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Step 1 · Basics · Topic 2 of 17 · 6 min

Typical value and spread

Mean, median and standard deviation: how to describe a group with one or two numbers, and when those numbers mislead.

In short#

  • The mean x̄ = Σxᵢ / n is sensitive to extreme values. The median is usually less affected by them.
  • The sample standard deviation SD = √(Σ(xᵢ − x̄)² / (n − 1)) describes how observations spread around the mean.
  • Quartiles Q1 and Q3 define the interquartile range, which covers the middle half of the distribution. Linza computes quartiles by linear interpolation.
  • CV = SD / x̄ × 100% is used for positive quantities on a ratio scale, where zero means the measured quantity is absent.

In plain words#

Five friends scored 3, 4, 4, 5 and 5 points. The mean is 4.2. Add a sixth friend with 20 points. The mean rises to 6.8, although nearly everyone scored about four. The median, the middle of the ordered values, changes little: from 4 to 4.5.

Spread shows how much values differ from each other. Two groups can have the same mean even though values are close together in one and widely scattered in the other. Standard deviation measures spread around the mean in the same units as the data. Extreme values have more influence because deviations are squared.

When appropriate, show individual observations alongside summary measures. The points reveal sample size and distribution shape, which a single bar for the mean can hide.

Try it yourself

How an extreme value affects summaries Drag points with the mouse. To use the keyboard, focus a point and press the arrow keys; hold Shift for larger steps. The values below update immediately.
010203040Mean

MeanMedianMean ± SD

n
6
Mean
11.50
Median
11.50
Standard deviation
2.43

Add an extreme value and compare how the mean and median change. Move the points and watch SD.

In Linza#

  1. In Analyses, choose descriptive statistics. The table shows n, mean, median, SD, SEM, 95% CI, minimum and maximum, quartiles and CV.
  2. For Bars or Dots and mean plots, show individual observations and choose SD, SEM or 95% CI error bars. A box plot shows the median and quartiles, with whiskers from the minimum to the maximum.
Try it in Linza Opens in a new tab with synthetic data. Work in the original tab remains unchanged.

Common mistakes#

  • With strongly skewed data, a single mean may poorly represent the group. Show the median, quartiles and data distribution.
  • A ± label that does not say whether it is SD or SEM.

Check yourself#

A group has the values 2, 3, 3, 4 and 38. What describes the typical value better?

Optional. Your answer is not saved.

Practice exercise: What is the mean strength of the new formulation, MPa? →

Sources#

  • Tukey J.W. (1977) Exploratory Data Analysis. Addison-Wesley. Search for publication Opens in a new tab
  • Weissgerber T.L., Milic N.M., Winham S.J., Garovic V.D. (2015) Beyond bar and line graphs: time for a new data presentation paradigm. PLoS Biology, 13(4). Open publication Opens in a new tab

References are for further reading. A method’s suitability depends on the study design and its assumptions.