2.02a Interpret single variable data: tables and diagrams

1. In a particular week, a dentist treats 100 patients. The length of time, to the nearest minute, for each patient's treatment is summarised in the table below.

Draw a histogram to illustrate these data.

5 The examination marks obtained by 1200 candidates are illustrated on the cumulative frequency graph, where the data points are joined by a smooth curve. \includegraphics[max width=\textwidth, alt={}, center]{11316ea6-3999-4003-b77d-bee8b547c1da-04_1335_1319_404_413} Use the curve to estimate

1. the interquartile range of the marks,
2. \(x\), if \(40 \%\) of the candidates scored more than \(x\) marks,
3. the number of candidates who scored more than 68 marks. Five of the candidates are selected at random, with replacement.
4. Estimate the probability that all five scored more than 68 marks. It is subsequently discovered that the candidates' marks in the range 35 to 55 were evenly distributed - that is, roughly equal numbers of candidates scored \(35,36,37 , \ldots , 55\).
5. What does this information suggest about the estimate of the interquartile range found in part (i)? \section*{June 2005}

2 Every day, George attempts the quiz in a national newspaper. The quiz always consists of 7 questions. In the first 25 days of January, the numbers of questions George answers correctly each day are summarised in the table below.

1. Draw a vertical line chart to illustrate the data.
2. State the type of skewness shown by your diagram.
3. Calculate the mean and the mean squared deviation of the data.
4. How many correct answers would George need to average over the next 6 days if he is to achieve an average of 5 correct answers for all 31 days of January?

5. The histogram shows the number of hours worked in a given week by a group of 64 freelance photographers.

1. Give a reason to justify the use of a histogram to represent these data. Given that 16 of these freelance photographers spent between 10 and 20 hours working in this week,
2. estimate the number that spent between 12 and 24 hours working in this week.
3. Find an estimate for the median time spent working in this week by these 64 freelance photographers. Charlie decides to model these data using a normal distribution. Charlie calculates an estimate of the mean to be 23.9 hours to one decimal place.
4. Comment on Charlie's decision to use a normal distribution. Give a justification for your answer.

Each year the total number of hours, \(x\), of sunshine in Kintoo is recorded during the month of June. The results for the last 60 years are summarised in the table.

\(x\)	\(30 \leqslant x < 60\)	\(60 \leqslant x < 90\)	\(90 \leqslant x < 110\)	\(110 \leqslant x < 140\)	\(140 \leqslant x < 180\)	\(180 \leqslant x \leqslant 240\)
Number of years	4	8	14	25	7	2

1. Draw a cumulative frequency graph to illustrate the data. [3]
2. Use your graph to estimate the 70th percentile of the data. [2]
3. Calculate an estimate for the mean number of hours of sunshine in Kintoo during June over the last 60 years. [3]

The manager of a company noted the times spent in 80 meetings. The results were as follows. Draw a cumulative frequency graph and use this to estimate the median time and the interquartile range. [6]

\includegraphics{figure_3} The birth weights of random samples of 900 babies born in country \(A\) and 900 babies born in country \(B\) are illustrated in the cumulative frequency graphs. Use suitable data from these graphs to compare the central tendency and spread of the birth weights of the two sets of babies. [6]

\includegraphics{figure_3} In an open-plan office there are 88 computers. The times taken by these 88 computers to access a particular web page are represented in the cumulative frequency diagram.

1. On graph paper draw a box-and-whisker plot to summarise this information. [4]

An 'outlier' is defined as any data value which is more than 1.5 times the interquartile range above the upper quartile, or more than 1.5 times the interquartile range below the lower quartile.

1. Show that there are no outliers. [2]

On a certain day in spring, the heights of 200 daffodils are measured, correct to the nearest centimetre. The frequency distribution is given below.

Height (cm)	\(4 - 10\)	\(11 - 15\)	\(16 - 20\)	\(21 - 25\)	\(26 - 30\)
Frequency	22	32	78	40	28

1. Draw a cumulative frequency graph to illustrate the data. [4]
2. 28\% of these daffodils are of height \(h\) cm or more. Estimate \(h\). [2]
3. You are given that the estimate of the mean height of these daffodils, calculated from the table, is 18.39 cm. Calculate an estimate of the standard deviation of the heights of these daffodils. [3]

Jim records the length, \(l\) mm, of 81 salmon. The data are coded using \(x = l - 600\) and the following summary statistics are obtained. $$n = 81 \quad \sum x = 3711 \quad \sum x^2 = 475181$$

1. Find the mean length of these salmon. [3]
2. Find the variance of the lengths of these salmon. [2]

The weight, \(w\) grams, of each of the 81 salmon is recorded to the nearest gram. The recorded results for the 81 salmon are summarised in the box plot below. \includegraphics{figure_2}

1. Find the maximum number of salmon that have weights in the interval $$4600 < w \leqslant 7700$$ [1]

Raj says that the box plot is incorrect as Jim has not included outliers. For these data an outlier is defined as a value that is more than \(1.5 \times\) IQR above the upper quartile \quad or \quad \(1.5 \times\) IQR below the lower quartile

1. Show that there are no outliers. [3]

A meteorologist measured the number of hours of sunshine, to the nearest hour, each day for 100 days. The results are summarised in the table below.

1. On graph paper, draw a histogram to represent these data. [5]
2. Calculate an estimate of the number of days that had between 6 and 9 hours of sunshine. [2]

Hospital records show the number of babies born in a year. The number of babies delivered by 15 male doctors is summarised by the stem and leaf diagram below. Babies \quad (4|5 means 45) \quad Totals 0 \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (0) 1|9 \quad \quad \quad \quad \quad \quad \quad \quad \quad (1) 2|1 6 7 7 \quad \quad \quad \quad \quad \quad (4) 3|2 2 3 4 8 \quad \quad \quad \quad \quad (5) 4|5 \quad \quad \quad \quad \quad \quad \quad \quad \quad (1) 5|1 \quad \quad \quad \quad \quad \quad \quad \quad \quad (1) 6|0 \quad \quad \quad \quad \quad \quad \quad \quad \quad (1) 7 \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (0) 8|6 7 \quad \quad \quad \quad \quad \quad \quad \quad (2)

1. Find the median and inter-quartile range of these data. [3]
2. Given that there are no outliers, draw a box plot on graph paper to represent these data. Start your scale at the origin. [4]
3. Calculate the mean and standard deviation of these data. [5]

The records also contain the number of babies delivered by 10 female doctors. 34 \quad 30 \quad 20 \quad 15 \quad 6 32 \quad 26 \quad 19 \quad 11 \quad 4 The quartiles are 11, 19.5 and 30.

1. Using the same scale as in part (b) and on the same graph paper draw a box plot for the data for the 10 female doctors. [3]
2. Compare and contrast the box plots for the data for male and female doctors. [2]

The 19 employees of a company take an aptitude test. The scores out of 40 are illustrated in the stem and leaf diagram below. \(2|6\) means a score of 26 \begin{align} 0 & | 7 & (1)
1 & | 88 & (2)
2 & | 4468 & (4)
3 & | 2333459 & (7)
4 & | 00000 & (5) \end{align} Find

1. the median score, [1]
2. the interquartile range. [3]

The company director decides that any employees whose scores are so low that they are outliers will undergo retraining. An outlier is an observation whose value is less than the lower quartile minus 1.0 times the interquartile range.

1. Explain why there is only one employee who will undergo retraining. [2]
2. On the graph paper on page 5, draw a box plot to illustrate the employees' scores. [3]

The birth weights, in kg, of 1500 babies are summarised in the table below.

Weight (kg)	Midpoint, \(x\)kg	Frequency, \(f\)
\(0.0 - 1.0\)	\(0.50\)	\(1\)
\(1.0 - 2.0\)	\(1.50\)	\(6\)
\(2.0 - 2.5\)	\(2.25\)	\(60\)
\(2.5 - 3.0\)		\(280\)
\(3.0 - 3.5\)	\(3.25\)	\(820\)
\(3.5 - 4.0\)	\(3.75\)	\(320\)
\(4.0 - 5.0\)	\(4.50\)	\(10\)
\(5.0 - 6.0\)		\(3\)

[You may use \(\sum fx = 4841\) and \(\sum fx^2 = 15889.5\)]

1. Write down the missing midpoints in the table above. [2]
2. Calculate an estimate of the mean birth weight. [2]
3. Calculate an estimate of the standard deviation of the birth weight. [3]
4. Use interpolation to estimate the median birth weight. [2]
5. Describe the skewness of the distribution. Give a reason for your answer. [2]

The stem-and-leaf diagram shows the values taken by two variables \(A\) and \(B\).

\(A\)		\(B\)
8, 7, 4, 1, 0	1	1, 1, 2, 5, 6, 8, 9
9, 8, 7, 6, 6, 5, 2	2	0, 3, 4, 6, 7, 7, 9
9, 7, 6, 4, 2, 1, 0	3	1, 4, 5, 5, 8
8, 6, 3, 2, 2	4	0, 2, 6, 6, 9, 9
6, 4, 0	5	2, 3, 5, 7
5, 3, 1	6	0, 1

Key: 3 | 1 | 2 means \(A = 13\), \(B = 12\)

1. For each set of data, calculate estimates of the median and the quartiles. [6 marks]
2. Calculate the 42nd percentile for \(A\). [2 marks]
3. On graph paper, indicating your scale clearly, construct box and whisker plots for both sets of data. [4 marks]
4. Describe the skewness of the distribution of \(A\) and of \(B\). [2 marks]

In a survey of natural habitats, the numbers of trees in sixty equal areas of land were recorded, as follows:

1. Construct a stem-and-leaf diagram to illustrate this data, using the groupings 5 - 9, 10 - 14, 15 - 19, 20 - 24, etc. [8 marks]
2. Find the three quartiles for the distribution. [4 marks]
3. On graph paper construct a box plot for the data, showing your scale and clearly indicating any outliers. [4 marks]

The back-to-back stem and leaf diagram shows the journey times, to the nearest minute, of the commuter services into a big city provided by the trains of two operating companies. Key: \(4|3|6\) means 34 minutes for Company \(A\) and 36 minutes for Company \(B\).

1. Write down the numbers needed to complete the diagram. [1 mark]
2. Find the median and the quartiles for each company. [6 marks]
3. On graph paper, draw box plots for the two companies. Show your scale. [6 marks]
4. Use your plots to compare the two sets of data briefly. [2 marks]
5. Describe the skewness of each company's distribution of times. [2 marks]

1000 houses were sold in a small town in a one-year period. The selling prices were as given in the following table:

Selling Price	Number of Houses	Selling Price	Number of Houses
Up to £50 000	60	Up to £150 000	642
Up to £75 000	227	Up to £200 000	805
Up to £100 000	305	Up to £500 000	849
Up to £125 000	414	Up to £750 000	1000

1. Name (do not draw) a suitable type of graph for illustrating this data. [1 mark]
2. Use interpolation to find estimates of the median and the quartiles. [6 marks]
3. Estimate the 37th percentile. [2 marks]

Given further that the lowest price was £42 000 and the range of the prices was £690 000,

1. draw a box plot to represent the data. Show your scale clearly. [4 marks]

In another town the median price was £149 000, and the interquartile range was £90 000.

1. Briefly compare the prices in the two towns using this information. [2 marks]

The following data were collected by counting the number of cars that passed the gates of a college in 60 successive 5 minute intervals.

12	20	19	31	32	35	37	29	26	27	20	17
15	9	8	11	13	17	17	21	25	27	28	25
30	32	37	40	45	45	44	47	42	41	36	38
35	34	30	30	27	26	29	24	23	21	21	18
16	16	19	22	26	28	23	17	15	10	12	13

1. Make a stem and leaf diagram for this data, using the groups \(5-9\), \(10-14\), \(\ldots\), \(45-49\). Show the total in each group and give a key to the diagram. [7 marks]
2. Find the three quartiles for this data. [4 marks]
3. On graph paper, draw a box plot for the data. [4 marks]
4. Describe the skewness of the distribution. [1 mark]

The length of time, in minutes, that visitors queued for a tourist attraction is given by the following table, where, for example, '\(20 -\)' means from 20 up to but not including 30 minutes.

Queuing time (mins)	\(0 -\)	\(10 -\)	\(15 -\)	\(20 -\)	\(30 -\)	\(40 - 60\)
Number of visitors	\(15\)	\(24\)	\(x\)	\(13\)	\(10\)	\(y\)

1. State the upper class boundary of the first class. [1 mark]

A histogram is drawn to represent this data. The total area under the histogram is \(36\) cm\(^2\). The '\(10 -\)' bar has width \(1\) cm and height \(9.6\) cm. The '\(15 -\)' bar is ten times as high as the '\(40 - 60\)' bar.

1. Find the values of \(x\) and \(y\). [7 marks]
2. On graph paper, construct the histogram accurately. [5 marks]

The following table gives the weights, in grams, of 60 items delivered to a company in a day.

Weight (g)	0 - 10	10 - 20	20 - 30	30 - 40	40 - 50	50 - 60	60 - 80
No. of items	2	11	18	12	9	6	2

1. Use interpolation to calculate estimated values of
   1. the median weight,
   2. the interquartile range,
   3. the thirty-third percentile.
   [7 marks]

Outliers are defined to be outside the range from \(2.5Q_1 - 1.5Q_2\) to \(2.5Q_2 - 1.5Q_1\).

1. Given that the lightest item weighed 3 g and the two heaviest weighed 65 g and 79 g, draw on graph paper an accurate box-and-whisker plot of the data. Indicate any outliers clearly. [5 marks]
2. Describe the skewness of the distribution. [1 mark]

The mean weight was 32.0 g and the standard deviation of the weights was 14.9 g.

1. State, with a reason, whether you would choose to summarise the data by using the mean and standard deviation or the median and interquartile range. [2 marks]

On another day, items were delivered whose weights ranged from 14 g to 58 g; the median was 32 g, the lower quartile was 24 g and the interquartile range was 26 g.

1. Draw a further box plot for these data on the same diagram. Briefly compare the two sets of data using your plots. [6 marks]

40 people were asked to guess the length of a certain road. Each person gave their guess, \(l\) km, correct to the nearest kilometre. The results are summarised below.

\(l\)	10-12	13-15	16-20	21-30
Frequency	1	13	20	6

1. 1. Use appropriate formulae to calculate estimates of the mean and standard deviation of \(l\). [6]
   2. Explain why your answers are only estimates. [1]
2. A histogram is to be drawn to illustrate the data. Calculate the frequency density of the block for the 16-20 class. [2]
3. Explain which class contains the median value of \(l\). [2]
4. Later, the person whose guess was between 10 km and 12 km changed his guess to between 13 km and 15 km. Without calculation state whether the following will increase, decrease or remain the same:
   1. the mean of \(l\), [1]
   2. the standard deviation of \(l\). [1]

The lengths, in centimetres, of 18 snakes are given below. 24 62 20 65 27 67 69 32 40 53 55 47 33 45 55 56 49 58

1. Draw an ordered stem-and-leaf diagram for the data. [3]
2. Find the mean and median of the lengths of the snakes. [2]
3. It was found that one of the lengths had been measured incorrectly. After this length was corrected, the median increased by 1 cm. Give two possibilities for the incorrect length and give a corrected value in each case. [2]

A camera records the speeds in miles per hour of 15 vehicles on a motorway. The speeds are given below. $$73 \quad 67 \quad 75 \quad 64 \quad 52 \quad 63 \quad 75 \quad 81 \quad 77 \quad 72 \quad 68 \quad 74 \quad 79 \quad 72 \quad 71$$

1. Construct a sorted stem and leaf diagram to represent these data, taking stem values of 50, 60, ... . [4]
2. Write down the median and midrange of the data. [2]
3. Which of the median and midrange would you recommend to measure the central tendency of the data? Briefly explain your answer. [2]