6.11 Spearman's rank coefficient
For each of the following graphs, determine whether Pearson's correlation coefficient or Spearman's rank coefficient is more suitable for the data set. Explain your choice.
For each of the following sets of data:
Add ranks for each value to the table.
Calculate Spearman's rank coefficient. Round your answer to two decimal places.
Describe the correlation between the two variables.
| x | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|
| y | 7 | 7.4 | 7.88 | 7.64 | 7.72 | 8.2 | 7.32 |
| x | 2 | 6 | 7 | 14 | 17 | 22 |
|---|---|---|---|---|---|---|
| y | 2.0 | 2.4 | 2.0 | 0.6 | 0.9 | 0.2 |
| x | 1 | 4 | 7 | 10 | 13 | 16 | 19 |
|---|---|---|---|---|---|---|---|
| y | -1 | -1.7 | -1.7 | -1.84 | -1.91 | -1.98 | -2.05 |
| x | 4 | 6 | 8 | 13 | 17 | 21 |
|---|---|---|---|---|---|---|
| y | 0.4 | 0.9 | 0.6 | 1.7 | 2.4 | 1.7 |
The following table displays results from an experiment:
| x | 1 | 4 | 5 | 8 | 9 | 11 | 13 | 15 | 18 | 19 |
|---|---|---|---|---|---|---|---|---|---|---|
| y | 2 | 4 | 6 | 8 | 12 | 24 | 30 | 46 | 52 | 64 |
Calculate Pearson's correlation coefficient. Round your answer to three siginificant figures.
Calculate Spearman's rank coefficient. Round your answer to three siginificant figures.
Describe the correlation between the variables.
A study was conducted to find the relationship between the age at which a child first speaks and their level of intelligence as teenagers. The following table shows the ages at which some teenagers first spoke, and their results in an aptitude test:
| \text{Age when first spoke, }x | 14 | 27 | 9 | 9 | 16 | 21 | 17 | 10 | 7 | 19 |
|---|---|---|---|---|---|---|---|---|---|---|
| \text{Aptitude test results, }y | 96 | 69 | 84 | 90 | 101 | 87 | 92 | 99 | 104 | 93 |
Construct a scatter plot for the data.
Add the ranks for each variable to the table of values.
Calculate Spearman's rank coefficient, to three significant figures.
A student was performing an experiment to study the relationship between the current and voltage through a resistor. He noted his results in the following table:
| \text{Current, }c | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| \text{Voltage, }v | 1 | 6 | 3 | 10 | 13 | 10 | 16 | 15 | 21 | 17 |
Construct a scatter plot for the data.
Add the ranks for each variable to the table of values.
Calculate Spearman's rank coefficient, to three significant figures.
The following table shows the time taken to finish a lap on a race track for various average speeds:
| \text{Speed, }s | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 | 65 |
|---|---|---|---|---|---|---|---|---|---|---|
| \text{Time. }t | 85 | 87 | 75 | 82 | 69 | 73 | 60 | 57 | 45 | 49 |
Construct a scatter plot for the data.
Calculate Spearman's rank coefficient, to three significant figures.
Describe the correlation between speed and time for this data.
Sean is a hotdog vendor. He records the maximum temperature of the day and the number of hotdogs sold. The results are displayed in the following table:
| \text{Maximum temperature (}\degree \text{C)}, x | 30 | 34 | 33 | 35 | 33 | 28 | 27 | 31 | 37 | 29 |
|---|---|---|---|---|---|---|---|---|---|---|
| \text{Number of hotdogs}, y | 18 | 38 | 26 | 40 | 24 | 8 | 20 | 35 | 43 | 38 |
Construct a scatter plot to represent the data in the table.
Calculate Spearman's rank correlation coefficient to two decimal places.
As the temperature increases, describe what happens to the sales.
The table lists the time taken to sprint 400 metres by runners who all ran in different temperatures as part of a study:
Construct a scatter plot to represent the data in the table.
How many runners were tested in the study?
Calculate Spearman's rank correlation coefficient, to three significant figures.
Describe the correlation between temperature and sprint time for the data.
Which data point represents an outlier?
Recalculate Spearman's rank correlation coefficient for this data, excluding the outlier. Round your answer to three significant figures.
| \text{Temperature }\\( \degree\text{C}) | \text{Time (sec)} |
|---|---|
| 5 | 60 |
| 2 | 67 |
| 10 | 48 |
| 8 | 69 |
| 1 | 65 |
| 7 | 49 |
| 6 | 57 |
| 4 | 53 |
| 3 | 59 |
| 9 | 52 |
The marks of 12 students in Maths and Sport were recorded in the following table:
Construct a scatter plot for the students' mark in Maths vs their mark in Sport.
Describe the correlation between students' marks in Maths and marks in Sport.
Which student's scores appear to represent an outlier?
Calculate Spearman's rank correlation coefficient for this data, excluding the outlier. Round your answer to three significant figures.
| Student | Mark in Maths | Mark in Sport |
|---|---|---|
| 1 | 63 | 44 |
| 2 | 92 | 74 |
| 3 | 60 | 52 |
| 4 | 79 | 70 |
| 5 | 88 | 67 |
| 6 | 81 | 60 |
| 7 | 61 | 73 |
| 8 | 91 | 86 |
| 9 | 72 | 84 |
| 10 | 42 | 93 |
| 11 | 66 | 57 |
| 12 | 92 | 92 |
The following table shows the average IQ of a random group of people against their height:
| \text{Height (cm)} | 140 | 145 | 150 | 155 | 160 | 165 | 170 | 175 | 180 | 185 |
|---|---|---|---|---|---|---|---|---|---|---|
| \text{IQ} | 103 | 95 | 98 | 111 | 85 | 89 | 108 | 145 | 110 | 93 |
Construct a scatter plot for this data.
Describe the relationship between IQ and height.
How tall is the person who appears to be an outlier?
Calculate Spearman's rank correlation coefficient for this data, excluding the outlier. Round your answer to three significant figures.