Describe shape or skewness of distribution

A question is this type if and only if it asks the student to identify or name the type of skewness (positive, negative) from a graph or summary statistics.

10 questions

OCR MEI S1 Q3
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.
    Age2030405065100
    Cumulative frequency (thousands)66012401810\(a\)24902770
    (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)1120650770590680610
  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.
OCR MEI S1 Q7
4 marks
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.
    Age2030405065100
    Cumulative frequency (thousands)66012401810\(a\)24902770
    (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)1120650770590680610
  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]
OCR MEI S1 Q7
4 marks
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.
    Age2030405065100
    Cumulative frequency (thousands)66012401810\(a\)24902770
    (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)1120650770590680610
  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]
OCR MEI AS Paper 2 2021 November Q2
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]
\includegraphics[alt={},max width=\textwidth]{2b9ce212-84e2-4817-be94-98e2adff12a3-03_534_1168_648_242} \captionsetup{labelformat=empty} \caption{Fig. 2.1}
\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.
OCR MEI Paper 2 2024 June Q3
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.
OCR MEI Paper 2 Specimen Q16
16 Fig. 16.1, Fig. 16.2 and Fig. 16.3 show some data about life expectancy, including some from the pre-release data set. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Life expectancy at birth 1974 for 193 countries} \includegraphics[alt={},max width=\textwidth]{e9f3a5f3-210b-453d-9ff5-8518340f5689-11_520_1671_479_182}
\end{figure} Life expectancy at birth 2014 for 222 countries \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Fig. 16.1} \includegraphics[alt={},max width=\textwidth]{e9f3a5f3-210b-453d-9ff5-8518340f5689-11_589_1477_1324_349}
\end{figure} Fig. 16.2
\includegraphics[max width=\textwidth, alt={}, center]{e9f3a5f3-210b-453d-9ff5-8518340f5689-11_207_828_2213_310} Increase in life expectancy from 1974 to 2014 (years) \begin{table}[h]
Increase in life expectancy for
193 countries from 1974 to 2014
Number of values193
Minimum- 4.618
Lower quartile6.9576
Median9.986
Upper quartile15.873
Maximum30.742
\captionsetup{labelformat=empty} \caption{Fig. 16.3}
\end{table} Source: CIA World
Factbook and
  1. Comment on the shapes of the distributions of life expectancy at birth in 2014 and 1974.
    1. The minimum value shown in the box plot is negative. What does a negative value indicate?
    2. What feature of Fig 16.3 suggests that a Normal distribution would not be an appropriate model for increase in life expectancy from one year to another year?
    3. Software has been used to obtain the values in the table in Fig. 16.3. Decide whether the level of accuracy is appropriate. Justify your answer.
    4. John claims that for half the people in the world their life expectancy has improved by 10 years or more.
      Explain why Fig. 16.3 does not provide conclusive evidence for John's claim.
  2. Decide whether the maximum increase in life expectancy from 1974 to 2014 is an outlier. Justify your answer. Here is some further information from the pre-release data set.
    Country
    Life expectancy
    at birth in 2014
    Ethiopia60.8
    Sweden81.9
    1. Estimate the change in life expectancy at birth for Ethiopia between 1974 and 2014.
    2. Estimate the change in life expectancy at birth for Sweden between 1974 and 2014.
    3. Give one possible reason why the answers to parts (i) and (ii) are so different. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Fig. 16.4 shows the relationship between life expectancy at birth in 2014 and 1974.} \includegraphics[alt={},max width=\textwidth]{e9f3a5f3-210b-453d-9ff5-8518340f5689-13_981_1520_333_292}
      \end{figure} Fig. 16.4 A spreadsheet gives the following linear model for all the data in Fig 16.4.
      (Life expectancy at birth 2014) \(= 30.98 + 0.67 \times\) (Life expectancy at birth 1974) The life expectancy at birth in 1974 for the region that now constitutes the country of South Sudan was 37.4 years. The value for this country in 2014 is not available.
    1. Use the linear model to estimate the life expectancy at birth in 2014 for South Sudan.
    2. Give two reasons why your answer to part (i) is not likely to be an accurate estimate for the life expectancy at birth in 2014 for South Sudan.
      You should refer to both information from Fig 16.4 and your knowledge of the large data set.
  3. In how many of the countries represented in Fig. 16.4 did life expectancy drop between 1974 and 2014? Justify your answer.
Edexcel S1 Q1
  1. Briefly describe what is meant by
    1. a statistical model,
    2. a refinement of a model.
    3. The random variable \(X\) has the discrete uniform distribution and takes the values \(\{ 1 , \ldots , n \}\). The standard deviation of of \(X\) is \(2 \sqrt { 6 }\). Find
    4. the mean of \(X\),
    5. \(\mathrm { P } \left( 3 \leq X < \frac { 1 } { 2 } n \right)\).
    6. The rainfall at a weather station was recorded every day of the twentieth century. One year is selected at random from the records and the total rainfall, in cm , in January of that year is denoted by \(R\) Assuming that \(R\) can be modelled by a normal distribution with standard deviation \(12 \cdot 6\), and given that \(\mathrm { P } ( R > 100 ) = 0 \cdot 0764\),
    7. find the mean of \(R\),
    8. calculate \(\mathrm { P } ( 75 < R < 80 )\).
    9. 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 visitors1524\(x\)1310\(y\)
  2. State the upper class boundary of the first class. A histogram is drawn to represent this data. The total area under the histogram is \(36 \mathrm {~cm} ^ { 2 }\). 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.
  3. Find the values of \(x\) and \(y\).
  4. On graph paper, construct the histogram accurately.
AQA AS Paper 2 2022 June Q11
11 Which of the terms below best describes the distribution represented by the boxplot shown in Figure 1? \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{11168e8f-5ba5-4d27-83ab-0327cc23d08c-14_154_831_927_584}
\end{figure} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{11168e8f-5ba5-4d27-83ab-0327cc23d08c-14_76_1143_1151_450}
\end{figure} Circle your answer.
even
negatively skewed
positively skewed
symmetric
AQA AS Paper 2 Specimen Q16
2 marks
16 The boxplot below represents the time spent in hours by students revising for a history exam.
\includegraphics[max width=\textwidth, alt={}, center]{f2bf5e19-98ba-4047-9023-3cfe20987e01-18_373_753_427_778} 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.
    [0pt] [1 mark] 16
  2. Use the information in the boxplot to explain why the distribution of revision times is negatively skewed.
    [0pt] [1 mark]
AQA Paper 3 2022 June Q12
1 marks
12 The box plot below shows summary data for the number of minutes late that buses arrived at a rural bus stop.
\includegraphics[max width=\textwidth, alt={}, center]{6ad3bac9-bf08-443d-8be2-b0c26209ffe8-17_515_1614_1384_212} Identify which term best describes the distribution of this data.
Circle your answer.
[0pt] [1 mark]
negatively skewed
normal
positively skewed
symmetrical