Averages and Range

Last year you found a mean straight from a frequency table, where every value was written down exactly. This year the data is grouped — sorted into class intervals like $150 \le h < 160$ — and that changes everything.

Inside a group such as "8 students between 150 and 160 cm" you no longer know the exact heights. They could all be 151 cm, or spread evenly, or anything in between. The exact total height is genuinely unknown.

Because the exact values are gone, you cannot find an exact mean. The best you can do is a sensible estimate — and the clever trick for that is to use the midpoint of each class. That is the heart of this year's work, and the next section shows you how.

Since you cannot use the real values, you make a fair assumption: treat every value in a class as if it sat at the midpoint of that class. The midpoint is just the average of the two class boundaries.

For the class $150 \le h < 160$, the midpoint is $\tfrac{150 + 160}{2} = 155$.

You then add a midpoint × frequency column, exactly like the value × frequency column you used last year.

Add the midpoint × frequency column: $435 + 1240 + 1815 + 1050 = 4540$. Add the frequency column: $3 + 8 + 11 + 6 = 28$.

$\text{estimated mean} = \frac{\text{total of (midpoint} \times \text{frequency)}}{\text{total frequency}} = \frac{4540}{28} \approx 162.1 \text{ cm}$

So the estimated mean height is about 162 cm. The answer is an estimate because each midpoint is a stand-in, not a real measurement.

With grouped data there is no single mode, because you do not know any exact value. Instead you find the modal class — the class interval with the highest frequency.

In the heights table, the frequencies are 3, 8, 11 and 6. The largest is 11, which sits next to the class $160 \le h < 170$. So the modal class is 160 ≤ h < 170 cm.

Be careful with two things. First, the answer is a class interval, not a number — you write "160 ≤ h < 170", not "11". Second, the modal class is the one with the highest frequency, which is not the same as the class containing the highest value.

The modal class tells you where the data is most concentrated — useful for spotting the typical group at a glance.

You cannot find an exact median from grouped data, but you can find the class interval that contains it. The method is the running total you met last year.

With 28 students, the median is at the middle position — between the 14th and 15th values once everything is in order. A grouped table is already in order, so build a running total (also called a cumulative frequency) and see which class the 14th and 15th values land in.

The running total reaches 11 after the second class, so values 4 to 11 are in $150 \le h < 160$. The next class pushes the total to 22, covering values 12 to 22 — and both the 14th and 15th values fall there. So the median lies in the class 160 ≤ h < 170 cm.

The headline Stage 9 skill is comparing two whole distributions fairly. The golden rule is unchanged: never compare on one measure alone. Use an average for the typical value and the range for the spread.

Picture two classes' grouped test results. Class A has an estimated mean of 64 with a range of 30; Class B has an estimated mean of 64 with a range of 12.

Both classes average the same score, so on that measure they tie. But Class B's range is far smaller, so its results are much more consistent — Class A has some very high and some very low marks.

A full comparison has two halves and links to the situation: "Both classes have an estimated mean of 64 marks, but Class B is more consistent because its range is much smaller, so its results are more predictable." Numbers plus a clear sentence — that is a complete comparison.

Estimating the mean from grouped data and comparing distributions are skills you will keep reaching for:

• Data representation — you will read frequencies from frequency diagrams, then estimate an average from them.
• Collecting and organising data — grouped frequency tables are exactly the data sets you summarise with these methods.
• IGCSE statistics — estimated mean, modal class and median class appear again, alongside cumulative frequency curves that build directly on running totals.
• In science and geography, you will summarise grouped measurements — rainfall, reaction times, populations — using an estimated average and a measure of spread.

Whenever someone reports a "typical" value for a large data set — average income, average rainfall, average score — they almost always worked from grouped data and an estimate, just as you now can.