About this chapter
Representations of data is where Statistics 1 asks you to read a diagram accurately and then say something sensible about it. Examiners regularly report two weaknesses: stem and leaf diagrams read the wrong way round, and comparisons between data sets that give no statistics or no context.
This chapter covers histograms and frequency density, estimating from part of a bar, stem and leaf diagrams including back-to-back ones, quartiles from diagrams, outliers using the rule the question gives, box plots, skewness, comparing two distributions, and recalculating after an anomaly is removed.
The ten questions
- Q1 (5 marks): race times in a histogram: find another bar's width and height from a given bar.
- Q2 (5 marks): the median and quartiles from a stem and leaf diagram, then show there is exactly one outlier.
- Q3 (5 marks): box plots for two schools: read the median and interquartile range, compare them, and describe skewness.
- Q4 (5 marks): two measures of skewness from summary statistics, then which average and spread to use.
- Q5 (5 marks): draw a stem and leaf diagram of museum visit times, then its median and interquartile range.
- Q6 (11 marks): a stem and leaf diagram with a missing leaf w and unknown quartiles, then outliers and a box plot.
- Q7 (11 marks): a histogram of distances to work: the total, part of a bar, the median by interpolation and the mean.
- Q8 (11 marks): a back-to-back stem and leaf diagram of pulse rates, compared for athletes and non-athletes.
- Q9 (10 marks): seedling heights: an outlier more than 2 standard deviations from the mean, removed, then the statistics recalculated.
- Q10 (12 marks): show exactly one outlier, draw a box plot, then compare two types of plant.
Key skills tested
Histograms. Area is proportional to frequency, so frequency density is frequency ÷ class width. Scale both the widths and the heights.
Estimating from histograms. Count part of a bar by area: two fifths of a bar's width holds two fifths of its frequency.
Stem and leaf diagrams. Read them with the key. On a back-to-back diagram, the left-hand leaves read outwards from the stem.
Quartiles from diagrams. Use the discrete rules: find n/4, n/2 and 3n/4, round up if not whole, and average two values if whole.
Outliers. Use exactly the rule the question gives, such as more than 1.5 times the interquartile range beyond a quartile, or more than 2 standard deviations from the mean. Then name the outliers.
Box plots. Draw the box from Q1 to Q3 with the median, whiskers to the most extreme values that are not outliers, and each outlier as a cross.
Skewness. Positive skew means Q3 − Q2 is greater than Q2 − Q1, or the mean is above the median. Two measures are 3(mean − median) ÷ standard deviation and (Q3 − 2Q2 + Q1) ÷ (Q3 − Q1).
Comparing distributions. Compare a measure of location and a measure of spread, with values and in context: median with interquartile range, or mean with standard deviation.
Cleaning data. Remove an anomaly only with a reason, then recalculate n, Σx and Σx².
Worked example
Question 2 from this chapter. The ages of 19 club members, in order, are 15, 18, 19, 21, 23, 24, 26, 26, 28, 30, 32, 32, 35, 37, 41, 44, 46, 63 and 72. (a) Find the median and quartiles. (b) An outlier is more than 1.5 times the interquartile range above Q3 or below Q1. Show that there is exactly one outlier.
(a) With n = 19, the median is the 10th value, 30. For Q1, 19 ÷ 4 = 4.75, so take the 5th value, 23. For Q3, 3 × 19 ÷ 4 = 14.25, so take the 15th value, 41.
(b) The interquartile range is 41 − 23 = 18. The upper limit is 41 + 1.5 × 18 = 68 and the lower limit is 23 − 27 = −4. Only 72 lies beyond these limits (63 does not), so 72 is the only outlier.
The final sentence naming the outlier is what earns the last mark.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- June 2024, Q1: quartiles from a stem and leaf diagram, showing there is only one outlier, a box plot, then a comparison with another data set, as in Q2 and Q10.
- June 2022, Q1: a stem and leaf diagram with a missing leaf, outliers by the rule given, then a box plot, as in Q6.
- June 2022, Q3(a), and June 2024, Q3(a): finding the width and height of a histogram bar from another bar, as in Q1.
Where marks are lost
- Reading the diagram backwards. In June 2024 some students read the stem and leaf diagram the wrong way round and got both quartiles wrong.
- Not naming the outlier. In both June 2022 and June 2024 students found the outlier limit but did not say which value was the outlier.
- The wrong outlier rule. In June 2022 some used the familiar 1.5 times the interquartile range rule instead of the one printed in the question.
- A comparison with no numbers. Too many June 2024 answers compared two data sets without naming a statistic, giving its values, or mentioning the context.
- Scaling only the height. In June 2022 a common wrong answer scaled a histogram bar's height by the frequencies and ignored the change in class width.
Common questions
What must a comparison include?
A named statistic for location and one for spread, their values for both groups, and a sentence in the context of the question.
Where do the whiskers of a box plot end?
At the most extreme values that are not outliers. Outliers are plotted separately as crosses.
Why use frequency density?
Because classes can have different widths. The area of each bar, not its height, shows the frequency.
Where this chapter leads
- Statistics 1 Chapter 7: The Normal Distribution: quartiles, the interquartile range and outliers for a normal model.
- Statistics 1 Chapter 5: Correlation and Regression: scatter diagrams and cleaning data.