Describe shape or skewness of distribution

3 The histogram shows the age distribution of people living in Inner London in 2001. \includegraphics[max width=\textwidth, alt={}, center]{b6d84f99-ee39-49c7-a5e8-05838efeef5a-2_804_1372_483_436} Data sourced from the 2001 Census, www.sta is \href{http://ics.gov.uk}{ics.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.

7 The histogram shows the age distribution of people living in Inner London in 2001. \includegraphics[max width=\textwidth, alt={}, center]{aabf9d8b-5f91-4a3b-bcf8-e46cb45127c4-4_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]

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]

2 Mia rolls a six-sided die 24 times and records the scores. She displays her results in a vertical line chart. This is shown in Fig. 2.1. \begin{figure}[h] \end{figure}

1. Describe the shape of the distribution. She repeats the experiment, but this time she rolls the die 50 times. Her results are displayed in Fig. 2.2. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Scores on a six-sided die} \includegraphics[alt={},max width=\textwidth]{2b9ce212-84e2-4817-be94-98e2adff12a3-03_476_1161_1617_242}
   \end{figure} Fig. 2.2 Her brother Kai rolls the same die 1000 times and displays his results in a similar diagram.
2. Assuming the die is fair, describe the distribution you would expect to see in Kai's diagram.

3 The histogram shows the amount spent on electricity in pounds in a sample of households in March 2023. \includegraphics[max width=\textwidth, alt={}, center]{8e48bbd3-2166-49e7-8906-833261f331ca-04_542_1276_1133_244}

1. Describe the shape of the distribution. A total of 16 households each spent between \(\pounds 60\) and \(\pounds 65\) on electricity.
2. Determine how many households were in the sample altogether.

11 Which of the terms below best describes the distribution represented by the boxplot shown in Figure 1? \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Circle your answer.
even
negatively skewed
positively skewed
symmetric

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 the Paris-Roubaix cycling race, there are a number of sections of cobbled road. The lengths of these sections, measured in metres, are illustrated in the histogram. \includegraphics{figure_1}

1. Find the number of sections which are between 1000 and 2000 metres in length. [2]
2. Name the type of skewness suggested by the histogram. [1]
3. State the minimum and maximum possible values of the midrange. [2]

The boxplot below represents the time spent in hours by students revising for a history exam. \includegraphics{figure_16}

1. Use the information in the boxplot to state the value of a measure of central tendency of the revision times, stating clearly which measure you are using. [1 mark]
2. Use the information in the boxplot to explain why the distribution of revision times is negatively skewed. [1 mark]

The box plot below shows summary data for the number of minutes late that buses arrived at a rural bus stop. \includegraphics{figure_12} Identify which term best describes the distribution of this data. Circle your answer. [1 mark] negatively skewed \quad\quad normal \quad\quad positively skewed \quad\quad symmetrical