2.02a Interpret single variable data: tables and diagrams

The heating quality of the coal in a sample of 50 sacks is measured in suitable units. The data are summarised below.

Heating quality (\(x\))	9.1 \(\leqslant x <\) 9.3	9.3 \(< x \leqslant\) 9.5	9.5 \(< x \leqslant\) 9.7	9.7 \(< x \leqslant\) 9.9	9.9 \(< x \leqslant\) 10.1
Frequency	5	7	15	16	7

1. Draw a cumulative frequency diagram to illustrate these data. [5]
2. Use the diagram to estimate the median and interquartile range of the data. [3]
3. Show that there are no outliers in the sample. [3]
4. Three of these 50 sacks are selected at random. Find the probability that
   1. in all three, the heating quality \(x\) is more than 9.5, [3]
   2. in at least two, the heating quality \(x\) is more than 9.5. [4]

The ages, \(x\) years, of the senior members of a running club are summarised in the table below.

Age (\(x\))	\(20 \leqslant x < 30\)	\(30 \leqslant x < 40\)	\(40 \leqslant x < 50\)	\(50 \leqslant x < 60\)	\(60 \leqslant x < 70\)	\(70 \leqslant x < 80\)	\(80 \leqslant x < 90\)
Frequency	10	30	42	23	9	5	1

1. Draw a cumulative frequency diagram to illustrate the data. [5]
2. Use your diagram to estimate the median and interquartile range of the data. [3]

Jane and Tahira play together in a basketball team. The list below shows the number of points that Jane scored in each of 30 games.

1. Construct a stem and leaf diagram for these data. [3 marks]
2. Find the median and quartiles for these data. [4 marks]
3. Represent these data with a boxplot. [3 marks]

Tahira played in the same 30 games and her lowest and highest points total in a game were 19 and 41 respectively. The quartiles for Tahira were 27, 31 and 35 respectively.

1. Using the same scale draw a boxplot for Tahira's points totals. [2 marks]
2. Compare and contrast the number of points scored per game by Jane and Tahira. [3 marks]

A College offers evening classes in GCSE Mathematics and English. In order to assess which age groups were reluctant to use the classes, the College collected data on the age in completed years of those currently attending each course. The results are shown in this back-to-back stem and leaf diagram. \includegraphics{figure_4} Key: \(1 | 3 | 2\) means age 31 doing Mathematics and age 32 doing English

1. Find the median and quartiles of the age in completed years of those attending the Mathematics classes. [4 marks]
2. On graph paper, draw a box plot representing the data for the Mathematics class. [3 marks]

The median and quartiles of the age in completed years of those attending the English classes are 25, 41 and 57 years respectively.

1. Draw a box plot representing the data for the English class using the same scale as for the data from the Mathematics class. [3 marks]
2. Using your box plots, compare and contrast the ages of those taking each class. [4 marks]

The radar diagrams illustrate some population figures from the 2011 census results. \includegraphics{figure_13} Each radius represents an age group, as follows: The distance of each dot from the centre represents the number of people in the relevant age group.

1. The scales on the two diagrams are different. State an advantage and a disadvantage of using different scales in order to make comparisons between the ages of people in these two Local Authorities. [2]
2. Approximately how many people aged 45 to 59 were there in Liverpool? [1]
3. State the main two differences between the age profiles of the two Local Authorities. [2]
4. James makes the following claim. "Assuming that there are no significant movements of population either into or out of the two regions, the 2021 census results are likely to show an increase in the number of children in Liverpool and a decrease in the number of children in Rutland." Use the radar diagrams to give a justification for this claim. [2]

Gareth has a keen interest in pop music. He recently read the following claim in a music magazine. In the pop industry most songs on the radio are not longer than three minutes.

1. He decided to investigate this claim by recording the lengths of the top 50 singles in the UK Official Singles Chart for the week beginning 17 June 2016. (A 'single' in this context is one digital audio track.) Comment on the suitability of this sample to investigate the magazine's claim. [1]
2. Gareth recorded the data in the table below.

   Length of singles for top 50 UK Official Chart singles, 17 June 2016
   2.5-(3.0)	3.0-(3.5)	3.5-(4.0)	4.0-(4.5)	4.5-(5.0)	5.0-(5.5)	5.5-(6.0)	6.0-(6.5)	6.5-(7.0)	7.0-(7.5)
   3	17	22	7	0	0	0	0	0	1

   He used these data to produce a graph of the distributions of the lengths of singles \includegraphics{figure_2} State two corrections that Gareth needs to make to the histogram so that it accurately represents the data in the table. [2]
3. Gareth also produced a box plot of the lengths of singles. \includegraphics{figure_3} He sees that there is one obvious outlier.
   1. What will happen to the mean if the outlier is removed?
   2. What will happen to the standard deviation if the outlier is removed? [2]
4. Gareth decided to remove the outlier. He then produced a table of summary statistics.
   1. Use the appropriate statistics from the table to show, by calculation, that the maximum value for the length of a single is not an outlier.

      Summary statistics
      Length of single for top 50 UK Official Singles Chart (minutes)
      Length of single	N	Mean	Standard deviation	Minimum	Lower quartile	Median	Upper quartile	Maximum
      	49	3.57	0.393	2.77	3.26	3.60	3.89	4.38

   2. State, with a reason, whether these statistics support the magazine's claim. [4]
5. Gareth also calculated summary statistics for the lengths of 30 singles selected at random from his personal collection.

   Summary statistics
   Length of single for Gareth's random sample of 30 singles (minutes)
   Length of single	N	Mean	Standard deviation	Minimum	Lower quartile	Median	Upper quartile	Maximum
   	30	3.13	0.364	2.58	2.73	2.92	3.22	3.95

   Compare and contrast the distribution of lengths of singles in Gareth's personal collection with the distribution in the top 50 UK Official Singles Chart. [3]

The table shows the increases, between 2001 and 2011, in the percentages of employees travelling to work by various methods, in the Local Authorities (LAs) in the North East region of the UK. \includegraphics{figure_4} The first two digits of the Geography code give the type of each of the LAs: 06: Unitary authority 07: Non-metropolitan district 08: Metropolitan borough

1. In what type of LA are the largest increases in percentages of people travelling by underground, metro, light rail or tram? [1]
2. Identify two main changes in the pattern of travel to work in the North East region between 2001 and 2011. [2]

Now assume the following.

• The data refer to residents in the given LAs who are in the age range 20 to 65 at the time of each census.
• The number of people in the age range 20 to 65 who move into or out of each given LA, or who die, between 2001 and 2011 is negligible.

1. Estimate the percentage of the people in the age range 20 to 65 in 2011 whose data appears in both 2001 and 2011. [2]
2. In the light of your answer to part (c), suggest a reason for the changes in the pattern of travel to work in the North East region between 2001 and 2011. [1]

In a study of reaction times, 25 participants completed a test where their reaction times (in milliseconds) were recorded. The results are shown in the stem-and-leaf diagram below: 20 | 3 5 7 9 21 | 0 2 5 6 8 22 | 1 3 4 5 7 9 23 | 0 2 5 8 24 | 1 4 6 7 25 | 2 5 Key: 21 | 0 represents a reaction time of 210 milliseconds

1. State the median reaction time. [1]
2. Calculate the interquartile range of these reaction times. [2]
3. Find the mean and standard deviation of these reaction times. [3]
4. State one advantage of using a stem-and-leaf diagram to display this data rather than a frequency table. [1]
5. One participant completed the test again and recorded a reaction time of 195 milliseconds. Add this result to the stem-and-leaf diagram and state the effect this would have on:
   1. the median
   2. the mean
   3. the standard deviation
   [4]
6. Explain why the interquartile range might be preferred to the standard deviation as a measure of spread in this context [2]

Paul drew a cumulative frequency graph showing information about the numbers of people in various age-groups in a certain region X. He forgot to include the scale on the cumulative frequency axis, as shown below. \includegraphics{figure_12}

1. Find an estimate of the median age of the population of region X. [1]
2. Find an estimate of the proportion of people aged over 60 in region X. [2]

Sonika drew similar cumulative graphs for another two regions, Y and Z, but she included the scales on the cumulative frequency axes, as shown below. \includegraphics{figure_12b}

1. Find an age group, of width 20 years, in which region Z has approximately 3 times as many people as region Y. [1]
2. State one advantage and one disadvantage of using Sonika's two diagrams to compare the populations in Regions Y and Z. [2]
3. Without calculation state, with a reason, which of regions Y or Z has the greater proportion of people aged under 40. [1]