Describing Statistical Relationships
We've already looked at scatterplots and how to describe relationships between different sets of variables.
Here is a summary of different types of linear relationship:
- Positive relationship: as one variable increases, the other variables also increases.
- Negative relationship: as one variable increases, the other variables decreases.
- No relationship: there is no correlation between the variables.
The closer the dots in a scatterplot are to a straight line, the stronger the correlation.
Once we can identify the type of relationship between two variables, we can make inferences about these variables and use them in everyday life. For example, say there is a positive relationship between study and results. From this, we can say that the more someone studies, the better they will do in their exams.
Let's look through some more examples now.
Examples
Question 1
Identify the type of relationship represented by the following scatter plot.
A) Negative linear B) No relationship C) Positive linear
Think: The values on the $x$x axis as representing one variable, and the values on the $y$y axis as representing another variable. How does the $y$y-variable change as the $x$x-variable increases?
Do: When we look at this graph, we can see that as $x$x-variable increases, the $y$y-variable also increases. So this graph is an example of an A) negative linear relationship.
QUESTION 2
The marks of $12$12 students in Maths and Sport were recorded in the following table.
| Student | Mark in Maths | Mark in Sport |
|---|---|---|
| $1$1 | $63$63 | $44$44 |
| $2$2 | $92$92 | $74$74 |
| $3$3 | $60$60 | $52$52 |
| $4$4 | $79$79 | $70$70 |
| $5$5 | $88$88 | $67$67 |
| $6$6 | $81$81 | $60$60 |
| $7$7 | $61$61 | $73$73 |
| $8$8 | $91$91 | $86$86 |
| $9$9 | $72$72 | $84$84 |
| $10$10 | $42$42 | $93$93 |
| $11$11 | $66$66 | $57$57 |
| $12$12 | $92$92 | $92$92 |
Which of the following scatterplots correctly represents the data from the table?
A
B
CIs the relationship between students' marks in Maths and Sport positive or negative?
Positive
ANegative
BIs the relationship between students' marks in Maths and Sport strong or weak?
Strong
AWeak
BWhich student has a score that represents an outlier?
Student 10
AStudent 4
BStudent 12
CNone of the students
D
Question 3
Scientists were looking for a relationship between the number of hours of sleep we receive and the effect it has on our motor and process skills. Some subjects were asked to sleep for different amounts of time, and were all asked to undergo the same driving challenge in which their reaction time was measured. The table shows the results, which are to be presented as a scatter plot.
| Amount of sleep (hours) | Reaction time (seconds) |
|---|---|
| $9$9 | $3$3 |
| $6$6 | $3.3$3.3 |
| $4$4 | $3.5$3.5 |
| $10$10 | $3$3 |
| $3$3 | $3.7$3.7 |
| $7$7 | $3.2$3.2 |
| $2$2 | $3.85$3.85 |
| $5$5 | $3.55$3.55 |
By moving the points, create a scatter plot for the observations in the table.
Loading Graph...According to the results, which of the following is true of the relationship between amount of sleep and reaction time?
As the amount of sleep decreases, the reaction time decreases.
AAs sleeping time decreases, reaction time improves.
BSleeping for longer improves reaction time.
CThe amount of sleep has no effect on the reaction time.
D