Summarise a Data Set, Step by Step

An ordered procedure on the set 1, 2, 3, 4, 5, 6, 8, 9: sort, count n = 8, find the median 4.5, compute the mean, then the range and the spread.

Step 1 — Sort. Write the values in ascending order: 1, 2, 3, 4, 5, 6, 8, 9. Sorting is not optional: the median and every percentile are defined on ordered data.

Step 2 — Count. n = 8, an even number. Because there is no single middle value, the median will be the arithmetic mean of the two middle values.

Step 3 — Find the median. The middle positions are the 4th and 5th: values 4 and 5. (4 + 5) / 2 = 4.5. This cut separates the higher half {5, 6, 8, 9} from the lower half {1, 2, 3, 4}.

Step 4 — Compute the mean. Sum the values: 1 + 2 + 3 + 4 + 5 + 6 + 8 + 9 = 38. Divide by n: 38 / 8 = 4.75.

Step 5 — Compute the range. Maximum 9 minus minimum 1 equals 8, in the original units.

Step 6 — Compute the spread. The variance is the average of the squared distances from 4.75, and the standard deviation is its square root. Report the standard deviation next to the mean, and the interquartile range next to the median.

• Sort before anything else — unsorted data breaks the median.
• Record n explicitly; even versus odd changes the median rule.
• Quote the mean with its standard deviation, or the median with its interquartile range.
• Check the minimum and maximum for outliers before quoting the range.

Further reading

• The Measures: Formulas and Worked Examples
• Mean vs Median with Skewed Data