Statistics

1. What is data?

Data means the facts or figures we collect for a definite purpose. When we just write numbers in the order we get them, it is called raw or ungrouped data. For example, the marks of 10 students — 23, 41, 56, 38, 41, 23, 56, 41, 38, 56 — are raw data. Data collected by the investigator herself (by surveying, measuring or counting) is called primary data, while data taken from a source someone else gathered (a book, website or report) is called secondary data. The range of data is the difference between the highest and lowest values, and it tells us how spread out the values are.

2. Presentation of data

Raw data is hard to read, so we organise it. The first step is often an array — arranging values in ascending or descending order. Next we build a frequency distribution table. The frequency of an observation is the number of times it occurs. Drawing small tally marks (groups of five, with the fifth crossing the bundle) makes counting fast and error-free. A table that lists each separate value with its frequency is an ungrouped frequency distribution.

3. Grouped frequency distribution

When data has many different values (say marks from 0 to 100), listing every value is clumsy. Instead we make class intervals (groups) like 0–10, 10–20, 20–30, and count how many observations fall in each. The size of a class is its class width (here 10). Each class has a lower class limit and an upper class limit. The midpoint of a class is the class mark, found by class mark = (lower limit + upper limit) ÷ 2. There are two styles: the exclusive (continuous) form, where 10–20 means 10 included but 20 excluded (20 goes into the next class), and the inclusive form like 1–10, 11–20, where both ends are included. To draw graphs we must first change inclusive form to exclusive form by adjusting the limits with half the gap between classes.

4. Graphical representation

Pictures make data instantly understandable. Bar graphs use bars of equal width with equal gaps; the height (length) of each bar shows the frequency. They are used for discrete, separate categories like favourite sports or modes of transport. A histogram looks like a bar graph but the bars touch each other with no gaps, because it is used for continuous class intervals; the area of each bar is proportional to its frequency. When the class widths are unequal we adjust the heights so the area stays proportional. A frequency polygon is made by joining the midpoints (class marks) at the tops of the histogram bars with straight line segments; we also add a class with zero frequency at each end so the polygon closes neatly on the horizontal axis. A frequency polygon can be drawn with or without first drawing the histogram.

5. Measures of central tendency

Often we want a single number that represents the whole data set — a typical or central value. These are the measures of central tendency: mean, median and mode.

Mean (average): add up all the observations and divide by how many there are. For values x₁, x₂, …, xₙ, the mean is the sum divided by n, written as the sum of all observations ÷ total number of observations.

Median: the middle value when the data is arranged in order. First arrange the data in ascending order. If the number of observations n is odd, the median is the value at position (n+1)÷2. If n is even, the median is the average of the two middle values, at positions n÷2 and (n÷2)+1. The median splits the data into two equal halves and is not affected by very large or very small extreme values.

Mode: the observation that occurs most often — the value with the highest frequency. A set can have one mode, more than one mode, or no mode at all.

6. Choosing the right measure

The mean uses every value, so it is the most common, but a single very large or very small value (an outlier) can pull it away from the centre. The median is safer when extreme values are present, because it only cares about position, not size. The mode is the best choice when we want the most popular or most frequent item, such as the shoe size a shop sells the most. Understanding which measure fits the situation is an important skill, not just calculating them.