight students got it.

The mode is the only one of the three measures that need not exist:

when every value occurs once, there is no mode;
when two values share the highest frequency, there are two (a bimodal set).

It has one advantage the others lack: it works where the data are not numbers at all. The commonest shoe size or the commonest car colour is a mode, and nobody can average colours.

When the mean misleads

Take a company of seven and their salaries, in thousands:

3,\ 4,\ 4,\ 5,\ 5,\ 6,\ 22

A single extreme value pushes the mean above what six of the seven earn. The median stays where the middle of the data really is.

Both measures:

\bar{x} = \frac{3 + 4 + 4 + 5 + 5 + 6 + 22}{7} = \frac{49}{7} = 7, \qquad \mathrm{Me} = 5

"People here earn 7 thousand on average" is true and misleading at the same time: six out of seven earn less. The reason is simple — the mean uses the size of every number, so one extreme value drags it across. The median sees only order: if the boss earned 220 thousand instead of 22, the median would still be 5.

That is the whole reason salaries, house prices and waiting times are reported as medians rather than means.

Three measures on one drawing

A dot plot shows it best, with one dot per observation. Take the set $2,\ 3,\ 3,\ 4,\ 13$ :

Four observations sit close together, the fifth far away. The median and the mode stay in the thick of the data; the mean drifts towards the outlier.

For this set:

\bar{x} = \frac{2 + 3 + 3 + 4 + 13}{5} = \frac{25}{5} = 5, \qquad \mathrm{Me} = 3, \qquad \mathrm{Mo} = 3

The gap between the mean and the median is information in itself: it says the data is skewed and that an outlier is sitting somewhere in it.

Which measure to use

measure	what it measures	when it is good
mean	the "shared out evenly" level	symmetric data, no extremes
median	the middle of the sorted data	data with outliers
mode	the most frequent value	categories, non-numeric data

The honest thing is to quote two of them. If the mean and the median are close, the data is reasonably symmetric; if they are far apart, neither one alone describes anything.

Exercises

The five kinds of question match the five things this lesson teaches. A prompt starting with $\bar{x}$ and a colon asks for the mean of the data that follows. A prompt of the form $\bar{x} = 8$ with an x inside the data asks for the missing value — use the total $n \cdot \bar{x}$ . The Median and Mode prompts give the data in random order, so the first step is always to sort it. Every answer is a number; with an even sample the median may end in a half — type it, for instance 6.5.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

x̄: 11, 10, 4, 17, 3

Common mistakes

A median without sorting — the middle number of an unsorted list is not the median, just an arbitrary value.
Dropping the half — with an even sample the median often ends in .5, and rounding it to something "nicer" is wrong.
Dividing by the wrong count — divide by the number of observations, not by the number of distinct values.
Taking the mode to be the largest value — the mode is the most frequent one, not the biggest.
Confusing the measures — "average" in everyday speech often means the median; a problem asks for whatever it names.
Trusting the mean alone — in a set with an outlier the mean describes the data worse than the median does.

Formula card

Topic: Mean, median and mode

Arithmetic mean
$\bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n}$
the sum of all the data divided by how many there are
Total from the mean
$x_1 + x_2 + \dots + x_n = n \cdot \bar{x}$
this is how a missing value is recovered when the mean is known
Median — an odd sample
$\mathrm{Me} = x_{\left(\frac{n+1}{2}\right)} \quad (n \ \text{odd})$
once sorted, it is exactly the middle number
Median — an even sample
$\mathrm{Me} = \frac{x_{\left(\frac{n}{2}\right)} + x_{\left(\frac{n}{2}+1\right)}}{2} \quad (n \ \text{even})$
the mean of the two middle values
Mode
$\mathrm{Mo} = \text{the value with the highest frequency}$
the number that occurs most often — there may be none at all
Weighted mean
$\bar{x}_w = \frac{w_1 x_1 + w_2 x_2 + \dots + w_k x_k}{w_1 + w_2 + \dots + w_k}$
every value counts as many times as its weight says

A frequency diagram of test grades: the bar above each grade shows how many students got it. The vertical guide stands where the mean falls, x̄ ≈ 3.71 — where no real grade exists at all.

The salaries of seven people, in thousands. The mean (7) sits above what six of the seven earn, because one extreme value pulls it up; the median (5) describes this company more honestly.

Frequently asked questions

Luminous histogram bars settling into the soft bell of a distribution, one outlying point brighter than the rest, against a cosmic nebula in violet and magenta

Statistics and probability

The average person does not exist

All of descriptive statistics rests on a single trade: you give up information about a crowd and receive one number in return. The trade is often worth making, but it is never free — and it pays to know exactly what gets lost in it. From the mortality tables of 1662, through Quetelet's average man, to the aircraft nobody ever saw.

16 min