2.02a Interpret single variable data: tables and diagrams

7 The histogram shows the age distribution of people living in Inner London in 2001. \includegraphics[max width=\textwidth, alt={}, center]{93bbc0cf-d3ad-4bc2-a6c6-36a3b8e394a9-4_805_1372_392_401} Data sourced from he 2001 Census, \href{http://www.statistics.gov.uk}{www.statistics.gov.uk}

1. State the type of skewness shown by the distribution.
2. Use the histogram to estimate the number of people aged under 25.
3. The table below shows the cumulative frequency distribution.

   Age	20	30	40	50	65	100
   Cumulative frequency (thousands)	660	1240	1810	\(a\)	2490	2770

   (A) Use the histogram to find the value of \(a\).
   (B) Use the table to calculate an estimate of the median age of these people. The ages of people living in Outer London in 2001 are summarised below.

   Age ( \(x\) years)	\(0 \leqslant x < 20\)	\(20 \leqslant x < 30\)	\(30 \leqslant x < 40\)	\(40 \leqslant x < 50\)	\(50 \leqslant x < 65\)	\(65 \leqslant x < 100\)
   Frequency (thousands)	1120	650	770	590	680	610

4. Illustrate these data by means of a histogram.
5. Make two brief comments on the differences between the age distributions of the populations of Inner London and Outer London.
6. The data given in the table for Outer London are used to calculate the following estimates. Mean 38.5, median 35.7, midrange 50, standard deviation 23.7, interquartile range 34.4.
   The final group in the table assumes that the maximum age of any resident is 100 years. These estimates are to be recalculated, based on a maximum age of 105, rather than 100. For each of the five estimates, state whether it would increase, decrease or be unchanged.
   [0pt] [4]

3 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. On the insert, draw a cumulative frequency diagram to illustrate the data.
2. Use your graph to estimate the median length of journey and the quartiles. Hence find the interquartile range.
3. State the type of skewness of the distribution of the data.

5 A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm , of these pears. \includegraphics[max width=\textwidth, alt={}, center]{056d3e9a-088d-4c97-9546-7cecb59b8727-3_815_1628_505_304}

1. Calculate the number of pears which are between 90 and 100 mm long.
2. Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate.
3. Calculate an estimate of the standard deviation.
4. Use your answers to parts (ii) and (iii) to investigate whether there are any outliers.
5. Name the type of skewness of the distribution.
6. Illustrate the data using a cumulative frequency diagram.

6 The times taken for 480 university students to travel from their accommodation to lectures are summarised below.

1. Illustrate these data by means of a histogram.
2. Identify the type of skewness of the distribution.

5 A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm , of these pears. \includegraphics[max width=\textwidth, alt={}, center]{99c502aa-2c9f-461d-9dc0-ed55e3df32a2-3_815_1628_505_304}

1. Calculate the number of pears which are between 90 and 100 mm long.
2. Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate.
3. Calculate an estimate of the standard deviation.
4. Use your answers to parts (ii) and (iii) to investigate whether there are any outliers.
5. Name the type of skewness of the distribution.
6. Illustrate the data using a cumulative frequency diagram.

4. A group of 100 adults recorded the amount of time, \(t\) minutes, they spent exercising each day. Their results are summarised in the table below. [You may use \(\sum \mathrm { f } x ^ { 2 } = 455\) 512.5]
A histogram is drawn to represent these data.
The bar representing the time \(0 \leqslant t < 15\) has width 0.5 cm and height 6 cm .

1. Calculate the width and height of the bar representing a time of \(60 \leqslant t < 120\)
2. Use linear interpolation to estimate the median time spent exercising by these adults each day.
3. Find an estimate of the mean time spent exercising by these adults each day.
4. Calculate an estimate for the standard deviation of these times.
5. Describe, giving a reason, the skewness of these data. Further analysis of the above data revealed that 18 of the 25 adults in the \(0 \leqslant t < 15\) group took no exercise each day.
6. State, giving a reason, what effect, if any, this new information would have on your answers to
   1. the estimate of the median in part (b),
   2. the estimate of the mean in part (c),
   3. the estimate of the standard deviation in part (d).

2. The stem and leaf diagram below shows the ages (in years) of the residents in a care home.

1. Find the median age of the residents.
2. Find the interquartile range (IQR) of the ages of the residents. An outlier is defined as a value that is either
   more than \(1.5 \times ( \mathrm { IQR } )\) below the lower quartile or more than \(1.5 \times ( \mathrm { IQR } )\) above the upper quartile.
3. Determine any outliers in these data. Show clearly any calculations that you use.
4. On the grid on page 5, draw a box plot to summarise these data.
   Ages

1. The table below shows the distances (to the nearest km ) travelled to work by the 50 employees in an office.

$$\text { [You may use } \left. \sum \mathrm { f } x = 394 , \quad \sum \mathrm { f } x ^ { 2 } = 6500 \right]$$ A histogram has been drawn to represent these data.
The bar representing the distance of \(3 - 5\) has a width of 1.5 cm and a height of 6 cm .

1. Calculate the width and height of the bar representing the distance of 6-10
2. Use linear interpolation to estimate the median distance travelled to work.
3. 1. Show that an estimate of the mean distance travelled to work is 7.88 km .
   2. Estimate the standard deviation of the distances travelled to work.
4. Describe, giving a reason, the skewness of these data. Peng starts to work in this office as the \(51 ^ { \text {st } }\) employee.
   She travels a distance of 7.88 km to work.
5. Without carrying out any further calculations, state, giving a reason, what effect Peng's addition to the workforce would have on your estimates of the
   1. mean,
   2. median,
   3. standard deviation
      of the distances travelled to work.

5. The values of daily sales, to the nearest \(\pounds\), taken at a newsagents last year are summarised in the table below.

1. Draw a histogram to represent these data.
2. Use interpolation to estimate the median and inter-quartile range of daily sales.
3. Estimate the mean and the standard deviation of these data. The newsagent wants to compare last year's sales with other years.
4. State whether the newsagent should use the median and the inter-quartile range or the mean and the standard deviation to compare daily sales. Give a reason for your answer.
   (2)

1 The following back-to-back stem-and-leaf diagram shows the annual salaries of a group of 39 females and 39 males. Key: 2 | 20 | 3 means \\(20200for females and \\)20300 for males.

1. Find the median and the quartiles of the females' salaries.
   You are given that the median salary of the males is \(\\) 24000\(, the lower quartile is \)\\( 22600\) and the upper quartile is \(\\) 25300$.
2. Draw a pair of box-and-whisker plots in a single diagram on the grid below to represent the data. \includegraphics[max width=\textwidth, alt={}, center]{adcf5ddd-5d49-45d1-b1fb-83d702c61082-02_994_1589_1736_310}

1 The stem-and-leaf diagram shows the heights, in metres to the nearest 0.1 m , of a random sample of trees of species \(A\). means 6.4 m

1. Find the median and interquartile range of the heights.
2. The heights, in metres to the nearest 0.1 m , of a random sample of trees of species \(B\) are given below. \(\begin{array} { l l l } 7.6 & 5.2 & 8.5 \end{array}\) 5.2
   6.3
   6.3
   6.8
   7.2
   6.7
   7.3
   5.4
   7.5
   7.4
   6.0
   6.7 In the answer book, complete the back-to-back stem-and-leaf diagram.
3. Make two comparisons between the heights of the two species of tree.

3 The table shows information about the numbers of people per household in 280900 households in the northwest of England in 2001.

1. Taking ' 5 or more' to mean ' 5 or 6 ', calculate estimates of the mean and standard deviation of the number of people per household.
2. State the values of the median and upper quartile of the number of people per household.

2 The masses, in grams, of 400 plums were recorded. The masses were then collected into class intervals of width 5 g and a cumulative frequency graph was drawn, as shown below. \includegraphics[max width=\textwidth, alt={}, center]{e5957185-5fe3-45d9-9ab3-c2aab9cbd8dd-3_1045_1401_358_333}

1. Find the number of plums with masses in the interval 40 g to 45 g .
2. Find the percentage of plums with masses greater than 70 g .
3. Give estimates of the highest and lowest masses in the sample, explaining why their exact values cannot be read from the graph.
4. On the graph paper in the answer book, draw a box-and-whisker plot to illustrate the masses of the plums in the sample.
5. Comment briefly on the shape of the distribution of masses.

1 The mean daily maximum temperatures at a research station over a 12-month period, measured to the nearest degree Celsius, are given below.

1. Construct a sorted stem and leaf diagram to represent these data, taking stem values of \(0,10 , \ldots\).
2. Write down the median of these data.
3. The mean of these data is 24.3 . Would the mean or the median be a better measure of central tendency of the data? Briefly explain your answer.

7 The birth weights of 200 lambs from crossbred sheep are illustrated by the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{4b259fe3-73ef-419f-85ad-1a3b1e6ea56e-4_917_1146_367_447}

1. Estimate the percentage of lambs with birth weight over 6 kg .
2. Estimate the median and interquartile range of the data.
3. Use your answers to part (ii) to show that there are very few, if any, outliers. Comment briefly on whether any outliers should be disregarded in analysing these data. The box and whisker plot shows the birth weights of 100 lambs from Welsh Mountain sheep. \includegraphics[max width=\textwidth, alt={}, center]{4b259fe3-73ef-419f-85ad-1a3b1e6ea56e-4_328_1616_1749_260}
4. Use appropriate measures to compare briefly the central tendencies and variations of the weights of the two types of lamb.
5. The weight of the largest Welsh Mountain lamb was originally recorded as 6.5 kg , but then corrected. If this error had not been corrected, how would this have affected your answers to part (iv)? Briefly explain your answer.
6. One lamb of each type is selected at random. Estimate the probability that the birth weight of both lambs is at least 3.9 kg .

6 The heights \(x \mathrm {~cm}\) of 100 boys in Year 7 at a school are summarised in the table below.

1. Estimate the number of boys who have heights of at least 155 cm .
2. Calculate an estimate of the median height of the 100 boys.
3. Draw a histogram to illustrate the data. The histogram below shows the heights of 100 girls in Year 7 at the same school. \includegraphics[max width=\textwidth, alt={}, center]{76283206-687f-45d6-9204-952d60843cf1-3_865_1349_1297_349}
4. How many more girls than boys had heights exceeding 160 cm ?
5. Calculate an estimate of the mean height of the 100 girls.

1 In a traffic survey, the number of people in each car passing the survey point is recorded. The results are given in the following frequency table.

1. Write down the median and mode of these data.
2. Draw a vertical line diagram for these data.
3. State the type of skewness of the distribution.

6 The birth weights in kilograms of 25 female babies are shown below, in ascending order.

1. Find the median and interquartile range of these data.
2. Draw a box and whisker plot to illustrate the data.
3. Show that there is exactly one outlier. Discuss whether this outlier should be removed from the data. The cumulative frequency curve below illustrates the birth weights of 200 male babies. \includegraphics[max width=\textwidth, alt={}, center]{6b886da6-3fb8-4b4c-b572-f4b770ae5a4c-3_929_1569_1450_248}
4. Find the median and interquartile range of the birth weights of the male babies.
5. Compare the weights of the female and male babies.
6. Two of these male babies are chosen at random. Calculate an estimate of the probability that both of these babies weigh more than any of the female babies.

5 At a tourist information office the numbers of people seeking information each hour over the course of a 12-hour day are shown below. $$\begin{array} { l l l l l l l l l l l l } 6 & 25 & 38 & 39 & 31 & 18 & 35 & 31 & 33 & 15 & 21 & 28 \end{array}$$

1. Construct a sorted stem and leaf diagram to represent these data.
2. State the type of skewness suggested by your stem and leaf diagram.
3. For these data find the median, the mean and the mode. Comment on the usefulness of the mode in this case.

8 The box and whisker plot below summarises the weights in grams of the 20 chocolates in a box. \includegraphics[max width=\textwidth, alt={}, center]{6015ae6c-bf76-4a0c-af0f-5c53f9c5ed2a-4_287_1177_319_427}

1. Find the interquartile range of the data and hence determine whether there are any outliers at either end of the distribution. Ben buys a box of these chocolates each weekend. The chocolates all look the same on the outside, but 7 of them have orange centres, 6 have cherry centres, 4 have coffee centres and 3 have lemon centres. One weekend, each of Ben's 3 children eats one of the chocolates, chosen at random.
2. Calculate the probabilities of the following events. A: all 3 chocolates have orange centres \(B\) : all 3 chocolates have the same centres
3. Find \(\mathrm { P } ( A \mid B )\) and \(\mathrm { P } ( B \mid A )\). The following weekend, Ben buys an identical box of chocolates and again each of his 3 children eats one of the chocolates, chosen at random.
4. Find the probability that, on both weekends, the 3 chocolates that they eat all have orange centres.
5. Ben likes all of the chocolates except those with cherry centres. On another weekend he is the first of his family to eat some of the chocolates. Find the probability that he has to select more than 2 chocolates before he finds one that he likes. \section*{END OF QUESTION PAPER} \section*{OCR
   Oxford Cambridge and RSA}

6 The temperature of a supermarket fridge is regularly checked to ensure that it is working correctly. Over a period of three months the temperature (measured in degrees Celsius) is checked 600 times. These temperatures are displayed in the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{7b92607f-1bf9-45f0-997b-fe76c88b5fcd-4_1054_1649_539_248}

1. Use the diagram to estimate the median and interquartile range of the data.
2. Use your answers to part (i) to show that there are very few, if any, outliers in the sample.
3. Suppose that an outlier is identified in these data. Discuss whether it should be excluded from any further analysis.
4. Copy and complete the frequency table below for these data.

   Temperature \(( t\) degrees Celsius \()\)	\(3.0 \leqslant t \leqslant 3.4\)	\(3.4 < t \leqslant 3.8\)	\(3.8 < t \leqslant 4.2\)	\(4.2 < t \leqslant 4.6\)	\(4.6 < t \leqslant 5.0\)
   Frequency			243	157

5. Use your table to calculate an estimate of the mean.
6. The standard deviation of the temperatures in degrees Celsius is 0.379 . The temperatures are converted from degrees Celsius into degrees Fahrenheit using the formula \(F = 1.8 C + 32\). Hence estimate the mean and find the standard deviation of the temperatures in degrees Fahrenheit.

1 Two students draw graphs to represent the numbers of pairs of shoes owned by members of their class. Andrew produces a bipartite graph, but gets it wrong. Barbara produces a completely correct frequency graph. Their graphs are shown below. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Draw a correct bipartite graph.
2. How many people are in the class?
3. How many pairs of shoes in total are owned by members of the class?
4. Which points on Barbara's graph may be deleted without losing any information? Charles produces the same frequency graph as Barbara, but joins consecutive points with straight lines.
5. Criticise Charles's graph.

13 The four pie charts illustrate the numbers of employees using different methods of travel in four Local Authorities in 2011. \includegraphics[max width=\textwidth, alt={}, center]{7298e7b9-ad52-480c-bc2b-8289aeab9ebb-10_1131_1077_347_242}

1. State, with reasons, which of the four Local Authorities is most likely to be a rural area with many hills.
2. Explain why pie charts are more suitable for answering part (a) than bar charts showing the same data.
3. Two of the Local Authorities represent urban areas.
   1. State with a reason which two Local Authorities are likely to be urban.
   2. One urban Local Authority introduced a Park-and-Ride service in 2006. Users of this service drive to the edge of the urban area and then use buses to take them into the centre of the area. A student claims that a comparison of the corresponding pie charts for 2001 (not shown) and 2011 would enable them to identify which Local Authority this was. State with a reason whether you agree with the student.