The scatter diagram and a statistical relation

A doctor examined 14 pupils, and wrote down two numbers for each: the height and the weight.

• The population: the pupils examined.
• The variables: the height and the weight – two quantitative variables, whose values are numbers.
• Each pupil is a pair of values, and in the diagram – a point: the height on the horizontal axis, the weight on the vertical axis.

A drawing like this is called a scatter diagram:

• The variable on the horizontal axis, $\displaystyle x$, is called the independent variable; the variable on the vertical axis, $\displaystyle y$, is called the dependent variable – the one we want to predict.
• The points rise from left to right: taller pupils usually weigh more.
• “Usually” – not always: one pupil is taller than another and weighs less. This is a statistical relation: a tendency of the data, not a law that holds for every pair.

• A positive relation: as $\displaystyle x$ grows, $\displaystyle y$ tends to grow – the points rise, like the height and the weight.
• A negative relation: as $\displaystyle x$ grows, $\displaystyle y$ tends to fall – the points fall. Heavier cars usually travel fewer kilometres on a litre of fuel:

• No relation: the points are scattered with no direction. The number of births in Jerusalem has nothing to do with the numbers drawn in the lotto:

Whether there is a relation, and its direction, can be told without any calculation: by looking at the diagram, or at a table of the pairs.

A perfect relation: $\displaystyle y$ is fixed entirely by $\displaystyle x$, and all the points lie on one line. When the line is straight, it is a perfect linear relation. The age in 2026 is $\displaystyle 2026$ minus the year of birth – a perfect, negative linear relation:

The relation between the radius of a circle and its area, $\displaystyle \pi r^2$, is perfect and positive, but it is not linear: the points lie on a curve.

In real data a perfect relation is rare. Later in the course the correlation coefficient will measure how close the points are to a straight line.

A statistical relation does not prove that a change in one variable causes a change in the other – a causal relation.

• At swimming pools, on days when a lot of ice cream is sold, a lot of towels also go missing. The ice cream does not make the towels go missing: on a hot day many people come to the pool, and they both buy more ice cream and forget more towels. The weather is a third factor that affects both variables.
• The relation may also be a coincidence, like two variables that simply both grow over the years.
• To show a causal relation, an experiment is needed, not only data that was collected.

Here are two tables of pairs.

Draw the two scatter diagrams.

For each table: is there a statistical relation? If so – positive or negative, perfect or not?

In each case – would you expect a relation between the two variables? If so – positive or negative, and perfect or not? Explain.

1. The amount of fuel in a car’s tank, and the number of kilometres it has travelled since it was last filled.
2. A family’s monthly income, and its electricity use.
3. The waiting time in a queue at the bank, and the number of clerks working at the time.
4. A pupil’s mark in maths, and the same pupil’s mark in physics.
5. The last digit of a person’s identity number, and the person’s height.

In each of these cases a positive statistical relation was found. Suggest an explanation of where the relation comes from.

1. The number of nursery schools in each town in Israel, and the town’s yearly income from parking fines.
2. The reading speed of primary school pupils, and their shoe size.

A study of men aged 18 to 54 found a positive relation between blood pressure and income.
What can be concluded? Choose and explain.

1. Of two men of these ages chosen at random, the one with the higher blood pressure is more likely to have the higher income.
2. The higher the blood pressure, the lower the income.
3. A high income causes high blood pressure.