TAHA Engineering Toolsby TAHA Knowledge Lab

Mathematics & Statistics

Mean, Median & Mode Calculator

Mean (average), median, mode, range, minimum, maximum, sum and count of a list of numbers.

Inputs

Right-skewed distribution curve with three vertical lines: the mode at the peak, the median to its right and the mean further right towards the tail.modemedianmeanlong tail →
For skewed data the three averages differ: the mode is at the peak, the median splits the data into two equal halves, and the mean is pulled towards the long tail. For symmetric data all three coincide. Diagram not to scale.

Separate the numbers with commas, spaces or semicolons (up to 5000 values). Use a dot for decimals, e.g. 2.5.

Result

Mean (average)
15
Median
15
Mode
15 (appears 3 times)
Range (max − min)
9
Minimum
11
Maximum
20
Sum
150
Count n
10values
Sorted data
11, 11, 12, 14, 15, 15, 15, 18, 19, 20
Working
Mean = sum ÷ n = 150 ÷ 10 = 15 Median = average of the 5th and 6th sorted values = (15 + 15) ÷ 2 = 15 Range = 20 − 11 = 9

For these 10 values the mean is 15 and the median is 15. Mode: 15 (appears 3 times).

Educational estimate — check the assumptions below.

About this calculator

The mean, median and mode are the three common measures of the “centre” of a data set. The mean is the ordinary average; the median is the middle value when the data are sorted, so it is not pulled by a few extreme values; the mode is the value that occurs most often. Together with the minimum, maximum and range they describe a set of measurements, test scores, cycle times or any other data.

For the spread of the data (standard deviation and variance), use the Standard Deviation & Variance calculator.

Formula

x̄ = (Σ xi) ÷ n

Median: sort the values. For odd n it is the middle value; for even n the average of the two middle values.

Mode: the value(s) with the highest count. If two or more different values all occur equally often, there is no mode.

Range = xmax − xmin

Worked example

Data: 12, 15, 11, 15, 18, 20, 15, 14, 11, 19 (n = 10).

  • Sum = 150, mean = 150 ÷ 10 = 15
  • Sorted: 11, 11, 12, 14, 15, 15, 15, 18, 19, 20 → median = (15 + 15) ÷ 2 = 15
  • 15 appears 3 times → mode = 15
  • Range = 20 − 11 = 9

Assumptions and limitations

  • Values are separated by commas, spaces or semicolons, so a decimal comma such as 2,5 is read as two values (2 and 5). Use a dot for decimals.
  • Values are compared exactly for the mode: 2 and 2.0 are the same value, 2 and 2.0001 are not.
  • The sum uses compensated (Neumaier) summation to keep full precision for long lists. Results are shown to 10 significant figures.

References

  • Montgomery, D. C. & Runger, G. C. Applied Statistics and Probability for Engineers — descriptive statistics.
  • NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.5.1 (measures of location).