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Is the average actually telling me the truth?

Get the mean, median, range and quartiles from a set of measurements, with the mean and median shown on the same axis so you can see when they disagree.

Mean, Median & Range · measure · Foundation · free, no account needed

Use this when

  • A report quotes an average and you want to know whether it describes anyone
  • Your data has a few very large values and you suspect they are distorting it
  • You need to describe a set of measurements to somebody who was not there

Mean, Median & Range

sandbox
Dot plot of 20 callback time values from 8 to 118. The mean is 25.25 and the median is 12.5. The mean sits to the right of the median.median 12.5mean 25.258118
Each dot is one measurement; dots stack where values repeat. Solid line: median. Dashed line: mean.

n

20

Mean

25.25

Median

12.5

Range

110

Minimum
8
Q1
10.75
Q3
15
Maximum
118

Right-skewedThe mean (25.25) is above the median (12.5). A few large values are pulling it up, so "average callback time 25.25" overstates what a typical case looks like — half of them are at or below 12.5. Waiting times, handling times and costs are nearly always shaped like this, which is why "average" so often describes nobody. Quote the median, or quote both — but do not quote the mean alone.

Range is max − min: it uses exactly two of your 20 values and gets less reliable as the sample grows, because a bigger sample has more chances to contain an extreme. Q1 to Q3 covers the middle half and does not have that problem.

One number per line, or label, number.

How this is calculated
  • Mean — sum ÷ n. Every value pulls on it, so one extreme value moves it.
  • Median — the middle value once sorted; the average of the two middle values when n is even. Only the position of the extremes matters, not how extreme they are.
  • Range — maximum − minimum.
  • Q1 and Q3 — the 25th and 75th percentiles, by the R7 rule (the one R, NumPy and Excel PERCENTILE.INC use). Packages disagree about quartiles, so the method is stated rather than assumed.
  • Standard deviation — the sample form, dividing by n − 1. Dividing by n understates the spread of a sample and is one of the most common errors in hand-built spreadsheets.
  • The skew note fires when the mean sits more than 0.2 spreads away from the median, where a spread is (Q3 − Q1) ÷ 1.349 — 3.15 here. That divisor is deliberately not the standard deviation: one extreme value inflates the SD faster than it moves the mean, so measured that way the gap is capped at 1/√n for a single outlier and from n = 26 upward nothing could ever trip it. The middle half of the data is the part the outlier is not in.

Source: Montgomery, D.C., Introduction to Statistical Quality Control 7e, Ch. 3; Hyndman, R.J. & Fan, Y. (1996), “Sample Quantiles in Statistical Packages”, The American Statistician 50(4), for the quantile definitions.

How this is calculated

Mean = sum / n. Median = middle value of the sorted data, or the mean of the two middle values when n is even. Range = max - min. Quartiles by the R7 rule (linear interpolation at h = (n-1)p), the definition used by R, NumPy and Excel PERCENTILE.INC. Standard deviation is the sample form, dividing by n-1.

Source: Montgomery, D.C., Introduction to Statistical Quality Control 7e, Ch. 3; Hyndman, R.J. & Fan, Y. (1996), Sample Quantiles in Statistical Packages, The American Statistician 50(4), for the quantile definitions.

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