Key facts
- The mean of 2, 4, 4, 5 and 10 is 25 ÷ 5 = 5.
- The median of those five values is 4. With an even number of values it is the mean of the two middle ones, so 1, 3, 5 and 7 also gives 4.
- The mode is the most frequent value: 4 in the first list. There is no mode at all when every value occurs equally often.
- Replacing the 10 with 90 lifts the mean from 5 to 21 but leaves the median at 4.
- Two group averages cannot be averaged unless the groups are the same size.
What the three averages measure
The mean, median and mode are three different answers to one question: what is typical here? The mean uses every value in the list. The median uses only each value's position once the list is sorted. The mode uses only how often each value repeats. Because they lean on different information, they often disagree, and the size and direction of that disagreement tells you about the shape of the data. The statistics section of the NumberUtils methodology sets out the conventions behind the calculations below.
| Average | How it is calculated | Use it when |
|---|---|---|
| Mean | Sum of all values ÷ number of values | Values are evenly spread and you need a figure that still adds back to a total |
| Median | Middle value of the sorted list, or the mean of the two middle values | One or two extreme values would distort the mean |
| Mode | The value occurring most often | You want the most common size, time or category |
The mean
The mean is the sum of the values divided by how many there are. For 2, 4, 4, 5 and 10 the sum is 25, so the mean is 25 ÷ 5 = 5. Two useful properties follow from that definition: multiply the mean by the count and you recover the total (5 × 5 = 25), and the mean moves whenever any single value moves. The NumberUtils average calculator performs exactly this calculation.
When the sum does not divide exactly, the mean is a decimal. The values 8, 9 and 8 sum to 25, and 25 ÷ 3 = 8.333…, which the calculator shows as 8.3333333333. Rounding a long repeating decimal for display is covered under rounding.
The median
Sort the values from smallest to largest and take the middle one. With an odd count that is a single value: in 2, 4, 4, 5 and 10 the third of five values is 4. With an even count there is no single middle value, so average the two middle ones. For 1, 3, 5 and 7 that is (3 + 5) ÷ 2 = 4, a number that appears nowhere in the data.
The median is the most stubborn of the three. Swap the 10 in 2, 4, 4, 5, 10 for 90 and the median stays at 4, because only the middle position matters. The mean, meanwhile, jumps to 105 ÷ 5 = 21.
The mode
The mode is whichever value occurs most often, so in 2, 4, 4, 5 and 10 it is 4, appearing twice. A list can have several modes: in 1, 1, 2, 2 and 3 the values 1 and 2 are tied, and the list is called bimodal. It can also have none. If every value occurs the same number of times — including the case where all values are distinct and each occurs once — then no value is more frequent than any other, and there is no mode at all. The NumberUtils mean, median and mode calculator reports those cases explicitly rather than picking an arbitrary value.
Choosing between them
Use the mean when the values are spread evenly around the middle and you need a figure that still relates to totals. Use the median when a few extreme values would drag the mean away from what most people experience. Use the mode when the most common value is what you actually need, such as the most frequent pack size or the most common delivery slot.
Consider four people earning £30,000 a year and one earning £230,000. The total is £350,000, so the mean is £350,000 ÷ 5 = £70,000. That figure describes nobody. The median is £30,000, the point in the middle of the sorted list, and the mode is also £30,000 because four of the five earn it. Reporting £70,000 as the typical salary would be misleading.
Income and house prices are the standard cases for the median, because both distributions have a long upper tail: a small number of very large values pulls the mean well above the experience of most people. Note also the direction of the disagreement. In the salary list the mean sits far above the median, and in 2, 4, 4, 5, 10 the mean of 5 sits just above the median of 4. A mean consistently above the median is a sign that high values are stretching the data.
Outliers and the range
The range is the largest value minus the smallest. For 2, 4, 4, 5 and 10 it is 10 − 2 = 8. The range is a measure of spread rather than an average, but it behaves like the mean in one respect: a single extreme value sets it completely. In the salary list the range is £230,000 − £30,000 = £200,000, which tells you one person earns far more than the rest and nothing about how tightly the other four are grouped.
This is the practical reason to calculate more than one average. Report the mean and the median together and the gap between them is itself information about the data; report the mean alone and a single odd value can pass for a fair summary.
Averaging averages
Two group averages cannot be averaged directly unless the groups are the same size. Take a class of 10 pupils with a mean of 60 and a class of 30 pupils with a mean of 80. The plain average of 60 and 80 is 70, which is wrong. The first group carries a total of 10 × 60 = 600 marks and the second a total of 30 × 80 = 2,400 marks, so the combined mean is 3,000 ÷ 40 = 75.
Weighting each mean by its group size gives the same answer: (10 × 60 + 30 × 80) ÷ (10 + 30) = 75. The unweighted figure of 70 is too low because the larger, higher-scoring group was given the same influence as the smaller one.
The reliable route is to work with totals: add up every value across both groups and divide by the combined count. That is precisely what the average calculator does when you enter every value, and it applies no weighting of its own. If you only have the group means, you cannot recover the combined mean without the group sizes.