Averages and Range

Imagine someone asks, "How tall is a Grade 6 student?" You cannot read out everyone's height — that would be a long list! Instead you give one number that represents the whole class. That number is an average.

An average is a typical value. It does not have to be a value anyone actually scored — it is a fair summary of the set.

There are three averages you need to know, and they each look at "typical" in a slightly different way:

• The mean spreads the total out equally, like sharing sweets fairly.
• The median finds the value sitting right in the middle.
• The mode picks the value that appears most often.

A fourth number — the range — is not an average at all. It tells you how spread out the data is. We will meet it later in this guide.

The mean is the average most people meet first. To find it, you add up all the values and then divide by how many values there are.

$\text{mean} = \frac{\text{total of all values}}{\text{number of values}}$

Think of it as sharing fairly. If five friends scored 6, 8, 5, 7 and 9 on a quiz, imagine pooling all their marks and sharing them out equally.

The total is $6 + 8 + 5 + 7 + 9 = 35$. There are 5 values, so the mean is $35 \div 5 = 7$.

Notice that nobody actually scored exactly 7 — the mean is a fair summary, not a real result. The mean uses every value, so it is sensitive: changing any one number changes the mean.

The median is the value in the middle once the numbers are placed in order from smallest to largest. Ordering first is essential — the median of an unsorted list is meaningless.

For the marks 5, 6, 7, 8, 9 (five values), the middle one is the 3rd, so the median is 7.

If there is an even number of values, there are two middle values — find the mean of those two. For 4, 6, 9, 11 the two middle values are 6 and 9, so the median is $(6 + 9) \div 2 = 7.5$.

The mode is simply the value that appears most often. In 3, 4, 4, 4, 7, 9 the mode is 4. A set can have no mode, or more than one mode, and the mode is the only average that works for words — you can have a modal favourite colour, but not a mean colour.

Two sets of data can have the same average but feel completely different. The range is the number that captures that difference — it measures how spread out the values are.

$\text{range} = \text{highest value} - \text{lowest value}$

For the marks 5, 6, 7, 8, 9 the highest is 9 and the lowest is 5, so the range is $9 - 5 = 4$.

A small range means the values are bunched together — consistent and reliable. A large range means the values are widely spread — less consistent.

The range is useful for comparing two sets. Two students might both average 7 in their tests, but one with a range of 1 is far steadier than one with a range of 8.

All three averages are useful, but they suit different situations. The big question is: is there an extreme value — one number much larger or smaller than the rest?

Look at these salaries in a small shop: $20, $22, $24, $23, $200 (the owner). The mean is $57.80 — but four of the five staff earn far less than that! One extreme value has pulled the mean upwards. The median here is $23, which describes a "typical" worker far better.

Here is a quick guide:

• Use the mean when the data is evenly spread with no extreme values.
• Use the median when there is an extreme value that would unfairly pull the mean.
• Use the mode for data that is words or categories, or when you want the most popular choice.

There is no single "correct" average — but there is usually a best one for the question being asked.

Knowing how to summarise a set of numbers opens up the rest of statistics:

• Data representation — when you read a bar chart or line graph, you will often calculate an average to describe it.
• Collecting and organising data — once you have gathered results in a frequency table, finding the mean, median and mode is the next step.
• Comparing groups — in science and geography you will compare two sets of results using an average and the range together.
• In everyday life, averages appear in sports statistics, school reports, weather summaries and game scores.

Whenever you see a single number describing a whole group — an average rainfall, an average score — remember you now know exactly how it was worked out, and how to question whether it is the fairest one.