## 23.5 Mean differences: Numerical summaries

Since the data are differences, a numerical should summarises the differences. Summarising the Before and After data is useful too, but summarising the differences is crucial because the RQ is about the differences below.

For the house insulation data, the appropriate numerical summary for paired data summarises the differences using means, standard deviations, and so on, as appropriate. (While a mean or a median may be appropriate for describing the data, the CI is about the mean, since the sampling distribution for the sample mean (under certain conditions) has a normal distribution which is best described using a mean.) A numerical summary of the energy savings from a calculator (Statistics Mode) or computer software gives the sample mean of the differences as $$\bar{d} = 0.54$$, and the standard deviation of the differences as $$s_d = 1.015655$$.

A formal numerical summary table is shown in Table 23.2.

TABLE 23.2: The mean, median, standard deviation and IQR for the energy consumption data (in MWh)
Mean Median Std dev IQR
Before 12.49 12.45 1.68 2.28
After 11.95 12.20 1.96 2.45
Energy savings 0.54 0.35 1.02 0.80