Averages & Range

Spec reference: Statistics
Key idea: Calculate mean, median, mode and range from lists, tables and grouped data.

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The four measures

Measure	Definition	Use
Mean	Sum ÷ count	Best overall average
Median	Middle value when ordered	When there are extreme values
Mode	Most frequent value	For non-numerical data
Range	Largest − Smallest	Measures spread

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Mean

$$\overline{x}=\frac{\sum x}{n}$$

Example

Find the mean of: 4, 7, 3, 8, 6, 5, 9

Sum $=42$, count $=7$

$$\overline{x}=\frac{42}{7}=\boxed{{\displaystyle 6}}$$

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Median

For $n$ values in order, the median is at position ${\displaystyle \frac{n+1}{2}}$.

Example

Find the median of: 2, 5, 7, 8, 11, 14, 17

$n=7$, position $=4$th value $=\boxed{{\displaystyle 8}}$

For an even count, the median is the mean of the two middle values.

Example

Median of: 3, 5, 8, 12

Position: between 2nd and 3rd values

$$\text{Median}=\frac{5+8}{2}=\boxed{{\displaystyle 6.5}}$$

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Mean from a frequency table

$$\overline{x}=\frac{\sum fx}{\sum f}$$

Example

Score ($x$)	Frequency ($f$)	$fx$
1	3	3
2	5	10
3	7	21
4	4	16
5	1	5
Total	20	55

$$\overline{x}=\frac{55}{20}=\boxed{{\displaystyle 2.75}}$$

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Estimated mean from grouped data

Use the midpoint of each class as the $x$ value.

Example

Height (cm)	Midpoint	Freq	$fx$
$150\le h<160$	155	5	775
$160\le h<170$	165	12	1980
$170\le h<180$	175	8	1400
Total		25	4155

$$\overline{x}=\frac{4155}{25}=\boxed{{\displaystyle 166.2\text{ cm}}}$$

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Exam tips

Watch out for

• Always sort the data before finding the median
• For grouped data, mean is only an estimate - you cannot find the exact mean
• Mode can have more than one value, or may not exist

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Test Yourself

Question 1 of 5

Mean of: 4, 7, 3, 9, 2?