Numerical description uses summary numbers to describe a set of observations: typical size, spread, and extreme values. These measures help compare series and check whether a value is unusually large or small.
Numerical description uses summary numbers to describe a set of observations: typical size, spread, and extreme values. These measures help compare series and check whether a value is unusually large or small.
Five weekly sales figures 10, 12, 11, 13, 14 can be summarised by a mean of 12 packets and a range of 4 packets.
A plot shows shape. Numbers answer questions such as: What is a typical value? How much do values vary? What were the lowest and highest observations? Both views are needed. Numerical summaries ignore time order unless they are calculated on specially chosen periods, so they cannot replace a time series plot.
Weekly packet sales: 10, 12, 11, 13, 14 (already in time order).
| Measure | Meaning | Interpretation |
|---|---|---|
| Mean | Arithmetic average of all observations | Typical level of the series |
| Median | Middle value after sorting | Typical level that is less affected by one extreme |
| Minimum / Maximum | Smallest and largest observations | The observed extremes |
| Range | Maximum minus minimum | Overall spread of the recorded values |
| Variance | Average of squared deviations from the mean | How widely values scatter; units are squared |
| Standard deviation | Square root of variance | Spread in the same units as the data |
| Quartiles | Values that split the ordered data into four parts | Where the lower, middle and upper observations lie |
Mean = (10 + 12 + 11 + 13 + 14) / 5 = 60 / 5 = 12.
Ordered values: 10, 11, 12, 13, 14. Median = 12.
Minimum = 10, maximum = 14, range = 14 − 10 = 4.
Deviations from the mean 12: −2, 0, −1, 1, 2. Squared deviations: 4, 0, 1, 1, 4. Sum = 10. Sample variance with divisor n − 1 = 10 / 4 = 2.5. Sample standard deviation = √2.5, which is a little more than 1.5.
The lower quartile of these five points is near 11 and the upper quartile is near 13. With only five observations, quartiles are a rough split, not a precise population claim.
The mean and median both equal 12, so the typical weekly sales in this tiny sample sit at 12 packets. The range of 4 packets shows modest spread. None of these numbers, by themselves, tell you that week 1 came before week 5. That order is seen on the time plot.
Numerical summaries cannot reveal all time-order patterns. A time series plot is also important.
The printed mean and median should match 12, and the range should be 4. Software may show extra decimal places for variance and standard deviation; use the same formula convention (sample or population) that your course requires.