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This page sets out the method for averages in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Averages topic page.
When you see a list of numbers and 'median', 'mode' or 'range', write the list in order before anything else.
When you see 'the mean of … is …' and a missing value or a new value is added, go through the total.
Add the values and divide by how many there are.
The mean of 3, 5 and 7 is 15 ÷ 3 = 5.
Order the list, then take the middle value.
With an even count, the median is the mean of the two middle values.
The mode is the value that appears most often.
A list can have more than one mode, or none, if nothing repeats.
Subtract the least value from the greatest.
The range of 3, 5 and 7 is 4. It is not an average.
The mean uses every value, so one extreme value pulls it.
The median resists a single extreme value.
If you add a value, the mean usually changes.
Recount how many values you are dividing by.
The mean matches a question that names the mean. Add every value, then divide by how many values there are. For 2, 3 and 7 the total is 12, so the mean is 12 ÷ 3 = 4. The count is three scores, not the biggest score.
Check by multiplying back: 4 × 3 = 12. Stopping at the total leaves a sum, not an average. The division is the step that turns the total into the mean.
Method: 1. Add every value in the list. 2. Count how many values you added. 3. Divide the total by that count. 4. Multiply the mean by the count to check the total.
The median matches a question that names the median. Order the list first. For 9, 2 and 6, the written middle is 2, but the ordered list is 2, 6, 9, so the median is 6.
The middle position is (n+1)/2. An odd count lands on one value. An even count lands halfway between two values, and you average them. For 4, 8, 10 and 16 the position is 2.5, so the median is (8+10)/2 = 9.
Method: 1. Write the values in order, least to greatest. 2. Count them and find the position (n+1) ÷ 2. 3. If the position is a whole number, read that value. 4. If it ends in .5, average the two values either side.
The mode is the value that appears most often. For the sizes 3, 5, 5, 8 and 9, the mode is 5. The greatest size is 9, but 9 appears only once, so it is not the mode.
There can be more than one mode. If 4 and 7 both appear most often, state both. If nothing repeats, there is no mode. When these worked examples use a mode, that mode is single.
Method: 1. List the different values. 2. Count how many times each value appears. 3. Pick the value whose count is the highest. 4. If two values tie, state both modes.
The range is the greatest value minus the least. For 6, 14 and 19 the range is 19 - 6 = 13. Adding the ends is not a spread. The range is not an average, so do not divide it.
An outlier pulls the mean more than the median. For 2, 4 and 6 the mean and the median are both 4. Add 18. The new mean is 30 ÷ 4 = 7.5 and the new median is (4+6)/2 = 5. The mean rose by 3.5 and the median by 1. Keep the outlier unless the question says to remove it.
Method: 1. Find the greatest value and the least value. 2. Subtract the least from the greatest for the range. 3. For a mean, keep an outlier inside the total and the count. 4. For a median, order the list and use the middle place.
The mean is the total divided by the number of values, so the total is the mean times the number of values. This is the key to a popular harder question: "The mean of 4 numbers is 9. Three of them are 5, 12 and 8. Find the fourth." The four numbers must total 4 × 9 = 36, the three known ones total 25, so the fourth is 36 - 25 = 11.
The same idea handles a new value being added. If the mean of 5 numbers is 8, their total is 40. If a sixth number brings the mean to 9, the new total is 6 × 9 = 54, so the sixth number is 54 - 40 = 14. Do not just add 1 to the mean or share the difference out. Always rebuild the old and new totals, then subtract.
Method: 1. Multiply the mean by the number of values to get the total. 2. Subtract the values you already know. 3. What is left is the missing value, or the total of the missing values. 4. Check by working out the mean of the complete set.
Tip: Write "total = mean x number of values" before you start. It is the line that earns the method mark.
The median is the middle value once the data is in order. Sort first, then count in from both ends. With an odd number of values there is one middle value. With an even number there are two middle values, and the median is half way between them, found by adding them and dividing by 2.
Take the times 14.2, 13.8, 15.1, 14.6, 13.9, 14.4. In order they are 13.8, 13.9, 14.2, 14.4, 14.6, 15.1. There are six values, so the median lies between the third and fourth: (14.2 + 14.4) ÷ 2 = 14.3. Taking the middle two from the unordered list would give 15.1 and 14.6, which is wrong. The median can be a number that is not in the data.
Method: 1. Write the values in order, smallest first. 2. Count the values and find the middle position. 3. If there are two middle values, add them and halve. 4. State the median with its unit.
The range is the largest value minus the smallest. It is one number, not two. If a question gives the range and the smallest value, the largest is smallest plus range. Watch the units: a smallest mass of 3.45 kg and a range of 820 g need the same unit before you add, so 820 g = 0.82 kg and the largest is 3.45 + 0.82 = 4.27 kg.
Questions such as "the mode is 7, the median is 9 and the range is 8, write down a possible set of five numbers" have many answers. Choose the median first, put it in the middle, then place two 7s so the mode is 7, then pick the largest number from the range. The set 7, 7, 9, 11, 15 works. Always check each condition in turn.
Method: 1. Convert to one unit before you add or subtract. 2. For a set with conditions, fix the median in the middle position first. 3. Add the mode value at least twice, and keep every other value to one use. 4. Check the mode, median and range of your set one at a time.
A comparison question usually gives you an average and the range for each set and says "compare the distributions". Write one comment about the average and one about the spread, and use the context. For example: "The median for Team B is higher, so Team B is slower on average. The range for Team B is smaller, so Team B is more consistent."
Quote the actual values you are comparing. A bare comment such as "Team B is better" gains nothing. The average is a typical value, so a higher median means larger values in that set. The range is the spread, so a smaller range means the values are closer together. An extreme value can stretch the range and pull the mean, but the median is hardly affected.
Method: 1. Write both medians or both means, say which is higher, and say what that means in context. 2. Write both ranges, say which is smaller, and say what that means. 3. Use words from the question, such as faster, taller or more consistent. 4. Keep to one comment about average and one about spread.
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