Classifying statistical data

Kinds of data

Statistics starts with data, and the first job is to say what kind of data you have. The kind of data decides how you can organise it, which averages make sense and which diagrams can show it.

Qualitative data (also called categorical data) is described in words, not numbers: favourite sport, eye colour, type of vehicle, country of birth. You can count how many items fall in each category, but you cannot do arithmetic with the categories themselves.

Quantitative data (numerical data) is numbers that come from counting or measuring. It comes in two kinds.

• Discrete data can only take particular separate values, usually whole numbers found by counting: the number of goals in a match, the number of people in a car, the number of pages in a book. Between 3 goals and 4 goals there is nothing.
• Continuous data is measured, and can take any value in a range: height, mass, time, temperature, distance. A height written down as 162 cm could really be 161.8 cm or 162.37 cm; the value recorded is rounded to the accuracy of the measuring instrument.

A quick test. Ask whether a value halfway between two possible values makes sense. Halfway between 3 and 4 children means nothing, so the number of children is discrete. Halfway between 3 kg and 4 kg is 3.5 kg, a real mass, so mass is continuous.

Two cases catch students out.

• Shoe size is discrete. Sizes come in fixed steps (5, 5.5, 6, and so on), so a size of 5.3 is impossible, even though not every size is a whole number. Discrete does not mean "whole numbers only"; it means separate values with nothing in between.
• Money is treated as discrete, because amounts go up in whole cents.

Age is a measurement of time, so it is continuous, even though we usually round it down and say "15" for anyone between their 15th and 16th birthdays.

Tally charts and frequency tables

Raw data, a list in the order it was collected, is hard to read. A frequency table sorts it out: each possible value (or category) gets a row, and its frequency is the number of times that value occurs.

A tally is the quickest way to count by hand.

1. Go through the list once, in order.
2. For each item, make one stroke in the row for its value, and cross the item off the list.
3. Group the strokes in fives: four strokes with the fifth drawn across them, shown here as |̶|̶|̶|̶. Groups of five are quick to add up.
4. Count each row's tally to give its frequency.

Working through the list once, rather than hunting for every 0, then every 1, and so on, is what stops items being missed or counted twice.

Always check the total. The frequencies must add up to the number of items in the list. If 20 matches were recorded, the frequency column must add up to 20. It takes a few seconds and catches almost every counting slip.

Grouping data into classes

When data takes many different values, a row for every value makes a long and unhelpful table. Instead the values are put into classes (groups), and the table records how many values fall in each class. This is a grouped frequency table.

For discrete data, the classes are ranges of whole numbers, such as 1 to 5, 6 to 10, 11 to 15. Every possible value is in exactly one class.

For continuous data, classes are written with inequalities, because there are no gaps between possible values. For heights of $h$ cm:

• $150⩽h<155$
• $155⩽h<160$
• $160⩽h<165$

A height of exactly 155 cm goes in the class $155⩽h<160$, because the first class needs $h$ to be less than 155. The two signs say exactly where a boundary value belongs, so nothing falls between two classes and nothing belongs to two classes.

Good classes:

• do not overlap, so no value belongs to two classes;
• leave no gaps, so every possible value belongs to a class;
• cover all the data, from the smallest value to the largest;
• are usually of equal width, which makes them easy to compare.

Labels such as "10 to 20, 20 to 30" break the first rule: a value of 20 fits both. Labels such as "10 to 19, 20 to 29" are fine for whole numbers, but leave a gap for continuous data: where would 19.6 go?

Grouping loses information. Once 162.4 cm is placed in $160⩽h<165$, the exact value is gone, which is why any average worked out from a grouped table is only an estimate.

Two-way tables

A two-way table sorts the same people or items in two ways at once, for example by year group and by favourite sport. Each cell counts the items that fit both its row heading and its column heading. An extra row and an extra column hold the totals.

Three facts let you fill in missing cells:

• each row adds up to its row total;
• each column adds up to its column total;
• the grand total, in the bottom right corner, equals both the sum of the row totals and the sum of the column totals.

Method. Look for a row or column with only one number missing. Find it by subtraction, then look again: each new number usually opens up another row or column. Finish by checking that every row and every column adds up.

When the information is given in words, draw the empty table first, with headings and a Total row and column. Put each fact into its cell, including totals you can work out at once ("the rest are juniors"), and then fill in the remaining cells one at a time.

Reading a two-way table needs care with the word of. "What fraction of the Year 11 students chose art?" divides by the Year 11 total, not by the grand total.