Mean, Median and Mode Calculator
Three different things are called the average and they answer different questions. When they disagree, the list is telling you something.
Mean
18.0000
- Mean
- 18.0000
- Median
- 15.5000
- Mode
- None — no value repeats
- Range
- 38.0000
- Count
- 6
- Sum
- 108.0000
- Smallest
- 4.0000
- Largest
- 42.0000
- The mean is the total shared out evenly. The median is the middle value once they are sorted. They differ whenever the list is lopsided, which is why house prices and salaries are almost always quoted as medians.
- A list can have more than one mode, and every value tied for most frequent is shown rather than one picked arbitrarily. A list where nothing repeats has no mode at all.
About mean, median and mode
The mean is the total shared out evenly. The median is the middle value once the list is sorted. The mode is whatever appears most often. All three are averages and they agree only when the data is symmetrical.
The gap between mean and median is the useful signal. A few values far from the rest pull the mean towards them and leave the median where it was, which is why house prices and salaries are almost always quoted as medians — a handful of very large values would otherwise describe a market nobody is in.
The mode is the only one that works on things that are not numbers. The most common shoe size, the most frequent answer to a survey question, the most popular colour: no mean exists for those and the mode does. A list can also have several modes or none at all, and every value tied for most frequent is shown rather than one picked arbitrarily.
The range — largest less smallest — is the crudest measure of spread there is, and it is entirely determined by the two most extreme values. It is worth glancing at alongside the averages, because two lists with the same mean and very different ranges are not similar in any useful way.
What it works out
- Mean, median, mode and range
- Count, sum, smallest and largest
- Lists separated by commas, spaces or new lines
The formula
Mean = Sum ÷ Count Median = the middle value once sorted
The list 4, 8, 15, 16, 23, 42 sums to 108. Divided by six values, the mean is 18.
The median needs the list sorted, which it already is. With six values there is no single middle one, so it is the average of the third and fourth — 15 and 16 — which is 15.5. With an odd count the median is simply the middle value.
The mean sits above the median here, and 42 is the reason. It is well clear of the rest and drags the mean up while leaving the median where it is. That is exactly the pattern that makes medians the right choice for incomes and house prices.
No value in this list repeats, so there is no mode at all. That is a legitimate answer rather than an error — the mode only exists where something occurs more than once.
- Mean
- The total divided by how many. What most people mean by "average".
- Median
- The middle value when sorted. Unaffected by extremes.
- Mode
- The most frequent value. There can be several, or none.
- Range
- Largest less smallest. A crude measure of spread.
A worked example
The list 4, 8, 15, 16, 23, 42.
That works out to 18.0000 .
- Mean
- 18.0000
- Median
- 15.5000
- Mode
- None — no value repeats
- Range
- 38.0000
- Count
- 6
- Sum
- 108.0000
- Smallest
- 4.0000
- Largest
- 42.0000
Questions
What is the difference between mean, median and mode?
The mean is the total divided by the count. The median is the middle value once sorted. The mode is the most frequent value. They agree only when the data is symmetrical.
When should I use the median instead of the mean?
When the data has extreme values. A few very large numbers pull the mean towards them and leave the median alone, which is why incomes and house prices are quoted as medians.
How do I find the median of an even-numbered list?
Average the two middle values. In a list of six sorted numbers, take the third and fourth and halve their sum.
Can a list have more than one mode?
Yes. If two values tie for most frequent, the list is bimodal and both are modes. Every tied value is shown here rather than one chosen arbitrarily.
What if no number repeats?
Then there is no mode, which is a real answer rather than an error. The mode only exists where something occurs more than once.
What is the range?
The largest value less the smallest. It is the simplest measure of spread and entirely determined by the two extremes, so it says nothing about what happens between them.
How should I separate the numbers?
However you like — commas, spaces, new lines or a mix. Anything that is not a number is ignored, so pasting a column out of a spreadsheet works.