IMAT Mathematics · Probability and Statistics

Mean, Median, Mode and Spread

The mean, median and mode describe the centre of a data set, the range and standard deviation describe its spread, and a mean is always a total in disguise.

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Why this matters

The statistics questions so far all come from the older papers (2015 to 2022): a mean after the data change, a combined mean for two groups, and a median or a largest value found from the mean and other facts. The ministry papers have not asked one yet, but averages are on the syllabus.

Concept 1 of 3: Mean, median, mode and range

The mean shares the total out equally; the median is the middle value once the data are in order; the mode is the most common value. The most useful trick is to turn a mean back into a total: if 5 numbers have mean 8, they add up to 40, whatever they are.

Definition

Mean $\bar{x} = \dfrac{\text{sum of the values}}{\text{number of values}}$ , so the total is mean $\times$ number.
Median: put the values in order; for an odd count take the middle one, for an even count the mean of the two middle ones.
Mode: the value that occurs most often (there can be more than one).
Range: largest value minus smallest value.

Mean

\bar{x} = \frac{\sum x}{n} \qquad \sum x = n\,\bar{x}

$\sum x$ sum of all the values
$n$ number of values

Worked example

Find the mean, median, mode and range of 7, 3, 9, 3, 12, 8.

Practice this conceptself-check · 4 quick reps

Order the data before finding the median

The median is the middle of the ordered list, not the middle of the list as written. With an even number of values it is the mean of the two middle values, which need not be one of the data.

Concept 2 of 3: Combined and weighted means, and how averages change when data change

Two groups pooled together give a mean pulled towards the larger group, because the larger group contributes more to the total. When every value is shifted by the same amount, the centre moves but the spread does not; when every value is scaled, centre and spread both scale.

Definition

Combined mean of two groups: add the totals, divide by the total number. It equals the simple average of the two means only when the groups are the same size.
A weighted mean works the same way: $\bar{x} = \dfrac{\sum w x}{\sum w}$ , with each value $x$ counted $w$ times.
Adding $c$ to every value adds $c$ to the mean, median and mode; the range and standard deviation do not change.
Multiplying every value by $k$ multiplies the mean, median, mode and range by $k$ (the spread by $|k|$ ).
Adding a new value above the mean raises the mean; below the mean lowers it.

Combined mean

\bar{x} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}

$n_1, n_2$ sizes of the two groups
$\bar{x}_1, \bar{x}_2$ their means

Worked example

Class A has 20 students with a mean mark of 64; class B has 30 students with a mean mark of 74. What is the mean mark of all 50 students?

Practice this conceptself-check · 4 quick reps

A combined mean is not the average of the two means

Pooling a group of 20 and a group of 30 weights the second more. Work through the totals; the simple average of the two means is right only when the groups are equal in size.

Concept 3 of 3: Frequency tables and standard deviation (in outline)

A frequency table is a shorthand for a long list: "value 2, frequency 6" means six 2s. The standard deviation measures how far the values typically sit from the mean: tightly bunched data have a small one, scattered data a large one.

Definition

In a frequency table, value $x$ occurs $f$ times. The number of values is $\sum f$ and the mean is $\dfrac{\sum f x}{\sum f}$ .
The median is the value in position $\tfrac{n + 1}{2}$ of the ordered list; use running totals of $f$ to find it. The mode is the value with the highest frequency.
Standard deviation $\sigma$ : the square root of the mean of the squared distances from the mean. Its square is the variance.
$\sigma = 0$ only when all values are equal. Adding a constant leaves $\sigma$ unchanged; multiplying by $k$ multiplies $\sigma$ by $|k|$ .

Mean of a frequency table and standard deviation

\bar{x} = \frac{\sum f x}{\sum f} \qquad \sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{n}}

$f$ frequency of the value $x$
$\sigma$ standard deviation

Worked example

A football team's goals in 20 matches: 0 goals in 4 matches, 1 goal in 7, 2 goals in 6, 3 goals in 3. Find the mean, the median and the mode.

Practice this conceptself-check · 4 quick reps

Shifting every value does not change the spread

Adding the same number to every value moves the mean but leaves every distance from the mean the same, so the standard deviation and the range do not change. Only scaling changes the spread.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Mean, median, mode and range
Mean
$\bar{x} = \frac{\sum x}{n} \qquad \sum x = n\,\bar{x}$
Combined and weighted means, and how averages change when data change
Combined mean
$\bar{x} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}$
Frequency tables and standard deviation (in outline)
Mean of a frequency table and standard deviation
$\bar{x} = \frac{\sum f x}{\sum f} \qquad \sigma = \sqrt{\frac{\sum (x - \bar{x})^2}{n}}$

Watch out for (3)

Order the data before finding the median
→ Mean, median, mode and range
A combined mean is not the average of the two means
→ Combined and weighted means, and how averages change when data change
Shifting every value does not change the spread
→ Frequency tables and standard deviation (in outline)

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