An average is a summary, not a verdict. When one value sits far from the rest, the mean can move a long way while the median barely flinches. This guide shows you how to see that shift, describe it, and avoid reading more into it than the numbers support.

Two summaries, two behaviors

The mean adds every value and divides by the count. The median sorts the values and takes the middle one. Both estimate a typical value, but they respond differently to extremes. A single large number is fully counted by the mean and can pull it upward; the median only cares about position, so it can sit still.

NIST's engineering statistics handbook shows this in its beam-deflection case study, where summary statistics are reported side by side: sample size, mean, median, minimum, maximum, range, and standard deviation [s1]. The same source later examines a set of plotted points and concludes that several specific points appear to be outliers [s1]. That pairing is the useful habit: report location summaries together, and treat extremes as something to notice, not instantly delete.

A worked demonstration (hypothetical)

The default dataset in the 13 Labs experiment is 2, 3, 3, 4, 5, 6, 100. These numbers are illustrative, not measurements of anything. Six of seven values are small and close together, so the median is 4, the middle value. The mean is 123 / 7, about 17.6 — well outside the cluster of ordinary values. The range runs from 2 to 100, a span of 98.

Now remove the 100. The remaining six values center tightly; the median sits between 3 and 4, and the mean drops into that small neighborhood too. The point is not that 100 is wrong. It is that the mean and the median tell different stories when an extreme is present, and you should know which summary you are quoting.

Why a median resists a single value

Sorting puts the extreme at one end. Changing how far out that end point sits changes the mean (it is summed) but does not change which value is in the middle (the median only reads rank order). That framing — center, spread, and marked extremes — is a sturdier summary than one number alone.

Where the experiment fits

The interactive tool on this site lets you edit a numeric dataset (3 to 200 finite values), compute mean, median, and range, and then remove an outlier to compare before and after. Open the experiment to try it; the same tool is embedded on the home page. Results appear in a real plotted dot distribution and an accessible results table, and you can download the exact dataset and settings as JSON so a comparison can be reproduced.

Sampling settings are bounded: sample size 2 to 100, repeats 50 to 500, with an explicit seed shown on screen. The tool resamples with replacement — meaning a drawn value goes back into the pool and can be drawn again. NIST describes bootstrap sampling plainly: generate subsamples, recompute the statistic, and repeat enough times to see the sampling distribution, noting that 500 subsamples is usually sufficient [s4]. The same page warns that the bootstrap is not appropriate for every statistic, and singles out the range as poorly suited because it depends heavily on the tails [s4]. Keep that caution in mind: repeated sampling of the mean is informative; repeated sampling of an extreme-sensitive statistic is less so.

What the spread of sample means shows

Drawing many small samples and plotting their means gives a picture of how much a typical value can wobble. NIST's propagation-of-error tables show the classic result that the standard deviation of an average shrinks with the square root of the number of measurements, and that approximations can be seriously in error when the sample is small [s5]. In the experiment, set a small sample size and compare the spread of sample means with a larger one. You are watching variability, not proving a difference.

Do not turn this into a significance test. The tool makes no p-values, no confidence intervals, and no predictions, and this guide does not either. A tight cluster of resampled means is a description of your own dataset, not evidence about a wider population.

A short checklist

  • Report the median alongside the mean whenever an extreme value is present.
  • Look at a plot before trusting a summary number.
  • Name the outlier procedure explicitly. NIST shows outlier removal improving a fit only modestly — about a 5% reduction in residual standard deviation — and calls the decision a judgment call [s3].
  • If you remove a value, keep the original dataset and record why it was excluded.
  • State sample size and sampling method; label any resampled output as illustrative.

Reading results without overclaiming

Descriptive language stays close to the data: "the mean is 17.6 and the median is 4," or "removing the value 100 moved the mean to 23/6, approximately 3.83." Words like "caused," "significant," and "predicts" require evidence this experiment does not generate. Continue with what a small sample can and cannot tell you and read the experiment without overclaiming for the full framing. If you want to know what this project is, see About.

Primary sources

  • NIST/SEMATECH, Engineering Statistics Handbook — beam deflections case study, summary statistics and outlier identification [s1]. eda.pdf
  • NIST/SEMATECH — bootstrap plot: resampling, 500 subsamples, cautions on range [s4]. eda.pdf
  • NIST/SEMATECH — outlier removal and modest fit improvement [s3]. eda.pdf
  • NIST/SEMATECH — propagation of error, 1/√N scaling, small-sample caveats [s5]. mpc.pdf

Published by NovCog. This guide is educational and contains no inferential significance claims.