Average Calculator

“Average” usually means the mean: add the numbers up and divide by how many there are. The median is the middle number once they're in order, and the mode is the number that turns up most often. The range is the gap between the biggest and smallest.

The formula

mean = sum of the numbers ÷ how many numbers
median = the middle number in order (or the mean of the two middle ones)
mode = the number that appears most often
range = largest − smallest
weighted average = sum of (value × weight) ÷ sum of weights
next number needed = target × (count + 1) − current sum

Worked example

For 6, 8, 3, 9, 8, 2, 7 and 5: the sum is 6 + 8 + 3 + 9 + 8 + 2 + 7 + 5 = 48, and there are 8 numbers, so the mean is 48 ÷ 8 = 6. In order they are 2, 3, 5, 6, 7, 8, 8, 9. With an even count the median is the mean of the two middle numbers: (6 + 7) ÷ 2 = 6.5. 8 appears twice, so the mode is 8, and the range is 9 − 2 = 7. To lift the average to 7 with one more number, you need 7 × 9 − 48 = 15.

Weighted average

Use it when some values count more than others. Homework 85 (worth 20%), a midterm of 72 (30%) and a final of 90 (50%) give (85 × 20 + 72 × 30 + 90 × 50) ÷ 100 = (1,700 + 2,160 + 4,500) ÷ 100 = 83.6. The plain average of 85, 72 and 90 would be 82.33, which ignores that the final counts most. The same sum gives an average share price: 10 shares at 100 and 30 at 80 cost (1,000 + 2,400) ÷ 40 = 85 each. If your weights are percentages that don't add to 100 yet, the calculator divides by what they do add to, which is your grade so far. To plan the score you need on a final, try the final grade calculator.

Averaging percentages the right way

You can't just average percentages that come from groups of different sizes. 80% of 50 questions is 40 right, 60% of 200 is 120 and 90% of 10 is 9. That's 169 right out of 260, which is 169 ÷ 260 = 65%. The plain average of 80, 60 and 90 is 76.67%, far too high because the big 200-question test counts the same as the tiny 10-question one. Only when every group is the same size do the two answers match.

Average of times

Times are turned into seconds, averaged, and turned back. Laps of 25:30, 24:10, 26:05 and 23:55 are 1,530 + 1,450 + 1,565 + 1,435 = 5,980 seconds; 5,980 ÷ 4 = 1,495 seconds = 24:55 a lap. Times of day are different, because the clock wraps round at midnight: bedtimes of 23:30, 00:15, 23:45 and 00:30 average to 00:00, not midday. The calculator places each time on a 24-hour clock face and averages the directions (a circular mean).

Geometric, harmonic and other means

• Geometric mean (the nth root of all the numbers multiplied together) suits growth rates and ratios. Growth factors of 2 and 8 have a geometric mean of √16 = 4. For yearly returns, see the CAGR calculator.
• Harmonic mean (count ÷ sum of 1 ÷ each number) suits rates over the same distance. Driving there at 60 km/h and back at 40 km/h averages 2 ÷ (1/60 + 1/40) = 48 km/h, not 50, because you spend longer at the slower speed. For trips with different distances or times use the speed, distance & time calculator.
• Quadratic mean (root mean square) squares, averages and takes the square root; it is used for electrical currents and errors.
• Trimmed mean drops a set share of the highest and lowest values first, so one odd result can't drag the average.
• Midrange is halfway between the smallest and largest.

For positive numbers that aren't all the same, harmonic < geometric < mean < quadratic mean.

Moving average

A moving average with a window of 3 averages numbers 1–3, then 2–4, then 3–5 and so on, smoothing out bumps so you can see the trend. On the example list the first is (6 + 8 + 3) ÷ 3 = 5.67 and the last is (2 + 7 + 5) ÷ 3 = 4.67. The running (cumulative) average is the mean of everything so far.

Mean, median or mode?

• Mean uses every number, so it's best when values are spread fairly evenly.
• Median ignores extreme values, so it's better for things like house prices or salaries where a few very large values pull the mean up.
• Mode works for the most common shoe size, score or answer. A list can have no mode, or several.

For how spread out your numbers are (variance and standard deviation), use the standard deviation calculator.