الأسبوع 1 · الارتباط: العلاقة الطردية والعكسية بين متغيرين وأشكال الانتشار ومقياس معامل الارتباط r من −1 إلى 1 · 3 دروس
1
Two variables that change together: direct and inverse correlation, and why a correlation is not a cause
Statistics Sec 3 Term 1, Unit 1, Lesson 1
GoalDecide whether two variables move in the same direction or in opposite directions, and explain why a correlation does not prove a cause.
New words
variable · a quantity that changes from one case to another, such as hours of revisiondirect (positive) correlation · the two variables tend to rise together and fall togetherinverse (negative) correlation · when one variable rises the other tends to fallcorrelation · a statistical way to describe the type and the degree of a relation between two variables
Explanation
In statistics we often record two variables for the same cases: the hours each pupil revised and the mark that pupil got, or the altitude of a town and its temperature. Correlation describes how the two variables move together. If larger values of x usually come with larger values of y (and smaller with smaller), the correlation is direct, also called positive. If larger values of x usually come with smaller values of y, it is inverse, or negative. If knowing x tells us nothing about y, there is no correlation. Notice the word “usually”: one pair that breaks the pattern does not cancel it. A strong correlation means that knowing one value helps us predict the other; a weak one means it helps very little. One warning matters as much as the definitions: two variables can move together without one causing the other. On hot days a beach kiosk sells more ice cream and more people swim, yet ice cream does not make anyone swim; the hot weather raises both. A correlation describes a pattern in the data; it does not tell us why the pattern is there.
Pair (x, y)
When x rises, y …
Type
hours of revision, quiz mark
usually rises
direct
altitude of a town, its air temperature
usually falls
inverse
price of a juice, bottles sold
usually falls
inverse
seat number, quiz mark
no pattern
none
Worked examples
Hours of training x
1
2
3
4
5
Goals y
3
4
6
7
9
Direct: a school team records the weekly hours of training and the number of goals scored in practice (out of 10 shots): (1 h, 3), (2 h, 4), (3 h, 6), (4 h, 7), (5 h, 9). As the hours rise, the goals rise: a direct correlation.
Altitude x (100 m)
1
3
5
8
10
Temperature y (°C)
28
25
22
18
15
Inverse: five towns, with altitude x (in hundreds of metres) and the average evening temperature y (°C): (1, 28), (3, 25), (5, 22), (8, 18), (10, 15). Higher towns are cooler, so when x rises y falls: an inverse correlation.
Day
Highest °C
Ice creams
Swimmers
1
27
40
25
2
30
55
38
3
33
70
52
4
36
90
70
5
39
110
85
A correlation is not a cause. Over five days a kiosk records the highest temperature, the ice creams sold and the swimmers. Ice creams and swimmers both rise with the temperature, so they are correlated with each other, but the third variable (the heat) explains both.
Class activity
Think of four pairs of things in your class or home (for example the number of people at home and the loaves of bread bought, or the age of a phone and its price). Write each pair as direct, inverse or none. For one direct pair, say whether one variable really causes the other or whether a third factor could be behind both.
Practice
1
A bus makes the same 60 km trip every day. Compare its average speed with the time it takes. Direct or inverse?
Inverse: a higher speed gives a shorter time.
2
A school shop raises the price of a notebook. Over the following months, the number of notebooks sold usually rises or falls? Which type of correlation is price against number sold?
It usually falls, so the correlation is inverse (a higher price lowers sales).
3
The number of pages in a book and the weight of the book: direct, inverse or none?
Direct: more pages usually mean more weight.
4
In winter a shop sees that umbrella sales and the number of customers in coats both rise. Does selling umbrellas make people wear coats? Explain.
No. Cold rainy weather raises both. The correlation is real but neither variable causes the other.
5
Give one pair of variables that you expect to have no correlation, and say why.
For example the number of letters in a pupil’s name and the pupil’s height: nothing links them, so knowing one does not help to predict the other.
