HISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS

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1 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 1 of 25 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier HISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS Version: 2.4 Date:

2 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 2 of 25 Histograms. This section deals with two additional ways of representing data; the histogram and the cumulative frequency diagrams. Example (1): A police officer has measured the speeds of 100 cars on a mobile speed camera on the hard shoulder of a part of the M60 motorway where the speed limit is 70 mph. The table of results is shown below. Speed (v) in mph Frequency 50 v v 5 60 v v v v v v 4 Two pupils, Amy and Beth, produced bar charts to illustrate the data. Amy used the police data in its original grouped form, to produce this bar chart: Notice the strong concentration of cars recorded travelling at mph, and the rapid tail-off at the higher, and especially the lower, speeds.

3 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 3 of 25 Beth, on the other hand, had regrouped the frequencies for all the cars driven slower than 65 mph, and those driven faster than 80 mph. Speed (v) in mph Frequency 50 v v v v v 13 Beth s bar chart appears to show a more uniform height for the bars, because of the way she had grouped the data.

4 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 4 of 25 Histograms versus bar charts. The bar charts from Example(1) appear to be different in shape, despite the fact that they represent the same data. Amy s bar chart shows a strong concentration of cars recorded at mph, with rapid tail-offs towards the higher and lower speeds. By contrast, Beth s bar chart seems to be more rectangular in shape with less concentration towards the middle range of speeds. Because of this, we have to use histograms to represent such data without bias. A histogram is a special type of bar chart, with the following characteristics: The frequency of the data is denoted by the area of each bar rather than by its height The bars are not separated The width of the bar corresponds to the width of the class interval The data is numeric, but can be continuous or discrete. A histogram is not merely a bar chart with the gaps between bars removed. Because frequency is now represented by area rather than height, we introduce the idea of frequency density. This can be obtained by dividing the actual frequency by the class width. Going back to the original police data, we produce this new table and histogram. Speed (v) in mph Frequency, f Class width, w Frequency density, f/w 50 v v v v v v v v The gaps between the bars have been removed, and the vertical axis now shows frequency density, but the shape of the distribution is no different from that of Amy s original bar chart.

5 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 5 of 25 Example (2) : Produce a histogram from Beth s data for motorway speeds. We calculate frequency densities, remembering that the class intervals are now of varying widths. Speed (v) in mph Frequency, f Class width, w Frequency density, f/w 50 v v v v v The resulting histogram is shown below. This histogram is less detailed than Amy s, due to the crude grouping of the data, but the overall shape is still the same. Notice how the mph and mph bars have the same area, given that they each represent 18 cars, even though their linear proportions are different.

6 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 6 of 25 Example (3): A garage issued a pledge to its customers whereby any brand new car purchased from them could be returned to have any faults fixed free of charge within the first year, under guarantee. The number of customer returns based on the time after purchasing the car was recorded below. Months after purchase Recalls Draw a histogram to represent the data. The data is continuous here, so the 0-1 class width is 1, the 4-6 class width is 2, and so on. Months after purchase Recalls (frequency), f Class width, w Frequency density, f/w

7 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 7 of 25 Example (4): The table below shows the distribution of marks (rounded to nearest unit) for 96 pupils taking Maths Paper 1. Plot the results on a histogram. Range of marks Frequency Frequency density Because the marks form a set of rounded discrete data, the true class intervals have a width of 10, such that the class is actually 49.5 (mark) < We therefore need to divide all the frequencies by 10 to get the frequency densities. Note how there is a peak at the modal 50-59% class interval and the intervals on either side of it. Because we are rounding to the nearest unit, the 50-59% class interval is strictly 49.5 (mark) < 59.5, but this is not an issue in this particular example.

8 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 8 of 25 Example (5): Plot a similar histogram to that in Example(4) for Maths Paper 2. How does it differ from the previous one? Range of marks Frequency Frequency density This time the modal range is now 60-69%, with less of a concentration about the centre than there we had with Paper 1. There also seems to be a skew towards the higher marks. There are ways of expressing this spread, called variance and standard deviation, but this is outside the scope of GCSE.

9 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 9 of 25 Example(6): An athletics club has analysed the results of the women s javelin throwing events over a season, and the following results obtained: Draw and describe the histogram corresponding to the results below: Distance (d) 30 d d d d d d 70 in metres Frequency Note that the data is continuous here, and also that the class intervals are of differing widths. We therefore need to work out frequency densities as follows: Distance (d) 30 d d d d d d 70 in metres Frequency Class width Frequency density

10 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 10 of 25 Cumulative frequency. Another method of displaying grouped data is the cumulative frequency diagram, which keeps a running total of frequencies. Example (7): Plot a cumulative frequency graph for the results of Maths Paper 1. Range of marks Frequency When we look at the frequency table, we can see that 1 pupil scored up to 9%, or 2 pupils scored up to 19%, or 4 pupils scored up to 29%, and so on until 96 pupils scored up to 99%. We therefore put the running totals in a cumulative frequency row as follows: Marks Frequency Cumulative frequency The results can then be put into a graph as shown below, joining points with a smooth curve. Note how cumulative totals are lined up at 9.5, 19.5, 29.5 and so forth on the data axis. This is because the class interval (say) can take in all values from 39.5 to 49.5 (when rounded).

11 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 11 of 25 Example (8): Plot a corresponding cumulative frequency graph for the results of Maths Paper 2. Compare and contrast it with the previous graph. Marks Frequency Firstly we work out the cumulative frequency at the end of each class interval : Marks Frequency Cumulative frequency Finally, plot the cumulative frequencies (2 for 0-9, 7 for 10-19, 15 for ) against 9.5, 19.5, on the data axis. The cumulative frequency graph has a somewhat gentler slope overall, particularly around the median, suggesting a lesser concentration of marks around it.

12 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 12 of 25 Interpreting cumulative frequency graphs finding the median and quartiles. We will now look at how to read off information from a cumulative frequency graph, in this case the results of Maths Paper 1 from Example (7). Additionally, we are told that the lowest mark was 7 and the highest mark was 95. There are marks for 96 pupils in all, so the median will correspond to the 48 th mark out of the 96. When using cumulative frequency graphs, we are not pedantically concerned with there being two middle numbers, the 48 th and 49 th. This is because we can, in any case, only estimate the median as we do not have the original data items. We can find the median by drawing a horizontal line along the halfway point (48) of the cumulative frequency axis to meet the curve, and then drawing a vertical from the curve to the data axis, meeting it at about 56 the estimated median. Other statistical points are the upper and lower quartiles. Those points are three quarters and one quarter of the way up the cumulative frequency axis, and can be found in a similar way to the median. The upper quartile corresponds to the 72 nd result, and meets the data axis at 65%. The lower quartile corresponds to the 24 th result, and meets the data axis at 46%. The inter-quartile range is the difference between the upper and lower quartiles, namely 19%.

13 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 13 of 25 The Box and Whisker Plot. The diagram at the bottom of the cumulative frequency plot is what is known as a box-and whisker plot, showing the quartile and median in a box and the two extreme values as whiskers. The vertical lines in the box-and-whisker plot coincide with the crucial vertical lines in the graph.. Box-and-whisker plots can show irregularities in a distribution, namely positive and negative skew. There are four main distribution patterns to recognise at GCSE; normal, rectangular, positive skew and negative skew. The histograms, cumulative frequency graphs and box-and-whisker plots are shown on the next two pages for comparison. Most histograms at GCSE display a symmetrical or normal distribution, characterised by a bellshaped histogram and a balanced cumulative frequency diagram and box plot, where the quartiles are closer to the median than they are to the extreme values. In addition, there is little difference between the mean and the median. In a rectangular distribution, there is no concentration of marks about the centre, all the columns of the histogram are equal in height, the cumulative frequency curve is a straight line, and the quartiles are equally distant from the median and the extremes. Again, the mean and median are practically coincident.

14 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 14 of 25 This time, we have an asymmetrical distribution where the median is less than the mean, and is closer to the lower quartile. This is evident in the histogram where the highest frequencies occur to the left of the central value. In addition, the whiskers are unequal in length. This is the mirror case to the previous one here, the median is greater than the mean, and is closer to the upper quartile, with the whiskers again unequal in length. We can see that the results for Maths Paper 1 show a symmetrical distribution overall. The length of the box compared with that of the whiskers can also give information about the spread of the data.

15 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 15 of 25 Example (9): Find the median, upper quartile, lower quartile and inter-quartile range for the cumulative frequency graph from Example (8) the Maths Paper 2 results. Also draw the box-andwhisker plot, assuming the lowest and highest marks are 14 and 97. The inter-quartile range is rather higher than with Maths Paper 1-24% compared to 19%. Also, the median and upper quartile are considerably higher than those for Maths Paper 1, and there seems to be a hint of negative skew.

16 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 16 of 25 Example (10): Plot a cumulative frequency graph for the women s javelin results. In addition, draw a box-and-whisker plot for the same data, given that the shortest distance thrown is 32 m and the longest 64 m. Describe the distribution. Data are as for Example (6). Distance (d) 30 d d d d d d 70 in metres Frequency Cumulative Frequency Because the data is assumed to be continuous, we line up the points at 30, 35, on the data axis. The distribution appears to have appreciable positive skew.

17 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 17 of 25 Example 10(a) : A throw in the women s javelin is considered by the AAA to be of Grade 1 standard if the length of throw is greater than 41 metres. How many of the throws in Examples (6) and (10) qualified for that standard, and what percentage of the total is that? It can be estimated from the cumulative frequency diagram that 22 out of the 60 throws recorded were less than or equal to 41 metres in distance. Hence an estimated (60 22), or 38, throws were of AAA Grade 1 standard. That equates to 63% of the total.

18 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 18 of 25 Estimating the median and quartiles from a histogram, without a cumulative frequency plot. Sometimes we might be asked to estimate the median and/or quartiles of a set of values directly from a histogram, without using a cumulative frequency plot. Example (11) (Recall Examples (6) and (10)) Once again, we have the table of women s javelin results, along with the histogram. Distance (d) 30 d d d d d d 70 in metres Frequency Freq.Density Cumulative Frequency It is not vital to complete the cumulative frequency table, but it is helpful when looking for the median and the quartiles. Estimate the median and quartiles for the resulting histogram.

19 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 19 of 25 To visualise the median on a histogram, we imagine a vertical line dividing the histogram into two equal areas. Because there are 60 values in all, the dividing line would occur at the point that the cumulative frequency reaches the halfway stage, i.e. on the 30 th result. The cumulative frequency reaches 18 at the end of the 35 d 40 class interval, i.e. at 40 metres, and 38 at the end of the 40 d 45 interval, or at 45 metres. The 30 th result would therefore lie in the 40 d 45 interval, which contains the 20 data items from the 19 th to the 38 th. In other words, the 30 th result 12, or 3 5, of the way, along is 20 the 40 d 45 interval, by interpolation. Now 5 3 of 5 (the class width) = 3, so the median is approximately , or 43 metres. (Actual frequencies are shown at the top of the columns, along with the split column containing the median.) The quartiles can be estimated in a similar way. Thus, the lower quartile or 15 th 7 result will be 10 way along the 35 d 40 class interval, or at of 5, or 38.5 metres (see lower left). of the The upper quartile or 45 th 7 result will be 10 i.e. at of 5, or 48.5 metres (see lower right). of the way along the 45 d 50 interval,

20 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 20 of 25 Sometimes an exam question on histograms will provide only skeleton data. Example (12): The following data apply to potato yields per hectare on 60 farms in Cheshire in Yield per hectare, y, in tonnes Frequency 20 y y y y y y 80 i) Use the data in the histogram to complete the table. ii) Use the data in the table to complete the histogram. iii) Estimate the number of farms whose potato yields were 35 tonnes per hectare or less. i) Firstly, we need to find an item of data common to both the histogram and the table to fix the vertical frequency density scale. The class interval 30 y 40 is common to both, and as its class width is 10 and its frequency is 12, its frequency density is therefore 1.2. We can now enter the values 0.0, 1.0, 2.0, 3.0 and 4.0 on the vertical frequency density tick marks. Next, we can read off the frequency density for the 20 y 30 class. Its value of 0.3 coupled with the class interval width of 10 corresponds to a frequency of 3. Similarly, for the 40 y 45 class, the frequency density reads as 3.6, which equates to an actual frequency of 18 given the class width of 5. The only missing entry for frequency for the table is that for the 60 y 80 interval. All the other frequencies sum to 56, and there are 60 farms in all, so the missing frequency must therefore be 4.

21 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 21 of 25 The completed data table now looks like this (calculated values in bold): Yield per hectare, y, in tonnes Frequency Class width Frequency density 20 y y y y y y ii) We now have all the required data to complete the histogram. iii) To estimate how many potato farms yielded 35 tonnes or less per hectare, we have to include: a) the whole of the 20 y 30 class, or 3 farms, plus b) half of the 30 y 40 class, or an estimate of 6 farms, since 35 is midway between 30 and 40. This gives an estimate of 9 farms in total yielding 35 tonnes of potatoes or less per hectare.

22 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 22 of 25 The next example is an omnibus exam-style question taking in a variety of points. Example(13): (Same data as Example (7) in Averages ) An electrical retailer has analysed the money spent by customers in the first 500 transactions in the January sales, excluding purchases greater than Spending, x, in Frequency Frequency density x < x x x x x 1000 (i) Complete the data table and partial histogram above. (ii) Use the histogram produced in (i) to estimate the median and the inter-quartile range. (i) First, we find the ratio between the frequency density and the actual frequency. It can be seen from the values in the row from the 0-50 class interval that the frequency density is given by dividing the actual frequency (70) by the class interval width (50). From there, we can find the actual frequency for the class interval, namely = 110. The class interval has a width of 100, so its frequency density is or The class interval has a width of 200, so its frequency density is 200 or 0.4. Similar working gives a frequency density of 0.3 for the class interval. We are told that 500 transactions have been analysed, and therefore the frequency for the class interval is the sum of all the previous frequencies subtracted from 500, or 80.

23 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 23 of 25 The completed frequency table and histogram therefore look like this, with a class width column added for information: Spending, x, in Frequency Class Width ( ) Frequency density x < x x x x x

24 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 24 of 25 To find the median and quartiles, and hence the inter-quartile range, we reckon up the appropriate areas on the histogram., using linear interpolation as shown in the diagram below Because there are 500 transactions in total, the median occurs on the 250 th transaction and the lower and upper quartiles on the 125 th and 375 th transactions. Scanning the histogram from left to right, the 70 th transaction occurs at the end of the 0-50 class interval and the 180 th at the end of the interval. The 125 th is exactly halfway along the interval, so the lower quartile is 75. Continuing the scan, the interval includes the 181 st to the 280 th transactions the 250 th one is seven-tenths of the way along, so the median is 170. Finally, the interval includes the 361 st to the 420 th transactions the 375 th is one-quarter of the way along, and so the upper quartile is 450. Therefore the median value of the transactions is 170, and the inter-quartile range is (450-75) or 375. Note the positive skew of the data the median of 175 is less than the mean of 286 calculated in Example 7 of the section Averages.

25 Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 25 of 25 More on Box Plots. Many examination questions concern comparative box plots. Example (14): The box plots below show the judges scoring trends for two weeks of the Strict Dance TV show over the last four years, with 60 results analysed in total for the first week of the show, and 40 results for the sixth week of the show. The totals of the judges scores are all whole numbers. i) Compare and contrast the box plots for the two weeks, with at least two supporting statements. ii) Jay assumes that 15 people scored more than 28 points in Week 1, and that 10 people scored 27 or less in Week 6. Is he correct? i) Here are a few possible answers: All the key values (extremes, quartiles and median) were higher in Week 6 than in Week 1, suggesting an overall score improvement across the board. For example, the median was up by 6 points. The inter-quartile range was lower in week 6 (6 points) than in week 1 (8), which suggests greater consistency in the middle half of the table. The overall range had not changed in the intervening five weeks. This could be due to outliers, but there is no actual data to confirm or deny that. ii) The upper quartile score for Week 1 was 28, suggesting that a quarter of 60, or 15, scores were higher than that. By similar reasoning, the lower quartile score for Week 6 was 27, suggesting that a quarter of 40, or 10 scores, were equal or lower. The only misgiving about Jay s assumption is the possibility of tied scores in the neighbourhood of the quartiles.

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