Average Calculator

Mean, median and mode: three kinds of average

“Average” usually means the mean, but there are three common ways to pick one number that represents a list, and each answers a slightly different question:

• The mean shares the total out equally: add everything up and divide by how many numbers there are.
• The median is the middle value once the numbers are sorted. Half the values are at or below it, half at or above it.
• The mode is the value that appears most often. A list can have several modes, or none at all when every value appears the same number of times.

The range, the gap between the largest and smallest value, is a first measure of how spread out the numbers are; the standard deviation calculator gives a fuller picture.

Paste numbers from a spreadsheet or type them separated by commas, spaces or new lines. Because the comma separates values, write decimals with a point (2.5) and leave out thousands separators. Switch to weighted average when values count unequally, such as course grades with different credit hours.

Formulas

x₁ … xₙ

the values; for the median they must be sorted from smallest to largest

n

how many values there are

wᵢ

the weight of value xᵢ; only the proportions between weights matter

When every value is positive, the calculator also shows the geometric mean (the nth root of the product, used for growth rates and ratios) and the harmonic mean (n divided by the sum of reciprocals, used for averaging rates such as speeds over equal distances).

Worked example: eight test scores

Scores: 72, 85, 90, 85, 68, 94, 77, 85.

1. Mean: the sum is 656, and 656 ÷ 8 = 82.
2. Median: sorted, the scores are 68, 72, 77, 85, 85, 85, 90, 94. With eight values the middle two are the 4th and 5th, both 85, so the median is (85 + 85) ÷ 2 = 85.
3. Mode: 85 appears three times, more than any other score, so the mode is 85.
4. Range: 94 − 68 = 26.

The mean is lower than the median because the two low scores (68 and 72) pull it down more than the high ones pull it up.

Weighted: coursework 92 (worth 30%), midterm 85 (30%) and final exam 78 (40%) give 92 × 30 + 85 × 30 + 78 × 40 = 8,430, and 8,430 ÷ 100 = 84.3. The plain mean of the three marks is 85; the heavier final exam pulls the weighted result down.

Which average should you use?

• Mean for data without extreme values, and whenever the total matters — the mean times the count gives back the sum.
• Median for skewed data such as incomes or house prices. For 30, 35, 40, 45 and 250 the mean is 80 but the median is 40, much closer to four of the five values.
• Mode for categories or repeated values, such as the most common shoe size sold.
• Don’t average averages unless the groups are the same size. A class of 50 averaging 80% and a class of 150 averaging 60% have an overall average of 65%, not 70% — use the weighted mode with the class sizes as weights.