Mean, median and mode · Statistics: data & averages

An average is one number that stands for a whole set of data. There are three, and they answer slightly different questions:

The mean shares the total out equally: add all the values, divide by how many there are.
The median is the middle value once the data is in order.
The mode is the value that appears most often.

Try it

Three kinds of averageDrag the dots

6.78mean (balance point)7median (middle value)7mode (most common)9range (largest − smallest)

The mean is where the dot plot would balance: add everything and divide by how many (3 + 5 + 5 + 6 + 7 + 7 + 7 + 9 + 12 = 61, ÷ 9). The median is the middle value once they're in order. The mode is the value that appears most often.

Drag the dots. The mean is the balance point: the triangle under the line is where a ruler holding these dots would balance. The median has half the dots on each side. Now press Add an unusually large value. One large value drags the mean towards it, but the median hardly moves. That’s why house prices and salaries are often described by the median.

The mean

\text{mean} = \frac{\text{sum of the values}}{\text{number of values}}

For $4, 7, 7, 9, 13$ : the sum is $40$ and there are 5 values, so the mean is $40 \div 5 = 8$ .

The total trick

Many mean questions give you a mean and ask about a missing or changed value. Turn the mean back into a total:

\text{total} = \text{mean} \times \text{number of values}

More: the total trick

The median

Put the values in order first. With $n$ values, the median is the value in position $\frac{n + 1}{2}$ .

Odd number of values: one middle value. In $2, 3, 5, 8, 9$ the median is $5$ .
Even number of values: two middle values, so take the number halfway between them. In $2, 3, 5, 8, 9, 12$ the middle two are $5$ and $8$ , so the median is $6.5$ .

The mode

The mode is the value with the highest frequency. There can be more than one mode (for example, $2, 2, 5, 5, 7$ has modes $2$ and $5$ ).

More: mean, median and mode of a list

Averages from a frequency table

A frequency table is a short way of writing a long list. “Score 4, frequency 10” means the score 4 appears 10 times.

Score $x$	1	2	3	4	5	6
Frequency $f$	2	6	12	10	6	4

Mean: multiply each value by its frequency, add, then divide by the total frequency: $\begin{aligned} \bar x = \frac{\sum fx}{\sum f} &= \frac{2 + 12 + 36 + 40 + 30 + 24}{40} \\ &= \frac{144}{40} = 3.6 \end{aligned}$
Median: there are 40 values, so the median is halfway between the 20th and 21st. Keep a running total of the frequencies: $2, 8, 20, 30, \ldots$ . The 20th value is a 3 and the 21st is a 4, so the median is $3.5$ .
Mode: the highest frequency is 12, so the mode is $3$ .

More: averages from a frequency table

Worked example · WAEC 2016

WAEC 2016 · Paper 2 · Q12 (a)

Marks	3	4	5	6	7	8	9
Frequency	1	4	3	5	2	$x$	2

The table shows the distribution of marks of students in a Mathematics test. If the mean of the distribution is 6, calculate the:

value of $x$ ;

Write the totals in terms of x
$\sum f = 1 + 4 + 3 + 5 + 2 + x + 2 = 17 + x$ .
$\sum fx = 3 + 16 + 15 + 30 + 14 + 8x + 18$ , each value times its frequency.
$\sum fx = 96 + 8x$ , adding the numbers.
Think first. What is $\sum f$ ? What is $\sum fx$ ?
Use the mean
$\frac{96 + 8x}{17 + x} = 6$
Think first. The mean is 6. Write an equation.
Solve
$96 + 8x = 102 + 6x \quad\Rightarrow\quad 2x = 6 \quad\Rightarrow\quad x = 3$
Check: $\sum f = 20$ and $\sum fx = 120$ , and $120 \div 20 = 6$ ✓.

Your turn

WAEC 2022 · Paper 2 · Q10 (a)

Age (years)	3	4	5	6	7	8	9	10
Number of children	2	6	5	$x$	6	9	8	5

The table shows the distribution of ages of a number of children in a school. If the mean of the distribution is 7, find the:

(a)
value of $x$ ;

Worked solution (try it first)

(a)

$\sum f = 2 + 6 + 5 + x + 6 + 9 + 8 + 5$
$= 41 + x$ and $\sum fx = 6 + 24 + 25 + 6x + 42 + 72 + 72 + 50$
$= 291 + 6x$ .
The mean is 7: $\frac{291 + 6x}{41 + x} = 7$ .
So $291 + 6x = 287 + 7x$ , which gives $x = 4$ .
Check: $\sum f = 45$ , $\sum fx = 315$ , and $\frac{315}{45} = 7$ .

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