Analysis and Interpretation

With the data organised and presented, the project reaches its analytical heart. Analysis and interpretation means applying the statistical tools you have studied to extract meaning from the tabulated data — to summarise it, measure how spread out it is, and detect relationships among the variables. Three families of tools, drawn from the chapters on central tendency, dispersion and correlation, do most of this work.

1. Measures of central tendency — the average. These summarise a whole set of values by a single representative figure. The most common is the arithmetic mean. For a frequency distribution it is

where $X$ is the value (or class mid-point) and $f$ its frequency. When the figures are large, the step-deviation method simplifies the arithmetic:

with $A$ an assumed mean and $c$ the common class width. The mean answers the question "what is the typical value?" — for instance, the average monthly household expenditure on a product.

2. Measures of dispersion — the variability. An average alone can hide wide differences, so we also measure how much the values scatter around it. A key measure is the standard deviation:

A small $\sigma$ means the observations cluster close to the mean; a large $\sigma$ means they are widely spread. Reporting the mean together with the standard deviation gives a far more honest picture than the average by itself.

3. Correlation — the relationship. Where two variables may move together — say income and expenditure — a measure of correlation shows whether, and how strongly, they are related. Karl Pearson's coefficient of correlation,

ranges from $-1$ to $+1$: a value near $+1$ indicates a strong positive relationship, near $-1$ a strong negative one, and near $0$ little linear relationship. …