Statistics: Centre, Spread & Displays — Print
Report a centre, a spread, and the shape. One number on its own has almost never told the whole story.
What the practice questions 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.
Five houses on a street sold for $220k, $240k, $250k, $270k — and $3 000 000.
What is the typical price on this street? The mean says $796k.
But not one of the five houses cost anything like $796k. Four of them sold for under $280k.
The mean was dragged away by a single house. Take the middle value instead and you get $250k, which describes the street honestly.
Move that mansion to $10 million and the mean leaps again — the median does not move at all.
A statistic that barely notices one extreme value is called resistant. Median and IQR are resistant; mean and standard deviation are not — which is exactly why house prices are always reported as medians.
Choosing mean or median
For roughly symmetric data with no outliers, the mean and median agree and either will do.
For skewed data or data with outliers, use the median. It is resistant — one extreme value barely moves it.
House prices are always reported as medians, because a few enormous houses would drag the mean somewhere unrepresentative.
Resistant means unmoved by outliers
Median and IQR are resistant. Mean and standard deviation are not. Pick the resistant pair whenever the data is skewed or has outliers.
Spread
Range is maximum minus minimum — simple, and ruined by a single outlier.
IQR is Q3 - Q1, the width of the middle half. It ignores the extremes entirely.
Standard deviation measures typical distance from the mean. Bigger means more spread out.
Match the pair
Report mean with standard deviation, or median with IQR. Mixing a mean with an IQR is not wrong, but it looks confused.
Shape and skew
Symmetric: the two halves roughly mirror each other, and mean ≈ median.
Skewed right: a long tail towards the high values, and the mean is pulled above the median.
Skewed left: a long tail towards the low values, and the mean sits below the median.
The tail names the skew
Skewed right means the tail points right, even though the bulk of the data is on the left. Follow the tail, not the peak.
Choosing a display
Dot plot: small data sets, shows every value.
Histogram: larger data sets, shows the shape of the distribution.
Box plot: shows the five-number summary and is excellent for comparing two or more groups side by side.
What a box plot hides
A box plot cannot show two peaks. Two very different distributions can have identical box plots, so use a histogram when shape matters.
Comparing groups
Compare centres, compare spreads, then compare shapes. Three sentences, in that order.
Say which group is higher on average and which is more consistent. "More consistent" means a smaller spread.
Always mention overlap. Two groups can have different medians while most of their values overlap completely.
Worked examples
Five-number summary
Find the five-number summary of 4, 7, 9, 11, 14, 18, 20, 25.
- Already in order. Minimum 4, maximum 25. The two extremes.
- Eight values, so the median is the average of the 4th and 5th: 11 + 142 = 12.5 An even count has no single middle.
- Lower half 4, 7, 9, 11, so Q1 = 7 + 92 = 8 The median of the bottom half.
- Upper half 14, 18, 20, 25, so Q3 = 18 + 202 = 19 IQR = 19 - 8 = 11.
4, 8, 12.5, 19, 25; IQR = 11
Which centre?
Salaries at a small firm are $40k, $42k, $45k, $47k and $650k. Which centre should be reported?
- Mean: 40 + 42 + 45 + 47 + 6505 = 8245 = $164.8k Add and divide.
- Median: the middle of five values is $45k. Third value in order.
- Four of the five people earn less than the mean. The $650k dragged it upwards.
- So report the median, $45k. It is resistant to that outlier.
The median, $45k
Comparing two groups
Class A has median 78 and IQR 8. Class B has median 74 and IQR 20. Compare them.
- Centre: A is higher, 78 against 74. Compare medians.
- Spread: A is much more consistent, IQR 8 against 20. A smaller IQR means less variation.
- So A scored higher on average and far more consistently. Both comparisons reported.
- But with B's spread, plenty of B students beat A's median. Overlap matters, and is easy to forget.
A is higher and more consistent, though the groups overlap
Words to know
- Mean
- The average: total divided by how many.
- Median
- The middle value once the data is in order.
- Interquartile range
- Q3 - Q1, the spread of the middle half.
- 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.
Mistakes to watch for
Finding the median without ordering the data first. The median is the middle of the sorted list; taking the middle of the list as written gives whichever value happened to be typed third.
Reporting the mean for obviously skewed data. One extreme value pulls the mean towards it and leaves it describing nobody. If mean and median disagree sharply, that gap is itself the finding.
Calling a distribution skewed left when the tail points right. The tail names the skew, not the peak. Skewed right means a long tail of high values, with most of the data bunched on the left.
Comparing centres and forgetting spreads. Two classes with the same median can be completely different: IQR 8 means everyone near the middle, IQR 22 means the group is all over the place.
The box plot was invented by John Tukey in 1970, along with the stem-and-leaf plot and the word software.
Tukey also coined bit, for binary digit, in a 1947 memo at Bell Labs.
His 1.5 × IQR rule for flagging outliers was chosen for convenience, not derived — he wanted something a person could do by hand, quickly.
Problem 1
Find the mean of 4, 8, 10, 14.
Worked answer
9
36 ÷ 4 = 9
Problem 2
Find the median of 3, 9, 4, 12, 7.
Worked answer
7
3, 4, 7, 9, 12
The middle value is 7
Problem 3
Find the median of 2, 5, 8, 11.
Worked answer
6.5
Middle two are 5 and 8
5 + 82 = 6.5
Problem 4
Find the range of 12, 5, 20, 9.
Worked answer
15
20 - 5 = 15
Problem 5
Find Q1 of 2, 4, 6, 8, 10, 12, 14, 16.
Worked answer
5
Lower half: 2, 4, 6, 8
4 + 62 = 5
Problem 6
Find the IQR of 1, 3, 5, 7, 9, 11, 13, 15.
Worked answer
8
Q1 = 3+52 = 4
Q3 = 11+132 = 12
12 - 4 = 8
Problem 7
Data: 20, 22, 24, 25, 400. Should you report the mean or the median?
Worked answer
median
400 is far from the rest
It drags the mean up to 98.2
The median, 24, is representative
Problem 8
The mean is 62 and the median is 70. Describe the skew.
Worked answer
skewed left
Mean below median
So the tail points left
Skewed left
Problem 9
The mean is 85 and the median is 74. Describe the skew.
Worked answer
skewed right
Mean above median
The tail points right
Skewed right
Problem 10
Which pair of statistics is resistant to outliers?
- Median and IQR
- Mean and standard deviation
- Mean and range
- Median and range
Worked answer
Median and IQR
Median ignores the extremes
IQR uses only the middle half
Mean, range and standard deviation all move a lot
Problem 11
Find the five-number summary of 5, 7, 9, 12, 15, 18, 21.
Worked answer
5, 7, 12, 18, 21
Min 5, max 21
Median 12
Q1 = 7, Q3 = 18
Problem 12
Which display best compares three groups at once?
- Side-by-side box plots
- A single dot plot
- A pie chart
- A scatter plot
Worked answer
Side-by-side box plots
Box plots show the five-number summary
Placing them side by side compares groups directly
Problem 13
Four numbers have a mean of 15. Three of them are 10, 14 and 20. Find the fourth.
Worked answer
16
Total = 60
10 + 14 + 20 = 44
60 - 44 = 16
Problem 14
Using the 1.5 × IQR rule with Q1 = 20 and Q3 = 32, is 55 an outlier? Answer yes or no.
Worked answer
yes
IQR = 12
1.5 × 12 = 18
Upper fence = 32 + 18 = 50
55 > 50, so yes
Problem 15
Group A: median 60, IQR 5. Group B: median 60, IQR 22. Which is more consistent?
Worked answer
A
A has the smaller IQR
So A is more consistent
The centres are identical
Problem 16
A data set has mean 50. Every value is increased by 8. Find the new mean and the new standard deviation, given the old one was 6.
Worked answer
mean 58, sd 6
Every value up 8, so the mean is 58
Distances between values are unchanged
So the standard deviation is still 6