Averages: Which One to Use and How to Calculate It

Three kinds of "average"

In everyday speech "the average" usually means the mean, but there are three common measures of the middle of a set of numbers, and each answers a slightly different question.

Mean: add all the values and divide by how many there are. It answers "if everything were shared out equally, how much would each get?"
Median: the middle value once the numbers are sorted. It answers "what does a typical value in the middle look like?"
Mode: the value that appears most often. It answers "what is the most common value?"

For neat, evenly spread data the three are close together. When the data has a few extreme values, they can be far apart, and choosing the wrong one gives a misleading picture.

The mean, step by step

Mean = sum of values ÷ number of values

Example: five test scores of 72, 85, 90, 64 and 89. The sum is 72 + 85 + 90 + 64 + 89 = 400. There are 5 scores, so the mean is 400 ÷ 5 = 80.

On a calculator, put the sum in brackets: (72 + 85 + 90 + 64 + 89) ÷ 5. Without the brackets, the calculator divides only the last number by 5, because division comes before addition. The order of operations guide explains why. You can type the whole line into the basic calculator.

A quick sanity check: the mean must lie between the smallest and largest values. Here 80 is between 64 and 90, so it is plausible. A mean of 8 or 800 would show a keying error at once.

Weighted averages

A plain mean treats every value as equally important. A weighted average gives some values more influence than others. Multiply each value by its weight, add the products, and divide by the total of the weights.

Weighted average = sum of (value × weight) ÷ sum of weights

Example: a grade point average with credit hours. Grade points are shown here on a 4-point scale, but the method is the same for any scale.

Course	Credits (weight)	Grade points	Credits × points
Course A	4	3.7	14.8
Course B	3	3.0	9.0
Course C	2	4.0	8.0
Course D	1	2.0	2.0
Total	10		33.8

The weighted average is 33.8 ÷ 10 = 3.38. A plain mean of the four grades would be (3.7 + 3.0 + 4.0 + 2.0) ÷ 4 = 3.175, which undervalues the big four-credit course where the grade was high. How your own school rounds or which courses it includes can differ, so treat this as the method, not an official calculation.

Example: weights given as percentages. Homework counts for 20%, a midterm for 30% and a final exam for 50%. Scores are 90, 75 and 82. The weighted result is 0.20 × 90 + 0.30 × 75 + 0.50 × 82 = 18 + 22.5 + 41 = 81.5. Because the weights add up to 100%, there is no need to divide at the end.

The average-of-averages trap

You cannot simply average two averages if the groups are different sizes. Class A has 20 students with a mean score of 70. Class B has 30 students with a mean of 80. The overall mean is not 75.

Total for class A: 20 × 70 = 1,400.
Total for class B: 30 × 80 = 2,400.
Overall: (1,400 + 2,400) ÷ 50 = 3,800 ÷ 50 = 76.

The larger class pulls the result towards its own mean. This is really a weighted average, with class sizes as the weights.

A related trap appears with speeds. You drive 120 km to a town at 60 km/h and return the same 120 km at 40 km/h. The average speed is not 50 km/h. The outward trip takes 2 hours and the return takes 3 hours, so the total is 240 km in 5 hours, which is 48 km/h. You spend longer at the slower speed, so it counts for more.

The median and why it resists extremes

To find the median, sort the values and take the middle one. With an even number of values, take the mean of the two middle ones.

3, 7, 8, 12, 50: the middle value is 8. The mean is 80 ÷ 5 = 16.
4, 6, 9, 15: the two middle values are 6 and 9, so the median is (6 + 9) ÷ 2 = 7.5.

In the first list, the single value 50 drags the mean up to 16, which is larger than four of the five values. The median ignores how extreme the 50 is. That is why reports on things with a few very large values, such as house sizes or journey times, often prefer the median.

Example: ten friends record their commute. Nine take 20 minutes, and one, who lives far away, takes 200 minutes. The mean is (9 × 20 + 200) ÷ 10 = 380 ÷ 10 = 38 minutes, yet nobody actually commutes for 38 minutes. The median is 20 minutes, which describes the group far better.

The mode and when it helps

The mode is the only "average" that works for categories as well as numbers. A shop that sold shoes in sizes 38, 39, 39, 40, 41, 39 and 42 has a mode of 39, and that is the size it should reorder most. A mean shoe size of 39.7 is useless on a shelf.

A set can have no mode (every value appears once) or several modes (two values tie for most common). Small data sets often have a misleading mode by chance, so it is most useful with plenty of data.

Measure	Best for	Can mislead when
Mean	Totals and fair shares; data without extremes	A few very large or small values exist
Median	Typical values in skewed data	You need a total or the data has big gaps
Mode	Most common choice or category	The data set is small or values rarely repeat

What score do I need next?

This is one of the most useful average calculations. Work with totals, not averages.

Needed score = target average × total number of tests − sum of scores so far

Example: your mean after 4 tests is 78. You want a mean of 80 after the 5th test. You need a total of 5 × 80 = 400, and you have 4 × 78 = 312 so far, so the next score must be 400 − 312 = 88.

If you wanted a mean of 85 instead, you would need 5 × 85 − 312 = 113. If the maximum score is 100, that target cannot be reached in one test, and it is better to know that early.

With weights: coursework worth 60% of the final result has an average of 72, and the exam is worth 40%. To finish with 75 overall, solve 0.6 × 72 + 0.4 × exam = 75. That is 43.2 + 0.4 × exam = 75, so the exam score must be (75 − 43.2) ÷ 0.4 = 79.5.

Updating an average without starting over

If you already know the mean of n values and add one more value x, the new mean is the old mean plus (x − old mean) ÷ (n + 1). With a mean of 78 over 4 tests and a new score of 88: 78 + (88 − 78) ÷ 5 = 78 + 2 = 80. This matches the calculation above.

Common mistakes

Forgetting brackets around the sum. 70 + 80 + 90 ÷ 3 is 180, not 80.
Averaging averages of different-sized groups. Go back to totals.
Averaging speeds or rates directly. Use total distance ÷ total time.
Using the mean for skewed data. Check the median as well.
Weights that do not add up. If weights are percentages, they should total 100%; otherwise divide by their sum.

Averages often involve percentages, so the percentage guide is a natural next step.

FAQ

Can the average be a value that never appears in the data?
Yes. The mean of 1 and 2 is 1.5, and the mean number of people in a household can be 2.4. The median of an even-sized set can also fall between two values.

Is a weighted average always more accurate?
Only when the weights reflect real importance or size. With equal weights, a weighted average is simply the ordinary mean.

Should I round the average?
Keep full precision while calculating, then round the final result to a sensible number of decimal places, as explained in the rounding guide.

Open the calculator: Calculator →