2.02a Interpret single variable data: tables and diagrams

1. The lengths, \(x \mathrm {~mm}\), of 50 pebbles are summarised in the table below.

A histogram is drawn to represent these data.
The bar representing the class \(32 \leqslant x < 36\) is 2.5 cm wide and 7.5 cm tall.

1. Calculate the width and the height of the bar representing the class \(30 \leqslant x < 32\)
2. Using linear interpolation, estimate the median of \(x\) The weight, \(w\) grams, of each of the 50 pebbles is coded using \(10 y = w - 20\) These coded data are summarised by $$\sum y = 104 \quad \sum y ^ { 2 } = 233.54$$
3. Show that the mean of \(w\) is 40.8
4. Calculate the standard deviation of \(w\) The weight of a pebble recorded as 40.8 grams is added to the sample.
5. Without carrying out any further calculations, state, giving a reason, what effect this would have on the value of
   1. the mean of \(w\)
   2. the standard deviation of \(w\)

3. The parking times, \(t\) hours, for cars in a car park are summarised below. $$\text { (You may use } \sum \mathrm { fm } = 182 \text { and } \sum \mathrm { fm } ^ { 2 } = 883 \text { ) }$$ A histogram is drawn to represent these data.
The bar representing the time \(1 \leqslant t < 2\) has a width of 1.5 cm and a height of 6 cm .

1. Calculate the width and the height of the bar representing the time \(4 \leqslant t < 6\)
2. Use linear interpolation to estimate the median parking time for the cars in the car park.
3. Estimate the mean and the standard deviation of the parking time for the cars in the car park.
4. Describe, giving a reason, the skewness of the data. One of these cars is selected at random.
5. Estimate the probability that this car is parked for more than 75 minutes.

1. The stem lengths of a sample of 120 tulips are recorded in the grouped frequency table below.

A histogram is drawn to represent these data.
The area of the bar representing the \(40 \leqslant x < 42\) class is \(16.5 \mathrm {~cm} ^ { 2 }\)

1. Calculate the exact area of the bar representing the \(42 \leqslant x < 45\) class. The height of the tallest bar in the histogram is 10 cm .
2. Find the exact height of the second tallest bar. \(Q _ { 1 }\) for these data is 45 cm .
3. Use linear interpolation to find an estimate for
   1. \(Q _ { 2 }\)
   2. the interquartile range. One measure of skewness is given by $$\frac { Q _ { 3 } - 2 Q _ { 2 } + Q _ { 1 } } { Q _ { 3 } - Q _ { 1 } }$$
4. By calculating this measure, describe the skewness of these data.

1. The time taken to complete a puzzle, in minutes, is recorded for each person in a club. The times are summarised in a grouped frequency distribution and represented by a histogram.

One of the class intervals has a frequency of 20 and is shown by a bar of width 1.5 cm and height 12 cm on the histogram. The total area under the histogram is \(94.5 \mathrm {~cm} ^ { 2 }\) Find the number of people in the club.

4. A restaurant owner is concerned about the amount of time customers have to wait before being served. He collects data on the waiting times, to the nearest minute, of 20 customers. These data are listed below.

1. Find the median and inter-quartile range of the waiting times. An outlier is an observation that falls either \(1.5 \times\) (inter-quartile range) above the upper quartile or \(1.5 \times\) (inter-quartile range) below the lower quartile.
2. Draw a boxplot to represent these data, clearly indicating any outliers.
3. Find the mean of these data.
4. Comment on the skewness of these data. Justify your answer.

2. The number of caravans on Seaview caravan site on each night in August last year is summarised in the following stem and leaf diagram.

1. Find the three quartiles of these data. During the same month, the least number of caravans on Northcliffe caravan site was 31. The maximum number of caravans on this site on any night that month was 72 . The three quartiles for this site were 38,45 and 52 respectively.
2. On graph paper and using the same scale, draw box plots to represent the data for both caravan sites. You may assume that there are no outliers.
3. Compare and contrast these two box plots.
4. Give an interpretation to the upper quartiles of these two distributions.

1. Over a period of time, the number of people \(x\) leaving a hotel each morning was recorded. These data are summarised in the stem and leaf diagram below.

For these data,

1. write down the mode,
2. find the values of the three quartiles. Given that \(\Sigma x = 1335\) and \(\Sigma x ^ { 2 } = 71801\), find
3. the mean and the standard deviation of these data. One measure of skewness is found using $$\frac { \text { mean - mode } } { \text { standard deviation } } \text {. }$$
4. Evaluate this measure to show that these data are negatively skewed.
5. Give two other reasons why these data are negatively skewed.

1. Summarised below are the distances, to the nearest mile, travelled to work by a random sample of 120 commuters.

For this distribution,

1. describe its shape,
2. use linear interpolation to estimate its median. The mid-point of each class was represented by \(x\) and its corresponding frequency by \(f\) giving $$\Sigma f x = 3550 \text { and } \Sigma f x ^ { 2 } = 138020$$
3. Estimate the mean and the standard deviation of this distribution. One coefficient of skewness is given by $$\frac { 3 ( \text { mean - median } ) } { \text { standard deviation } } .$$
4. Evaluate this coefficient for this distribution.
5. State whether or not the value of your coefficient is consistent with your description in part (a). Justify your answer.
6. State, with a reason, whether you should use the mean or the median to represent the data in this distribution.
7. State the circumstance under which it would not matter whether you used the mean or the median to represent a set of data.

1. A teacher recorded, to the nearest hour, the time spent watching television during a particular week by each child in a random sample. The times were summarised in a grouped frequency table and represented by a histogram.

One of the classes in the grouped frequency distribution was 20-29 and its associated frequency was 9. On the histogram the height of the rectangle representing that class was 3.6 cm and the width was 2 cm .

1. Give a reason to support the use of a histogram to represent these data.
2. Write down the underlying feature associated with each of the bars in a histogram.
3. Show that on this histogram each child was represented by \(0.8 \mathrm {~cm} ^ { 2 }\). The total area under the histogram was \(24 \mathrm {~cm} ^ { 2 }\).
4. Find the total number of children in the group.

2. The age in years of the residents of two hotels are shown in the back to back stem and leaf diagram below. Abbey Hotel \(8 | 5 | 0\) means 58 years in Abbey hotel and 50 years in Balmoral hotel Balmoral Hotel For the Balmoral Hotel,

1. write down the mode of the age of the residents,
2. find the values of the lower quartile, the median and the upper quartile.
3. 1. Find the mean, \(\bar { x }\), of the age of the residents.
   2. Given that \(\sum x ^ { 2 } = 81213\) find the standard deviation of the age of the residents. One measure of skewness is found using $$\frac { \text { mean - mode } } { \text { standard deviation } }$$
4. Evaluate this measure for the Balmoral Hotel. For the Abbey Hotel, the mode is 39 , the mean is 33.2 , the standard deviation is 12.7 and the measure of skewness is - 0.454
5. Compare the two age distributions of the residents of each hotel.

1. The marks of a group of female students in a statistics test are summarised in Figure 1

\begin{figure}[h] \end{figure}

1. Write down the mark which is exceeded by \(75 \%\) of the female students. The marks of a group of male students in the same statistics test are summarised by the stem and leaf diagram below.

   Mark	(2|6 means 26)	Totals
   1	4	(1)
   2	6	(1)
   3	447	(3)
   4	066778	(6)
   5	001113677	(9)
   6	223338	(6)
   7	008	(3)
   8	5	(1)
   9	0	(1)

2. Find the median and interquartile range of the marks of the male students. An outlier is a mark that is
   either more than \(1.5 \times\) interquartile range above the upper quartile or more than \(1.5 \times\) interquartile range below the lower quartile.
3. In the space provided on Figure 1 draw a box plot to represent the marks of the male students, indicating clearly any outliers.
4. Compare and contrast the marks of the male and the female students.

4. The following table summarises the times, \(t\) minutes to the nearest minute, recorded for a group of students to complete an exam. $$\text { [You may use } \sum \mathrm { f } t ^ { 2 } = 134281.25 \text { ] }$$

1. Estimate the mean and standard deviation of these data.
2. Use linear interpolation to estimate the value of the median.
3. Show that the estimated value of the lower quartile is 18.6 to 3 significant figures.
4. Estimate the interquartile range of this distribution.
5. Give a reason why the mean and standard deviation are not the most appropriate summary statistics to use with these data. The person timing the exam made an error and each student actually took 5 minutes less than the times recorded above. The table below summarises the actual times.

   Time (minutes) \(t\)	\(6 - 15\)	\(16 - 20\)	\(21 - 25\)	\(26 - 30\)	\(31 - 40\)	\(41 - 55\)
   Number of students f	62	88	16	13	11	10

6. Without further calculations, explain the effect this would have on each of the estimates found in parts (a), (b), (c) and (d).

1. The table shows the time, to the nearest minute, spent waiting for a taxi by each of 80 people one Sunday afternoon.

1. Write down the upper class boundary for the \(2 - 4\) minute interval. A histogram is drawn to represent these data. The height of the tallest bar is 6 cm .
2. Calculate the height of the second tallest bar.
3. Estimate the number of people with a waiting time between 3.5 minutes and 7 minutes.
4. Use linear interpolation to estimate the median, the lower quartile and the upper quartile of the waiting times.
5. Describe the skewness of these data, giving a reason for your answer.

1. A random sample of 35 homeowners was taken from each of the villages Greenslax and Penville and their ages were recorded. The results are summarised in the back-to-back stem and leaf diagram below.

Key: 7 | 3 | 1 means 37 years for Greenslax and 31 years for Penville
Some of the quartiles for these two distributions are given in the table below.

1. Find the value of \(a\) and the value of \(b\). An outlier is a value that falls either $$\begin{aligned} & \text { more than } 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { above } Q _ { 3 } \\ & \text { or more than } 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { below } Q _ { 1 } \end{aligned}$$
2. On the graph paper opposite draw a box plot to represent the data from Penville. Show clearly any outliers.
3. State the skewness of each distribution. Justify your answers. \includegraphics[max width=\textwidth, alt={}, center]{8270bcae-494c-4248-8229-a72e9e84eab0-03_930_1237_1800_367}

6. The times, in seconds, spent in a queue at a supermarket by 85 randomly selected customers, are summarised in the table below. A histogram was drawn to represent these data. The \(30 - 60\) group was represented by a bar of width 1.5 cm and height 1 cm .

1. Find the width and the height of the \(70 - 80\) group.
2. Use linear interpolation to estimate the median of this distribution. Given that \(x\) denotes the midpoint of each group in the table and $$\sum f x = 6460 \quad \sum f x ^ { 2 } = 529400$$
3. calculate an estimate for
   1. the mean,
   2. the standard deviation,
      for the above data. One measure of skewness is given by $$\text { coefficient of skewness } = \frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } }$$
4. Evaluate this coefficient and comment on the skewness of these data.

1. As part of their job, taxi drivers record the number of miles they travel each day. A random sample of the mileages recorded by taxi drivers Keith and Asif are summarised in the back-toback stem and leaf diagram below.

Key: 0184 means 180 for Keith and 184 for Asif
The quartiles for these two distributions are summarised in the table below.

1. Find the values of \(a , b\) and \(c\). Outliers are values that lie outside the limits $$Q _ { 1 } - 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) \text { and } Q _ { 3 } + 1.5 \left( Q _ { 3 } - Q _ { 1 } \right) .$$
2. On graph paper, and showing your scale clearly, draw a box plot to represent Keith's data.
3. Comment on the skewness of the two distributions.

7. A college organised a 'fun run'. The times, to the nearest minute, of a random sample of 100 students who took part are summarised in the table below.

1. Give a reason to support the use of a histogram to represent these data.
2. Write down the upper class boundary and the lower class boundary of the class 40-44.
3. On graph paper, draw a histogram to represent these data. END

3. The frequency distribution for the lengths of 108 fish in an aquarium is given by the following table. The lengths of the fish ranged from 5 cm to 90 cm .

1. Calculate estimates of the three quartiles of the distribution.
2. On graph paper, draw a box and whisker plot of the data.
3. Hence describe the skewness of the distribution.
4. If the data were represented by a histogram, what would be the ratio of the heights of the shortest and highest bars?

5. In a survey unemployed people were asked how many months it had been, to the nearest month, since they were last employed on a full-time basis. The data collected is summarised in this stem and leaf diagram.

1. Write down the values needed to complete the totals column on the stem and leaf diagram.
2. State the mode of these data.
3. Find the median and quartiles of these data. Given that any values outside of the limits \(\mathrm { Q } _ { 1 } - 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) and \(\mathrm { Q } _ { 3 } + 1.5 \left( \mathrm { Q } _ { 3 } - \mathrm { Q } _ { 1 } \right)\) are to be regarded as outliers,
4. determine if there are any outliers in these data,
5. draw a box plot representing these data on graph paper,
6. describe the skewness of these data and suggest a reason for it.

1. (a) Explain the difference between a discrete and a continuous variable.

A random number generator on a calculator generates numbers, \(X\), to 3 decimal places, in the range 0 to 1 , e.g. 0.386 . The variable \(X\) may be modelled by a continuous uniform distribution, having the probability density function \(\mathrm { f } ( x )\), where $$\begin{array} { l l } \mathrm { f } ( x ) = 1 & 0 < x < 1 \\ \mathrm { f } ( x ) = 0 & \text { otherwise } \end{array}$$ (b) Explain why this model is not totally accurate.
(c) Sketch the cumulative distribution function of \(X\).

3 Answer part (i) of this question on the insert provided. A taxi driver operates from a taxi rank at a main railway station in London. During one particular week he makes 120 journeys, the lengths of which are summarised in the table.

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.

4 Answer part (i) of this question on the insert provided. A taxi driver operates from a taxi rank at a main railway station in London. During one particular week he makes 120 journeys, the lengths of which are summarised in the table.

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.

9 The finance department of a retail firm recorded the daily income each day for 300 days. The results are summarised in the histogram. \includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-6_689_1575_488_246}

1. Find the number of days on which the daily income was between \(\pounds 4000\) and \(\pounds 6000\).
2. Calculate an estimate of the number of days on which the daily income was between \(\pounds 2700\) and \(\pounds 3600\).
3. Use the midpoints of the classes to show that an estimate of the mean daily income is \(\pounds 3275\). An estimate of the standard deviation of the daily income is \(\pounds 1060\). The finance department uses the distribution \(\mathrm { N } \left( 3275,1060 ^ { 2 } \right)\) to model the daily income, in pounds.
4. Calculate the number of days on which, according to this model, the daily income would be between \(\pounds 4000\) and \(\pounds 6000\).
5. It is given that approximately \(95 \%\) of values of the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\) lie within the range \(\mu \pm 2 \sigma\). Without further calculation, use this fact to comment briefly on whether the proposed model is a good fit to the data illustrated in the histogram.

5. The following grouped frequency distribution summarises the number of minutes, to the nearest minute, that a random sample of 200 motorists were delayed by roadworks on a stretch of motorway.

1. Using graph paper represent these data by a histogram.
2. Give a reason to justify the use of a histogram to represent these data.
3. Use interpolation to estimate the median of this distribution.
4. Calculate an estimate of the mean and an estimate of the standard deviation of these data. One coefficient of skewness is given by $$\frac { 3 ( \text { mean } - \text { median } ) } { \text { standard deviation } } .$$
5. Evaluate this coefficient for the above data.
6. Explain why the normal distribution may not be suitable to model the number of minutes that motorists are delayed by these roadworks.