Two-Variable Statistics (Senior 6 Basic Mathematics): free lessons | Ganzaa

Two-Variable Statistics

Lessons

Senior 6 · Basic Mathematics

Two-Variable Statistics

4 lessons

Lesson 1 of 4

Bivariate data, scatter diagrams and types of correlation

25 min

Two measurements on the same person

A Basic Mathematics teacher in Huye asked eight students in her class two things: how many hours they had revised in the week before a test, and what mark they then scored. Every student gives a pair of numbers. The question she wants answered is not about revision alone or marks alone, but about how the two move together.

Student	A	B	C	D	E	F	G	H
Revision time, $x$ (hours)	2	3	4	5	6	7	8	9
Test mark, $y$ (%)	35	42	40	55	58	62	70	75

Illustrative data, made up for this lesson.

In earlier years you summarised one variable at a time with a mean, a median and a standard deviation. This unit studies two variables measured on the same individuals.

Definition

Bivariate data is a set of ordered pairs $(x_{i}, y_{i})$ , $i = 1, 2, \ldots, n$ , where both values are measured on the same individual or occasion.

The independent (explanatory) variable $x$ is the one we think may explain or predict the other. The dependent (response) variable $y$ is the one we want to predict.

Example: for student A the pair is $(2, 35)$ : 2 hours of revision and a mark of 35%.

Drawing a scatter diagram

A scatter diagram (scatter plot) shows each pair as one point on a coordinate grid.

Put the independent variable on the horizontal axis and the dependent variable on the vertical axis.
Choose a scale for each axis that covers the smallest and largest values. An axis need not start at zero, but if it does not, say so clearly.
Label both axes with the quantity and its unit.
Plot each pair as a single dot or cross. Do not join the points.
Give the diagram a title.

The points rise from the bottom left to the top right and lie close to a straight line. Students who revised longer tended to score higher.

Definition

Correlation is the relationship between two variables in bivariate data: how strongly, and in which direction, one changes as the other changes.

Positive correlation: as $x$ increases, $y$ tends to increase.
Negative correlation: as $x$ increases, $y$ tends to decrease.
No correlation: the points show no pattern.

Example: the revision data show a strong positive correlation.

Describing a correlation

A full description of a scatter diagram gives three things: the direction (positive or negative), the form (linear, if the points follow a straight line, or non-linear, if they follow a curve) and the strength (strong if the points lie close to the line or curve, weak if they are widely spread).

Pattern of points	Description
All points exactly on a rising straight line	Perfect positive linear correlation
Points close to a rising line	Strong positive linear correlation
Points loosely around a falling line	Weak negative linear correlation
Points spread with no trend	No (zero) correlation
Points close to a curve	Non-linear relationship; a straight-line measure does not describe it well

Khan Academy: describing a bivariate relationship by its direction, form (linearity) and strength. Watch how the same three words are used for every scatter plot.

Worked example

Worked example 1. A tomato seller at Nyabugogo market recorded the price per kilogram $x$ (Frw) and the mass sold $y$ (kg) on five days (illustrative): $(200, 50)$ , $(300, 44)$ , $(400, 35)$ , $(500, 30)$ , $(600, 22)$ . Describe the correlation.

Solution. Price is the independent variable, so it goes on the horizontal axis. As $x$ rises by 100 Frw each time, $y$ falls: 50, 44, 35, 30, 22. The falls are 6, 9, 5 and 8 kg, roughly equal, so the points lie close to a falling straight line.

The data show a strong negative linear correlation: the higher the price, the less was sold.

Worked example

Worked example 2. For ten adults a researcher records height (cm) and the last digit of their phone number. Predict the scatter diagram and explain.

Solution. The last digit of a phone number is assigned with no regard to the owner's body, so there is no reason for it to rise or fall with height. The points should be scattered with no trend: no correlation. If a small sample happened to show a slight trend, that would be chance, not a relationship.

Warning

Two common mistakes

Joining the dots. A scatter diagram is not a line graph; the order of the students means nothing.
Reading correlation as cause. A strong correlation shows that two variables move together, not that one causes the other. Ice-cream sales and cases of sunburn both rise in hot weather; the heat drives both.

Tip

Cover the diagram with a pencil laid along the points. If the pencil slopes up, the correlation is positive; if down, negative. If the points hug the pencil, it is strong; if you could turn the pencil in any direction, there is no correlation.

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Practice

State which variable is independent: (a) the age of a car and its resale value; (b) the mass of fertiliser applied to a plot and the yield of beans.
Which type of correlation do you expect between (a) the altitude of a place in Rwanda and its average temperature; (b) the number of people in a household and its weekly spending on food; (c) a student's shoe size and her mark in Kiswahili?
Plot the pairs $(1, 9)$ , $(2, 8)$ , $(3, 8)$ , $(4, 6)$ , $(5, 5)$ , $(6, 3)$ and describe the correlation fully.
Explain why a scatter diagram of the revision data above must not have its points joined.
A study finds a strong positive correlation between the number of fire engines sent to a fire and the cost of the damage. A student concludes that fire engines cause damage. Criticise this conclusion.
Describe a scatter diagram whose points lie close to a curve that rises and then falls. Why would the words "positive" or "negative" alone mislead?

Show answersHide answers

(a) The age of the car. (b) The mass of fertiliser.
(a) Negative: higher places are colder. (b) Positive. (c) No correlation.
The points fall from left to right close to a straight line: strong negative linear correlation.
Each point is a separate student; the order of students has no meaning, so a line through them would show a path that does not exist.
Large fires both need more engines and cause more damage. The size of the fire is a third variable driving both; the correlation does not show cause.
The relationship is non-linear. Over part of the range $y$ rises and over the rest it falls, so a single direction word, or a straight-line measure of correlation, hides the real pattern.

Important

Describe every scatter diagram by direction, form and strength. Correlation measures association, not cause.

Sources and credits

Text, data, diagrams and exercises: Written by Ganzaa. All data are illustrative.

Video: Bivariate relationship linearity, strength and direction, Khan Academy, embedded from YouTube.

1Two measurements on the same person

2Drawing a scatter diagram

3Describing a correlation

Two measurements on the same person

Drawing a scatter diagram

Describing a correlation