Measures of Central Tendency

Mean (arithmetic average): $\bar{x} = \frac{\text{sum of all values}}{\text{number of values}} = \frac{\sum x}{n}.$

Affected by EVERY value, including outliers.

Median. Sort the data; the middle value:

• If $n$ is ODD: the middle value (position $(n+1)/2$).
• If $n$ is EVEN: average the two middle values.

LESS affected by outliers — a good summary for skewed data.

Mode. The most frequently occurring value. There can be:

• ONE mode (unimodal).
• TWO modes (bimodal).
• NO mode (every value distinct).

The ONLY measure that works for purely categorical data.

Worked example. Dataset: $4, 7, 7, 8, 9, 10, 22$.

• Mean: $(4 + 7 + 7 + 8 + 9 + 10 + 22) / 7 = 67/7 \approx 9.57$.
• Median: middle value of 7 sorted values is the 4th = $8$.
• Mode: $7$ (appears twice).
• Notice: the mean is dragged up by the outlier 22. The median better represents 'typical'.

Range: max − min = $22 - 4 = 18$.

When data are summarised in a frequency table, the mean is:

$\bar{x} = \frac{\sum f x}{\sum f}.$

where $f$ is the frequency of each value $x$.

Worked example. Shoe-size data:

Size $x$	Frequency $f$	$f \cdot x$
5	3	15
6	7	42
7	12	84
8	6	48
9	2	18
Total	$\sum f = 30$	$\sum fx = 207$

Mean: $\bar{x} = 207 / 30 = 6.9$.

Mean from grouped data. When data are in intervals, use the MIDPOINT of each interval as $x$.

Time interval (min)	Midpoint $x$	Frequency $f$	$fx$
0-5	2.5	4	10
5-10	7.5	10	75
10-15	12.5	6	75
Total	—	20	160

Mean: $\bar{x} = 160/20 = 8$ minutes.

(This is an ESTIMATE because we don't know exact values within each interval.)

Use the MEAN when:

• Data is roughly symmetric.
• All values matter.
• No extreme outliers.

Use the MEDIAN when:

• Data is skewed.
• Outliers are present.
• You want the 'typical' value.

Use the MODE when:

• Data is categorical.
• You want the MOST COMMON value.
• Multiple peaks of interest (bimodal data).

Worked example. A salary dataset: most employees earn $30,000-$60,000, but the CEO earns $5 million.

• Mean: dragged up by the CEO — gives a misleading 'average'.
• Median: gives the typical employee salary — much more meaningful.

This is exactly why house prices and incomes are usually reported as MEDIANS, not means.

Worked example (criterion D). A teacher reports the class average on a test as a mean of 65. Sketch what the actual distribution might look like, and discuss whether the mean is a good summary.

• Could be: all students near 65 (low spread, mean is excellent).
• Or: half students at 90, half at 40 (bimodal — mean of 65 is misleading; no student actually got 65).
• The mean alone doesn't tell you the SHAPE. Always look at the distribution.

A grade-7 answer recognises the LIMITATIONS of summary statistics and the importance of looking at the full distribution.