12.2 Mean, Median, Mode & Range

From Paper D onwards, ICAS expects you to calculate and interpret the three averages — mean, median and mode — plus the range. Papers E–F lift the difficulty: instead of just computing an average, you may be asked which average best represents a data set, or to find a missing value when the mean is known. All calculations are chosen to work with mental or short written arithmetic, since personal calculators are not allowed.

The Four Measures

Worked example. Find the mean, median, mode and range of: 4, 7, 7, 8, 14.

• Mean: (4 + 7 + 7 + 8 + 14) ÷ 5 = 40 ÷ 5 = 8
• Median: the data is already ordered; the middle of 5 values is the 3rd value, 7
• Mode: 7 appears twice, more than any other value
• Range: 14 − 4 = 10

Notice the mean (8) is higher than the median (7) because the large value 14 pulls it up. ICAS loves testing exactly this idea.

Median with an Even Count

When there is an even number of values, no single middle value exists. The median is the mean of the two middle values.

Worked example. Find the median of 3, 5, 9, 12, 15, 20. With 6 values, the middle pair is the 3rd and 4th values: 9 and 12. Median = (9 + 12) ÷ 2 = 21 ÷ 2 = 10.5.

Two warnings. First, the data must be ordered before you pick the middle — an unordered list is the oldest ICAS trap in the book. Second, the median of an even count may not be a value in the data set at all (10.5 appears nowhere in the list), and that is perfectly fine.

Choosing the Right Average

Each average tells a different story, and ICAS asks which one suits a situation:

• Use the mean when values are fairly balanced and every value should count equally, such as average test scores across a class.
• Use the median when the data contains an extreme value. If four houses in a street sell for around $600 000 and one sells for $3 million, the median describes the 'typical' house far better than the mean, which the mansion drags upwards.
• Use the mode for categories or for the most popular choice — the most common shoe size a shop should stock, or the favourite sport in a survey. The mode is the only average that works for non-numerical data.

Working Backwards from the Mean

A favourite multi-step ICAS question gives you the mean and all but one value.

Worked example. The mean of five test scores is 14. Four of the scores are 11, 16, 12 and 17. Find the missing score.

Step 1: the total of all five scores must be 5 × 14 = 70. Step 2: the known scores add to 11 + 16 + 12 + 17 = 56. Step 3: the missing score is 70 − 56 = 14.

The key move is turning the mean back into a total: total = mean × count. The same idea handles questions like 'what score does Mia need on her next test to bring her mean up to 20?' — work out the total she needs, subtract what she has.

Averages from Frequency Tables

When data is presented in a frequency table, each value must be weighted by its frequency.

• Mean: 20 ÷ 15 = 1⅓ (total goals ÷ total players)
• Median: the 8th of 15 ordered values — counting down the frequencies (3, then 9, then 13, then 15), the 8th value is 1
• Mode: 1, because it has the highest frequency (6)
• Range: 3 − 0 = 3

The classic error is averaging the values 0, 1, 2, 3 to get 1.5, which ignores the frequencies entirely.

Comparing Data Sets

Papers E–F ask you to compare two data sets using both an average and the range. For example, two classes might both have a mean spelling score of 17, but Class A has a range of 4 while Class B has a range of 18. Class A's scores are consistent; Class B's are spread from very low to very high. Averages alone can hide this — the range tells you how much you can trust the average as a description of the group.

Common Traps

• Forgetting to order the data before finding the median.
• Halving the two middle values instead of adding then halving.
• Reporting 'no mode' when a set has two modes — a set can be bimodal, like 2, 2, 5, 7, 7.
• Confusing range (a single number, highest − lowest) with writing 'from 4 to 14'. ICAS treats the range as one number.