POP

E9.3 Averages and Measures of Spread

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01 — Explanation

Which average should I use — mean, median or mode?

Measure	Definition	Best used when
Mean	$\bar{x}=\dfrac{\sum x}{n}$	Data is fairly symmetric, no extreme outliers
Median	Middle value once ordered	Data has outliers or is skewed
Mode	Most frequent value	Categorical data, or a "most popular" value is needed

For $4,7,4,9,12,4,8$ : mean $=\dfrac{48}{7}=6.86$ ; ordering gives $4,4,4,7,8,9,12$ , so the median (4th of 7 values) is $7$ ; the mode is $4$ (appears three times); the range is $12-4=8$ .

Always sort the data before picking out the median — the middle value of an unsorted list is meaningless.

02 — Explanation

How do I find the quartiles and interquartile range from a raw list?

\text{IQR} = Q_3 - Q_1

Once the data is sorted, $Q_1$ and $Q_3$ can be found by splitting the ordered list into halves. With an even number of values, split exactly in half and find the median of each half; with an odd number, exclude the middle (overall median) value first, then split the remainder.

Example: $12$ ordered values: $3,5,6,8,9,11,12,14,15,17,19,22$ .

Lower half: $3,5,6,8,9,11$ → $Q_1=\dfrac{6+8}{2}=7$ . Upper half: $12,14,15,17,19,22$ → $Q_3=\dfrac{15+17}{2}=16$ .

\text{IQR} = 16-7=9

Unlike the range, the IQR ignores the most extreme $25\%$ of values at each end, so a single outlier can't distort it.

03 — Explanation

How do I estimate the mean of grouped data?

Once data is grouped into class intervals, individual values are lost, so the mid-interval value stands in for every item in that class:

\bar{x} \approx \frac{\sum(f \cdot x)}{\sum f}

Example: masses of $80$ apples.

Mass ( $g$ )	$f$	Mid-interval $x$	$f\times x$
$100<m\le120$	$12$	$110$	$1320$
$120<m\le140$	$30$	$130$	$3900$
$140<m\le160$	$26$	$150$	$3900$
$160<m\le180$	$12$	$170$	$2040$

\bar{x} = \frac{1320+3900+3900+2040}{80} = \frac{11160}{80} = 139.5\text{ g}

Examiner Tip

Multiply frequency by the mid-interval value, $\frac{\text{lower bound}+\text{upper bound}}{2}$ — not by the class width, which is a completely different quantity used elsewhere (in histograms). Also, because individual masses are replaced by a mid-interval estimate, the answer must be called an estimate of the mean, never the exact mean.

04 — Explanation

How do I find the modal class?

The modal class is the whole class interval with the highest frequency — never a single number. In the table above, $30$ is the highest frequency, so the modal class is $120<m\le140$ .

The same "split into halves" method for $Q_1$ and $Q_3$ shown earlier works for a list of any size — with an odd number of values like $9$ , exclude the overall median itself before splitting the remaining $8$ values into an upper and lower half of $4$ each.

Examiner Tip

"Estimate the mean" and "find the mean" are different instructions — the word "estimate" is the signal that the data has been grouped and the mid-interval method is required, not a hint that any level of inaccuracy is acceptable.