Averages and Range

In Grade 6 you met four ways of summarising a list of numbers. Stage 8 uses all of them, so it is worth a quick refresher.

• The mean is the fair share: add every value, then divide by how many there are.
• The median is the middle value once the data is in order.
• The mode is the value that appears most often.
• The range is the highest value minus the lowest — it measures spread, not a typical value.

The new skill this year is finding all four from a frequency table — a table that groups repeated values — and then using them to compare two sets of data. That is what the rest of this guide is about.

A frequency table lists each value once with how many times it occurred. If 12 students scored 3 goals, you do not write 3 twelve times — you write the value 3 with a frequency of 12.

To find the mean, you cannot just add the values in the first column. You must use the frequencies. The trick is an extra value × frequency column.

Add up the value × frequency column to get the grand total of all the goals: $0 + 7 + 12 + 9 = 28$. Add up the frequency column to find how many players there are: $4 + 7 + 6 + 3 = 20$.

$\text{mean} = \frac{\text{total of (value} \times \text{frequency)}}{\text{total frequency}} = \frac{28}{20} = 1.4$

So the mean is 1.4 goals per player. Notice you divide by the total frequency, not by the number of rows.

The mode from a frequency table is the easiest measure: it is the value with the highest frequency. In the goals table, the frequency 7 is the largest, and it sits next to the value 1 — so the modal number of goals is 1.

Be careful: the mode is the value, not the frequency itself. The answer is 1 goal, not 7.

The median needs a little more thought. With 20 players, the median is between the 10th and 11th value once everything is in order. A frequency table is already in order, so you find which row the 10th and 11th values land in using a running total.

The first row reaches a running total of 4, so values 1 to 4 are all 0 goals. The next row adds 7, reaching 11 — so values 5 to 11 are all 1 goal. Both the 10th and 11th values land here, so the median is 1 goal.

The range from a frequency table catches people out, because it is tempting to look at the frequency column. Do not — the range is always highest value − lowest value.

In the goals table, the values go from 0 to 3. So the range is:

$\text{range} = 3 - 0 = 3 \text{ goals}$

The frequencies (4, 7, 6, 3) play no part in the range. They tell you how often each value occurred, not how spread out the values are.

A small range tells you the values are bunched close together; a large range means they are widely spread. You will use the range a great deal in the next section, where you compare two sets of data.

The most important Stage 8 skill is comparing two sets of data fairly. The golden rule: compare an average and the range — never just one of them.

• The average tells you which set is generally higher or lower.
• The range tells you which set is more consistent (small range) or more varied (large range).

Look at two students' test scores out of 10:

Both have a mean of 7, so on average they score the same. But Maya's range is 2 and Ben's range is 8. Maya is far more consistent — her scores are bunched together, while Ben's swing wildly.

A strong comparison always has two parts and links to the situation: "Both students average 7 marks, but Maya is more consistent because her range is much smaller, so her results are easier to rely on." Numbers plus a clear sentence — that is what a full comparison looks like.

Finding averages from frequency tables and comparing data sets are skills you will reach for again and again:

• Data representation — you will read frequencies straight off bar charts, stem-and-leaf diagrams and frequency tables, then summarise them with an average.
• Collecting and organising data — grouped frequency tables and two-way tables give you data that you then summarise using these measures.
• Comparing groups — in science and geography you will compare experiment results and survey data using an average together with the range.
• In everyday life, comparisons like this decide which phone has more reliable battery life, or which team is the steadier performer.

Whenever two groups are compared — two classes, two teams, two months of weather — remember that a fair comparison needs both a typical value and a measure of spread.