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Welcome! Today we are going to look at how a measure of center summarizes a data set, and when the mean is the right choice for the job.
In statistics we summarize a data set with a measure of center. The mean, the median, and the mode each describe that center differently, so which one we report depends on the shape of the distribution.
The mean is the balance point of a distribution: the distances of the values above it add up to the distances of the values below it. Because it uses every value, one extreme outlier can pull it a long way.
We write the mean as x̄ = (Σx) / n: add all n data values, then divide by n. If a distribution is strongly skewed, report the median instead, because it resists outliers.
Find the mean of these four test scores: 15, 17, 12, and 16. Then decide whether the mean or the median better describes this data set.
- Count the data values: here n = 4.
- Add all the scores together: 15 + 17 + 12 + 16 = 60.
- Divide the total sum by the number of scores: 60 ÷ 4 = 15.
- The mean is x̄ = 15. The median of 12, 15, 16, 17 is 15.5, so both centers agree closely and no outlier is distorting the mean.
