NumberUtils

Mean, Median & Mode Calculator

Enter a list of numbers to find their mean, median, mode and range all at once.

Values may be whole numbers or decimals, and they can be entered in any order: the list is sorted internally wherever a calculation depends on position.

Separate numbers with commas or spaces.

Mean

4.8

Median: 4

Mode: 4

Range: 6

Mean = sum ÷ count. Median = middle value once sorted. Mode = most frequent value(s). Range = largest − smallest.

How it works

The mean (average) adds up every value and divides by how many there are.

The median is the middle value once the list is sorted, or the average of the two middle values if there is an even number of values.

The mode is whichever value (or values) occurs most often. If every value occurs equally often — including when no value repeats at all — there is no mode.

The range is the largest value minus the smallest.

Worked through 12, 4, 9, 15, 10: sort them to 4, 9, 10, 12, 15; add them for 50 and divide by 5 for a mean of 10; take the third of the five values for a median of 10; look for a repeated value and find none, so there is no mode; subtract the smallest from the largest for a range of 11.

The mean is also available on its own as the average calculator, and the guide to mean, median and mode sets out which of the three to quote. Results are rounded to 12 significant digits and displayed to at most 10 decimal places, so a mean of 50 ÷ 6 is reported as 8.3333333333.

Formula: Mean = sum ÷ count. Median = middle value once sorted. Mode = most frequent value(s). Range = largest − smallest.

Examples

A list with a repeated value

For 2, 4, 4, 6, 8: the mean is 4.8, the median is 4, the mode is 4 (it appears twice), and the range is 8 − 2 = 6.

An even number of decimal values with no repeats

For 2.5, 4, 5, 6, 7.5, 11: the mean is 6 (36 ÷ 6), the median is 5.5 (the two middle values, 5 and 6, averaged), there is no mode because every value occurs once, and the range is 11 − 2.5 = 8.5.

A list with two modes

For 2, 2, 4, 6, 6, 11, 18: the mean is 7 (49 ÷ 7), the median is 6 (the fourth of seven values), the modes are 2 and 6 (each occurs twice), and the range is 18 − 2 = 16.

Common uses

  • Summarising a set of marks, scores or readings in a single line for a report.
  • Comparing the mean cost per item across two quoted totals.
  • Using the median as a yardstick for whether one measurement is typical of the rest.

Things to keep in mind

  • The mean can be pulled higher or lower by a few unusually large or small values in the list; the median is not affected the same way, which is why the two can differ noticeably.
  • A list can have more than one mode (it is "multimodal") when two or more values are tied for the highest frequency.
  • A list where every value is different has no mode, since no single value occurs more often than the rest.
  • The list must contain at least one value: an empty list has no mean, median, mode or range.
  • Values are compared exactly as entered, so two readings that differ at all, such as 2.4 and 2.4001, count as different values when the mode is found.
  • Negative values are accepted. The mean, median and range use the signed values as entered, so a list that crosses zero is calculated rather than rejected.

Frequently asked questions

What is the difference between mean, median and mode?

The mean is the arithmetic average. The median is the middle value once the numbers are sorted. The mode is the value that appears most often. All three can be different for the same list of numbers.

Why does my list have no mode?

A mode requires at least one value to repeat more often than the others. If every value in your list is different, or every value repeats the same number of times, there is no single most-frequent value.

How many values do I need to enter?

At least one. A list of a single value gives a mean, median and range equal to that value, and no mode, because one value occurring once is not a repeat.

Do the values have to be entered in order, and can they be decimals?

Neither is required. The list is sorted internally for the median and the range, so the order of entry changes none of the four results, and decimals are handled in the same way as whole numbers.

Background and method

  • Mean, median and mode The three common averages and the range, how each is calculated, which to use when, and how outliers and ties change the answer.
  • Methodology: Averages The exact rules, input limits and rounding this calculator follows.
Average Calculator Find the arithmetic mean of a list of numbers.