The statistical problem solving cycle
Data are numbers in context and the goal of statistics is to get information from those data, usually through problem solving. A procedure or paradigm for statistical problem solving and scientific enquiry is illustrated in the diagram. The dotted line means that, following discussion, the problem may need to be re-formulated and at least one more iteration completed.
Descriptive statistics
Given a sample of $$n$$ observations $$x_1, x_2, \ldots, x_n$$ we define the sample mean to be \[\bar{x}=\frac{x_1+x_2+\ldots+x_n}{n}=\frac{\sum{x_i}}{n}\] and the corrected sum of squares by \[s_{xx}=\sum{\left(x_i-\bar{x}\right)^2}=\sum{x_i^2}-n\bar{x}^2=\sum{x_i^2}-\frac{\left(\sum x_i\right)^2}{n}\] $$\frac{s_{xx}}{n}$$ is sometimes called the mean squared deviation. An unbiased estimator of the population variance, $$\sigma^2$$, is provided by calculating $$s^2=\frac{s_{xx}}{(n-1)}$$. The sample standard deviation is $$s$$. In calculating $$s^2$$, the divisor $$(n-1)$$ is called the degrees of freedom (df). Note that $$s$$ is also sometimes written $$\hat{\sigma}$$.
If the sample data are ordered from smallest to largest then the:
- minimum (Min) is the smallest value;
- lower quartile (LQ) is the $$\frac{1}{4}(n+1)$$-th value;
- median (Med) is the middle [or the $$\frac{1}{2}(n+1)$$-th] value;
- upper quartile (UQ) is the $$\frac{3}{4}(n+1)$$-th value;
- maximum (Max) is the largest value.
These five values constitute a five-number summary of the data. They can be represented diagrammatically by a box-and-whisker plot, commonly called a boxplot.
A boxplot
Grouped frequency data
If the data are given in the form of a grouped frequency distribution where we have $$f_i$$ observations in an interval whose mid-point is $$x_i$$ then, if $$\sum{f_i}=n$$ \[ \bar{x}=\frac{\sum{f_i x_i}}{\sum{f_i}}=\frac{\sum{f_i x_i}}{n}\] and \[s_{xx}=\sum{f_i\left(x_i-\bar{x}\right)^2}=\sum{f_i x_i^2}-\frac{\left(\sum f_i x_i\right)^2}{n}\]