Averages
Mode, median, mean and range
An average is a single value that describes a whole set of data. In this unit you meet three averages — the mode, median and mean — and the range, which tells you how spread out the data is. Together they help you describe and compare sets of data.
The three averages
There are three kinds of average: • Mode — the value that appears most often. • Median — the middle value when the data is put in order. • Mean — the total of all the values divided by how many there are. Each one gives a 'typical' value in a different way.
Finding the mode
The mode is the value that appears most often. Tally or count how many times each value appears and pick the one that comes up the most. Sometimes a set has two values that both appear the most often. Then the data is 'bimodal' and both values are the mode.
Finding the median
Always put the numbers in order first. • If there is an odd number of values, the median is the single middle value. • If there is an even number of values, the median is halfway between the two middle values — add them together and divide by 2. The median can be a value that is not in the data.
Finding the mean
Add all the values together to get the total, then divide the total by how many values there are. You can use mental methods, written methods or a calculator. Example: for 5, 6 and 7 the total is 18 and there are 3 values, so the mean is 18 ÷ 3 = 6.
The range
The range is not an average — it tells you how spread out the data is. To find it, subtract the lowest value from the highest value. A small range means the values are close together (consistent). A large range means they are more spread out.
Choosing and comparing
Averages and the range help you describe and compare sets of data and make decisions. Be careful: one value that is much larger or smaller than the rest can pull the mean away from the typical value. When that happens, the median often describes the data better.
Worked examples
Finding the mode (bimodal)
- Look at 51 mm, 47 mm, 51 mm, 53 mm, 59 mm, 59 mm.
- Count how often each value appears: 51 appears twice and 59 appears twice.
- Two values are tied as the most common, so the data is bimodal.
Answer: The modes are 51 mm and 59 mm.
Finding the median (even number of values)
- Put the numbers in order: 1, 4, 5, 7, 8, 8.
- There are 6 values, so there is no single middle number.
- Take the two middle numbers, 5 and 7, add them and divide by 2: (5 + 7) ÷ 2.
Answer: The median is 6.
Finding the mean
- Add all six heights: 138 + 140 + 136 + 141 + 142 + 137 = 834.
- Count how many heights there are: 6.
- Divide the total by the number of heights: 834 ÷ 6.
Answer: The mean height is 139 cm.
Finding the range
- Look at Sarah's practice times: 10, 8, 2, 10, 20 minutes.
- Find the highest value (20) and the lowest value (2).
- Subtract the lowest from the highest: 20 − 2.
Answer: The range is 18 minutes.
Choosing the best average
- Seven charity buckets collected $34, $0, $42, $51, $0, $263, $37.
- The mean is 427 ÷ 7 = $61, but one bucket ($263) is much larger than the rest and pulls the mean up.
- The median is the middle amount, $37, which stays close to most of the buckets.
Answer: The median ($37) best describes a typical bucket, because the $263 bucket makes the mean too high.
Where you see this in real life
- A shop finds the mode top size sold so it knows which size to stock more of.
- Book review scores out of 10 are combined into an average to show what a typical reader thinks.
- Comparing the mean scores of bowlers or the throws of javelin throwers to decide who is best.
- Using the range of practice times to see which child practises most consistently.
Key words
- Average
- A single value that describes a whole set of data, such as the mode, median or mean.
- Mode
- The value that appears most often in a set of data.
- Bimodal
- Data with two values that both appear the most often, so there are two modes.
- Median
- The middle value when the data is put in order (halfway between the two middle values if there is an even number).
- Mean
- The total of all the values divided by how many values there are.
- Range
- The highest value take away the lowest value; it shows how spread out the data is.