Mathematics 9709/63 — May/June 2012
Cambridge AS Level · Probability & Statistics 1 · worked solutions for every part, with the mark scheme
Topics Representation of Data · Discrete Random Variables · Probability · Permutations and Combinations · The Normal Distribution
Ashfaq and Kuljit have done a school statistics project on the prices of a particular model of headphones for MP3 players. Ashfaq collected prices from 21 shops. Kuljit used the internet to collect prices from 163 websites.
Name a suitable statistical diagram for Ashfaq to represent his data, together with a reason for choosing this particular diagram.
Approach
For Ashfaq's 21 shop prices, choose a diagram that works well with a relatively small data set and displays the shape of the distribution while preserving the data.
Working
Ashfaq has only 21 prices. A stem-and-leaf diagram is suitable because
- it shows all 21 raw prices;
- it displays the shape, spread and range of the data;
- it makes it easy to read off the median.
A box-and-whisker plot is also acceptable, since it shows the shape/spread/range and allows the median to be read directly.
Answer
A stem-and-leaf diagram (or box-and-whisker plot), because the data set is small and the diagram shows the overall shape/spread/range and makes the median easy to identify.
Stem-and-leaf diagram (or box-and-whisker plot); reason: it shows shape/spread/range and makes the median easy to identify with a small data set.
Walkthrough
Ashfaq has collected 21 prices from shops. Because 21 is a relatively small number of data values, both a stem-and-leaf diagram and a box-and-whisker plot are sensible choices. A stem-and-leaf diagram is often preferred because it keeps every original price visible while also giving a quick visual impression of the shape, spread and range; it also makes finding the median straightforward. A box-and-whisker plot is another acceptable answer because it summarises the shape/spread/range and allows the median to be read from the box.
Key Takeaways
The key skill is matching the diagram to the size of the data set and to the information you want to show. For small data sets, a stem-and-leaf diagram is useful because it preserves the data values. The reason must justify the diagram, not just name it.
Common Mistakes
- Naming a diagram without giving a reason.
- Giving a reason that does not connect to the diagram, e.g. saying a histogram is good because it shows every individual value.
- Choosing a diagram that is unsuitable for a small data set.
Things to Be Careful About
The reason must depend on the diagram chosen. The mark scheme accepts stem-and-leaf, box-and-whisker, or histogram in part (i) but a histogram alone would not normally be given full credit unless the reasoning is adapted. For a stem-and-leaf diagram, the key points are that the data set is small enough to display all values and the median is easy to calculate.
Name a suitable statistical diagram for Kuljit to represent her data, together with a reason for choosing this particular diagram.
Approach
For Kuljit's 163 website prices, choose a diagram suited to a large data set, where listing every individual value is not practical.
Working
Kuljit has 163 prices, which is far too many for a stem-and-leaf diagram to be useful. A histogram is suitable because
- it copes well with a large number of data values;
- it shows the shape and spread of the distribution;
- it allows the modal class to be identified.
Alternative acceptable diagrams are a cumulative frequency graph or a box-and-whisker plot, since both also show the shape/spread and make the median easy to find.
Answer
A histogram (or cumulative frequency graph / box-and-whisker plot), because with 163 prices the large amount of data is best summarised by showing shape/spread/modal class.
Histogram (or cumulative frequency graph / box-and-whisker plot); reason: with a large number of prices it shows shape/spread/modal class and is easier to interpret than listing all values.
Walkthrough
Kuljit has 163 prices from websites, a much larger data set than Ashfaq's 21. A stem-and-leaf diagram would become too long and unwieldy. A histogram groups the data into intervals and is therefore appropriate for a large set; it shows the shape and spread and clearly indicates the modal class. Cumulative frequency graphs and box-and-whisker plots are also accepted because they summarise a large set effectively and make the median easy to read.
Key Takeaways
The suitability of a diagram depends on the size of the data set and what feature of the data is important. Histograms are good for large data sets because values are grouped; they show shape, spread and modal class without showing every individual value.
Common Mistakes
- Choosing a stem-and-leaf diagram for 163 values, which would be too long and impractical.
- Naming the diagram but failing to give a reason that refers to the large number of prices.
- Providing a reason such as 'it shows all the data' for a histogram, which is not true.
Things to Be Careful About
The reason must be specific to the chosen diagram. If you choose a histogram, mention grouping, shape/spread, or modal class. If you choose a cumulative frequency graph or box-and-whisker plot, mention that it handles a large number of prices and shows the median/shape. A diagram that preserves every individual value is not a sensible reason here because there are 163 values.
The rest of this paper
5 more questions- Q2Representation of Data5M
- Q3Permutations and Combinations9M
- Q4Discrete Random Variables · Probability10M
- Q5Probability · Discrete Random Variables10M
- Q6The Normal Distribution12M