Statistics

Data is a collection of facts in the form of numbers, such as the marks of every student in a class. When first collected it is called raw or ungrouped data and it looks like a jumble. The first job of statistics is to organise it. We arrange values in order, count how many times each value appears, and write these counts as the frequency. A frequency distribution table lists each value (or class interval) beside its frequency, turning a messy list into a tidy summary you can actually read.

A picture shows the pattern faster than a table. A bar graph uses bars of equal width separated by gaps; the height of each bar shows the frequency, and it suits discrete data like favourite sports. A histogram is used for continuous, grouped data such as heights or marks in class intervals. Its bars touch with no gaps because the data flows continuously, and the area of each bar represents its frequency. A frequency polygon joins the midpoints of the tops of the histogram bars to give a smooth outline of the distribution.

Often we want one number that represents the whole data; this is a measure of central tendency. The mean is the arithmetic average: add all the observations and divide by how many there are. The median is the middle value when the data is arranged in order; it splits the data into two equal halves and is not pulled around by a few very large or very small values. The mode is the value that occurs most often. Different situations call for different averages, which is why we learn all three.

The mean uses every value, so it is the most informative, but a single extreme value (an outlier) can drag it far from the centre. Suppose four people earn ordinary wages and one earns a fortune; the mean income looks high even though most people earn little. Here the median, the middle income, describes the typical person better. The mode is useful when we care about the most common case, such as the most frequent shoe size a shop should stock. Knowing the strengths of each average is the real skill.