Mean, median and mode calculator with range and standard deviation

How to use the mean, median and mode calculator

1. Paste your numbers into Numbers (separated by commas, spaces or new lines). A column copied from a spreadsheet works.
2. Read Mean, Median and Mode at the top.
3. Look further down for the count, sum, minimum, maximum, range, standard deviations, variances, quartiles and interquartile range.
4. Use the Sorted list and the Frequency table to check the median and mode by eye.

What it does and when to use it

The mean, median and mode are three ways to describe a typical value. The mean is the ordinary average. The median is the middle value and is not pulled about by one very large or small number. The mode is the most common value and is the only one that works for categories such as shoe sizes.

Students use this for statistics homework and coursework data; teachers use it to summarise marks. When all values are positive, the tool also gives the geometric and harmonic means. To see how far one value lies from the mean in standard deviations, take the results to the z-score calculator. For the percent error of an averaged lab result, average first and then compare.

How it works

• Mean = sum of values ÷ count. The sum uses a compensated method so that 0.1 + 0.2 + 0.3 comes out as exactly 0.6.
• Median: sort the values; take the middle one, or the average of the two middle ones.
• Mode: count each value; report those with the highest count, if it is more than 1.
• Range = maximum − minimum.
• Variance: σ² = Σ(x − mean)² ÷ n for a population, s² = Σ(x − mean)² ÷ (n − 1) for a sample. The standard deviation is the square root.
• Q1 and Q3: medians of the lower and upper halves.

Step by step for 4, 8, 15, 16, 23, 42, 8:

1. Sort: 4, 8, 8, 15, 16, 23, 42. There are 7 values.
2. Mean: 116 ÷ 7 = 16.571428…
3. Median: the 4th value of 7, which is 15.
4. Mode: 8 appears twice, every other value once, so the mode is 8.
5. Range: 42 − 4 = 38.
6. Quartiles: lower half 4, 8, 8 gives Q1 = 8; upper half 16, 23, 42 gives Q3 = 23.

The tool matches: Mean: 16.57142857, Median: 15, Mode: 8 (×2), range 38, Q1 / Q3 = 8 / 23.

Worked examples

Default list. As above. The single large value 42 pulls the mean (16.57) above the median (15).

Textbook standard deviation. 2 4 4 4 5 5 7 9 has mean 5, median 4.5 and mode 4. The tool shows Standard deviation (population σ): 2 and a sample s of 2.138089935.

No repeats. 3, 7, 1, 9 gives mean 5, median 5 and Mode: none (no value repeats).

Two modes. 1 1 2 2 3 gives Mode: 1, 2 (×2) and mean 1.8.

Limits and tips

• Results are shown to 10 significant figures. Round them for a report with the rounding calculator.
• The frequency table appears when there are 100 distinct values or fewer.
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