Statistics: Centre, Spread & Displays — Drills
Statistics: Centre, Spread & Displays has 14 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills cover
These questions use 10 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Mean — Two kinds of centre: the balance point, and the middle value.
- Median — Two kinds of centre: the balance point, and the middle value.
- Range — The set of outputs that actually occur.
- Scatter plot — A graph of paired data, one dot per case.
- Outlier — A dot far away from the rest of the pattern.
- Consistent — Has a solution / has none.
- Standard deviation — A measure of typical distance from the mean.
- Skew — A long tail on one side of the distribution.
- Five-number summary — Minimum, Q1, median, Q3, maximum.
- Dot plot — A chart with one dot per value, stacked above a number line.
Mean
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the mean of 22, 7, 19, 24, 4, 27.
Worked answer
103/6
Sum: 103
103 ÷ 6 = 1036
Problem 2
Find the mean of 17, 23, 12, 5.
Worked answer
57/4
Sum: 57
57 ÷ 4 = 574
Median
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the median of 26, 24, 21, 40, 31.
Worked answer
26
Ordered: 21, 24, 26, 31, 40
The middle value is 26.
Problem 2
Find the median of 20, 5, 13, 15, 24, 29.
Worked answer
17.5
Ordered: 5, 13, 15, 20, 24, 29
Middle two are 15 and 20, so 17.5.
Range
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the range of 41, 4, 57, 8, 11.
Worked answer
53
57 - 4 = 53
Problem 2
Find the range of 32, 51, 46, 54.
Worked answer
22
54 - 32 = 22
Quartile
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find Q3 of 4, 8, 12, 16, 20, 24, 28, 32.
Worked answer
26
Upper half: 20, 24, 28, 32
Median of that half: 26
Problem 2
Find Q1 of 5, 8, 11, 14, 17, 20, 23, 26.
Worked answer
9.5
Lower half: 5, 8, 11, 14
Median of that half: 9.5
Iqr
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the IQR of 2, 6, 10, 14, 18, 22, 26, 30.
Worked answer
16
Q1 = 8
Q3 = 24
24 - 8 = 16
Problem 2
Find the IQR of 3, 7, 11, 15, 19, 23, 27, 31.
Worked answer
16
Q1 = 9
Q3 = 25
25 - 9 = 16
Choosing the method
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Data: 20, 22, 24, 25, 700. Should you report the mean or the median?
Worked answer
median
700 is far from the rest.
It drags the mean up to 158.2.
The median, 24, is representative.
Problem 2
Data: 23, 25, 27, 28, 600. Should you report the mean or the median?
Worked answer
median
600 is far from the rest.
It drags the mean up to 140.6.
The median, 27, is representative.
Skew
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
The mean is 74 and the median is 86. Describe the skew.
Worked answer
skewed left
Mean below median.
So the tail points left.
Skewed left.
Problem 2
The mean is 64 and the median is 54. Describe the skew.
Worked answer
skewed right
Mean above median.
So the tail points right.
Skewed right.
Resistant
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Which pair of statistics is resistant to outliers?
- Median and range
- Mean and range
- Mean and standard deviation
- Median and IQR
Worked answer
Median and IQR
The median ignores the extremes entirely.
The IQR uses only the middle half.
Mean, range and standard deviation all move a lot.
Fivenum
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the five-number summary of 8, 11, 14, 17, 20, 23, 27.
Worked answer
8, 11, 17, 23, 27
Min 8, max 27
Median 17
Q1 = 11, Q3 = 23
Problem 2
Find the five-number summary of 2, 5, 8, 11, 14, 17, 23.
Worked answer
2, 5, 11, 17, 23
Min 2, max 23
Median 11
Q1 = 5, Q3 = 17
Display
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Which display best compares three groups at once?
- A pie chart
- Side-by-side box plots
- A single dot plot
- A scatter plot
Worked answer
Side-by-side box plots
Box plots show the five-number summary.
Side by side, they compare groups directly.
Finding the missing value
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
6 numbers have a mean of 21. 5 of them are 15, 33, 23, 23, 9. Find the last one.
Worked answer
23
Total = 126
15 + 33 + 23 + 23 + 9 = 103
126 - 103 = 23
Problem 2
4 numbers have a mean of 18. 3 of them are 25, 24, 30. Find the last one.
Worked answer
-7
Total = 72
25 + 24 + 30 = 79
72 - 79 = -7
Outlier
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Using the 1.5 × IQR rule with Q1 = 28 and Q3 = 42, is 73 an outlier? Answer yes or no.
Worked answer
yes
IQR = 14
1.5 × 14 = 21
Upper fence = 42 + 21 = 63
73 > 63, so yes.
Problem 2
Using the 1.5 × IQR rule with Q1 = 12 and Q3 = 24, is 38 an outlier? Answer yes or no.
Worked answer
no
IQR = 12
1.5 × 12 = 18
Upper fence = 24 + 18 = 42
38 ≤ 42, so no.
Comparing
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Group A: median 85, IQR 7. Group B: median 85, IQR 16. Which is more consistent?
Worked answer
A
A has IQR 7; B has IQR 16.
7 < 16, so A is more consistent.
The centres are identical.
Problem 2
Group A: median 53, IQR 6. Group B: median 53, IQR 23. Which is more consistent?
Worked answer
A
A has IQR 6; B has IQR 23.
6 < 23, so A is more consistent.
The centres are identical.
Effect
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A data set has mean 32 and standard deviation 8. Every value is increased by 10. Find the new mean and standard deviation.
Worked answer
mean 42, sd 8
Every value up 10, so the mean becomes 42.
The distances between values are unchanged.
So the standard deviation is still 8.
Problem 2
A data set has mean 46 and standard deviation 7. Every value is increased by 3. Find the new mean and standard deviation.
Worked answer
mean 49, sd 7
Every value up 3, so the mean becomes 49.
The distances between values are unchanged.
So the standard deviation is still 7.