How to calculate average, and when it misleads
To calculate an average, add up all the values, then divide by how many values there are. That gives the mean, the number every value would be if the total were shared out equally.
An average is one number standing in for a whole list, and the trouble starts when you forget what it hides. Below are the add-and-share method, a missing value, percentages and the test traps. More practice is in the numerical practice hub.
Mean average: share the total out equally
The mean is what you get when every value is levelled to the same size. Add the values to get the total, count them, and divide the total by the count. Every value counts, including a zero.
Formula
Mean = (Total) divided by (Count)
- Total is every value added together.
- Count is how many values you added, zeros included.
Worked example
A recruiter logged new sign-ups over five working days: 12, 0, 9, 15 and 9. What was the mean per day?
The zero day looks like nothing to add, so it gets skipped and the total is shared across 4 days. That averages a different set of days, and gives 11.25.
- Add the days: 12 + 0 + 9 + 15 + 9 = 45
- Share the total across all 5 days, the quiet one included: 45 ÷ 5 = 9
The mean was 9 sign-ups a day.
Check it: 5 days of 9 is 45, the total you started with.
How to work out an average backwards, from a mean you are given
A test often gives you the mean and hides one value. Turn the mean back into a total by multiplying it by the full count, take away the values you can see, and what is left is the missing one.
Formulas
Total = Mean × Count
Missing = Total − Known
- Count includes the missing value, and Known is the values you can see added together.
Worked example
Four mock tests average 72 marks. You have the first three scores, 70, 65 and 80. What did the fourth test score?
Multiplying 72 by 3 because only three scores are in front of you gives a total of 216 and a fourth score of just 1. The total belongs to all four tests.
- The total for four tests: 72 × 4 = 288
- The three known scores: 70 + 65 + 80 = 215
- What is left for the fourth: 288 − 215 = 73
The fourth test scored 73 marks.
Check it: 70 + 65 + 80 + 73 is 288, and 288 ÷ 4 is 72.
How to calculate average percentage when the groups differ in size
Percentages hide the size of the group behind them, so adding them up and dividing by two only works when both groups are the same size. Turn each percentage back into a head count, add the counts, and take one percentage at the end.
Formula
Overall = (Total passed) divided by (Total tested)
- Each group's passed count is its percentage times its size.
Worked example
Site A screened 50 applicants and 40 passed, which is 80 percent. Site B screened 100 and 30 passed, which is 30 percent. What was the pass rate across both sites?
Averaging 80 and 30 gives 55 percent. That treats a site with 50 applicants as if it had as many as the site with 100, and it overstates the pass rate.
- Applicants who passed: 40 + 30 = 70
- Applicants screened: 50 + 100 = 150
- Pass rate: 70 ÷ 150 = 0.47, which is about 47 percent
The pass rate across both sites was about 47 percent.
Check it: the bigger site passed only 30 percent, so an answer near 55 had to be too high.
Which average is the question asking for?
A table in a test may say average and mean nothing more, or it may name the type. The three below answer different questions, and one stray figure can pull the mean a long way from where most of the values sit.
- Mean. The total shared out equally. For salaries of 28, 30, 31, 33 and 120 thousand, the total is 242, and 242 ÷ 5 = 48.4 thousand.
- Median. The middle value once they are in order. Five values put it at position (5 + 1) ÷ 2 = 3, so the median salary is the third one, 31 thousand, and the single 120 thousand barely moves it.
- Mode. The value that turns up most often, which suits a most common size or answer. In shoe sizes 7, 8, 8, 8 and 9, size 8 is 3 of the 5, so 3 ÷ 5 = 0.6 of them, and 8 is the mode.
- Even count. With four values the median is halfway between the two middle ones. For 28, 30, 31 and 33, that is (30 + 31) ÷ 2 = 30.5.
- All four measures on one list, and when each one wins: mean, median and mode
Average questions in a test: where does the answer move?
Under a clock, the sum is rarely the hard part. The marks go when an extra value arrives and the reader forgets how many values the mean now has to cover.
When one more value joins, the count changes as well as the total. Build the old total, build the new total from the new mean and the new count, and the difference is the value that joined.
Formulas
Total = Mean × Count
Value = New − Old
- Old and New are the totals before and after the extra value, each built from its own mean and its own count.
Worked example
A depot averaged 14 parcels a day over 6 days. After day 7 the average is 15. How many parcels went out on day 7?
Answering 15 reads the new average as the new day. The day has to pay for the rise across all seven days, so it sits well above 15.
- Parcels over the first 6 days: 14 × 6 = 84
- Parcels over 7 days at 15 a day: 15 × 7 = 105
- The difference is day 7: 105 − 84 = 21
21 parcels went out on day 7.
Check it: 84 + 21 is 105, and 105 ÷ 7 is 15.
Want to try these under a clock? The button below opens the numerical practice page. Its 12-question diagnostic runs on a 12-minute clock, and the clock starts only when you press start on that page.
Which other method does your question need?
- Groups of different sizes, each with its own mean: weighted average questions
- A share written out of one hundred: percentages, step by step
- Every numerical formula, one worked example each: the formulas reference
Want to try these under a clock? The button below opens the numerical practice page. Its 12-question diagnostic runs on a 12-minute clock, and the clock starts only when you press start on that page.
How we write, check and mark the practice questions is set out on the Methodology page.