Statistics
How To Calculate The Average (Mean, Median And Mode)
Asking for the average is really asking which of three different summaries you want, and they disagree whenever the data is lopsided.
Quick Answer
mean = sum / count
- sum
- Total of all the values in the list
- n
- How many values the list contains
- mean
- Sum divided by count, the arithmetic average
- median
- Middle value once the list is sorted
- mode
- The most frequently occurring value
The mean adds every value and divides by how many there are. The median sorts the list and takes the middle value, or the average of the two middle values when the count is even. The mode is whichever value appears most often. For the list 4, 8, 15, 16, 23 and 42 the sum is 108, the count is 6, the mean is 18 and the median is 15.5.
What Is The Average (Mean, Median And Mode)?
The word average is used for at least three different numbers, and they can point in opposite directions. The mean is the sum divided by the count. The median is the middle value once the list is sorted. The mode is the value that appears most often. Choosing the wrong one is how a statistic ends up technically correct and completely misleading.
The mean is what most people picture when they say average, and it uses every value in the list. That is its strength and its weakness. A single extreme value drags the mean towards it, so a list of mostly modest numbers with one enormous one produces a mean that no member of the list comes close to.
The median ignores how large the extreme values are and cares only about their position. Sorting the list and taking the middle value produces a number that at least half the list sits below. For income, house prices and waiting times, the median is almost always the more honest summary, which is why official statistics usually report it alongside the mean.
With an even number of values there is no single middle entry, so the median is the average of the two central ones. A list of six numbers has its third and fourth values in the middle, and the median is halfway between them. That is why a list of six integers can have a median of 15.5.
The mode is the only average that works for data which is not numeric. Asking for the most common eye colour or the most popular model of car has no meaningful mean or median, but the mode answers the question directly. With numeric data the mode is useful when the values cluster into distinct groups, such as shoe sizes.
A list can have no mode at all. If every value appears exactly once there is no most frequent entry, and reporting one would be arbitrary. A list can also have several modes, which happens whenever two or more values tie for the highest count, and the honest answer is to report all of them.
The three averages coincide on a perfectly symmetric list with a single peak, which is why the distinction is easy to forget until the data stops being tidy. For 4, 8, 15, 16, 23 and 42 the mean is 18 and the median is 15.5, and the gap between them already shows the pull of the larger values even on a short list.
The range, the largest value minus the smallest, is not an average but is worth reporting beside one. Two lists can share a mean and have completely different spreads, and the range is the quickest way to see that. A mean of 18 from values between 4 and 42 tells a very different story from a mean of 18 from values between 17 and 19.
The mean has a property the others lack: it is the value that minimises the total squared distance to every point in the list. That is why it appears throughout statistics, from standard deviation to regression. The median instead minimises the total absolute distance, which is why it resists outliers so well.
Reporting all three is usually the right move when the audience matters. The mean is what most formulas consume, the median is what most people mean, and the mode reveals whether the data has distinct clusters. A single number labelled average is a decision, and it should be a deliberate one.
Formula
mean = sum / n
Add every value and divide by how many there are. Uses all the data, so extreme values pull it towards them.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| sum | Sum | same unit as the data | Total of every value in the list. |
| n | Count | count | How many values the list contains. |
| mean | Mean | same unit as the data | The arithmetic average. |
median = middle value of the sorted list
Sort the list, then take the middle entry. With an even count, average the two central values. Resists extreme values.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| n | Count | count | How many values the list contains. |
| median | Median | same unit as the data | Middle value after sorting, or the average of the two middle values. |
range = largest - smallest
A measure of spread rather than of centre. Two lists can share a mean and have very different ranges.
| Symbol | Meaning | Unit | Notes |
|---|---|---|---|
| x_max | Largest value | same unit as the data | The maximum in the list. |
| x_min | Smallest value | same unit as the data | The minimum in the list. |
| R | Range | same unit as the data | How far the data stretches. |
How To Calculate The Average (Mean, Median And Mode)
- 1
Count the values
The count is the denominator of the mean and it decides whether the median is a single value or the average of two. For the list 4, 8, 15, 16, 23 and 42 the count is 6.
- 2
Add them all up
Summing gives 108 for that list. Include every value, including any zeros, because a zero is a real observation rather than a missing entry.
- 3
Divide the sum by the count for the mean
108 divided by 6 is 18. This is the arithmetic average and it uses every value, so one extreme entry moves it.
- 4
Sort the list for the median
Ordered, the list is 4, 8, 15, 16, 23 and 42. With six values there is no single middle entry, so take the third and fourth values, 15 and 16, and average them to get 15.5.
- 5
Tally the values for the mode
Count how often each value appears. If one appears more than all the others it is the mode; if several tie, report all of them; if every value is unique, say there is no mode rather than picking one arbitrarily.
Examples
Example 1: The list 4, 8, 15, 16, 23, 42
- Values
- 4, 8, 15, 16, 23, 42
| Step | Calculation | Result |
|---|---|---|
| Count | six values | 6 |
| Sum | 4 + 8 + 15 + 16 + 23 + 42 | 108 |
| Mean | 108 / 6 | 18 |
| Median | (15 + 16) / 2 | 15.5 |
| Range | 42 - 4 | 38 |
Result: For the list 4, 8, 15, 16, 23 and 42 the mean is 18, the median is 15.5 and the range is 38, which is the largest value minus the smallest.
Example 2: A short list with one outlier
- Values
- 10, 12, 11, 13, 1000
| Step | Calculation | Result |
|---|---|---|
| Count | five values | 5 |
| Sum | 10 + 12 + 11 + 13 + 1000 | 1046 |
| Mean | 1046 / 5 | 209.2 |
| Median | middle of 10, 11, 12, 13, 1000 | 12 |
Result: Adding one value of 1000 lifts the mean of 10, 12, 11 and 13 to 209.2, while the median only moves to 12, because position rather than size is what it measures.
Example 3: Shoe sizes with a clear mode
- Values
- 8, 9, 9, 9, 10, 10, 11
| Step | Calculation | Result |
|---|---|---|
| Count | seven values | 7 |
| Sum | 8 + 9 + 9 + 9 + 10 + 10 + 11 | 66 |
| Mean | 66 / 7 | 9.428571428571429 |
| Median | fourth value of the sorted list | 9 |
| Mode | most frequent value | 9 |
Result: For the shoe sizes 8, 9, 9, 9, 10, 10 and 11 the mean is 9.428571428571429, the median is 9 and the mode is also 9, which is what a cluster of identical values produces.
Calculator
Prefer a full-width tool? Open the The Average (Mean, Median And Mode) calculator page.
Common Mistakes
Reporting the mean when the data is skewed
A single extreme value drags the mean far from the typical case. Income, house prices and response times are all routinely skewed, and quoting only the mean there describes a person who does not exist.
Forgetting to sort before taking the median
The median is the middle of the ordered list, not the middle of the list as given. Taking the middle entry of an unsorted list returns an arbitrary value that happens to sit in that position.
Averaging the two middle values incorrectly when the count is odd
With an odd count there is exactly one middle value and no averaging is needed. Averaging it with a neighbour produces a value that is not in the data at all.
Inventing a mode when every value is unique
If no value repeats there is no mode. Picking the smallest, the largest or the first is arbitrary, and reporting one as the mode misleads anyone who assumes it means something.
Confusing the mean with the median in a formula
Statistical formulas that say mean almost always want the arithmetic mean, but plain English average often means the median. Reading the definition rather than the label prevents a silent substitution.
Dropping zeros or blanks from the list
A zero is a measurement, not an absence. Removing zeros raises the mean and shifts the median, while leaving genuinely missing entries in as zeros drags both down. Decide which is which before summing.
FAQ
What is the difference between mean, median and mode?
The mean adds all the values and divides by the count. The median is the middle value once the list is sorted. The mode is the value that appears most often. For 4, 8, 15, 16, 23 and 42 the mean is 18 and the median is 15.5.
Which average should I use?
Use the mean when the data is roughly symmetric and every value should count equally. Use the median when extreme values would distort the picture, as with income or house prices. Use the mode for categories or for data that clusters into groups.
How do I find the median with an even number of values?
Sort the list, take the two middle values and average them. With six values the third and fourth entries are the middle ones, so 15 and 16 give a median of 15.5. The median need not be a value that appears in the list.
Can a list have more than one mode?
Yes. If two or more values tie for the highest count the list is multimodal and all of the tied values are modes. If every value appears exactly once there is no mode at all.
Why is the mean pulled by outliers but the median is not?
The mean uses the size of every value, so one enormous entry adds a large amount to the total while changing the count by only one. The median uses only the position of the values, so moving an extreme entry further out does not change which value sits in the middle.
Is the average always between the smallest and largest values?
Yes, for the mean and the median both. The mean is a weighted position between the extremes and the median is literally one of the middle values. A mode is also one of the observed values. An average outside the range of the data signals an arithmetic error.
References
- [1]Wikipedia, Mean, median and mode — https://en.wikipedia.org/wiki/Average
- [2]Khan Academy, Mean, median, and mode review — https://www.khanacademy.org/math/statistics-probability/summarizing-quantitative-data/mean-median-basics/a/mean-median-and-mode-review
- [3]Math is Fun, How to Find the Mean, Median and Mode — https://www.mathsisfun.com/mean.html
- [4]Australian Bureau of Statistics, Median — https://www.abs.gov.au/statistics/understanding-statistics/statistical-terms-and-concepts/median