For each statement about misleading graphs and charts, write down whether it is True or False.
(i)A pictogram gives a fair impression of the data only if every symbol represents the same fixed amount, with any partial symbol drawn strictly in proportion.(1)
(ii)A pie chart drawn with a 3D tilt effect always represents exactly the same information as the equivalent flat (2D) pie chart, just in a different style.(1)
(iii)In a histogram with unequal class widths, plotting frequency (instead of frequency density) on the vertical axis will always give a fair impression of how concentrated the data is in each class.(1)
(iv)Two line graphs plotted from identical data can look very different if one graph stretches its vertical axis and the other does not, even though none of the underlying figures have changed.(1)
(Total for Question 1 is 4 marks)
2
A student is investigating whether a histogram drawn with unequal class widths gives a fair impression of some grouped data.
List four checks the student could use to help decide whether the histogram might be misleading. At least one of your checks must relate specifically to histograms with unequal class widths.
(Total for Question 2 is 4 marks)
3
Two garden centres, Oakdale and Ferrytown, recorded their weekend plant sales (number of plants sold). Oakdale sold 64 plants and Ferrytown sold 70 plants. The bar chart shows this data, with the vertical axis starting at 60.
(a)Write down the number of plants sold at Oakdale and at Ferrytown.(1)
(b)Explain why this bar chart could give a misleading impression of the difference in sales between the two garden centres.(2)
(c)Suggest one specific change the garden centres could make to redraw this chart fairly, and explain what effect this change would have on how the bars compare.(2)
(Total for Question 3 is 5 marks)
4
A phone company records the number of app downloads (thousands) over five months.
Month
Jan
Feb
Mar
Apr
May
Downloads (1000s)
20
22
24
26
29
Graph R plots this data with the five months spaced equally along the horizontal axis. Graph S plots the same data and the same vertical axis, but the horizontal gap between April and May is drawn much narrower than the equal gaps used between the earlier months.
(a)Work out the percentage increase in downloads from April to May.(2)
(b)Explain how compressing the horizontal gap between April and May, while keeping the same vertical axis, could make this increase look more dramatic than it really is.(2)
(c)Suggest how the company should space the months on the horizontal axis to give a fair impression of the trend, and explain why.(2)
(Total for Question 4 is 6 marks)
5
A charity draws a poster to show how much money it raised in two years, using coin pictures. In Year 1 it raised £2,000, shown by a coin of radius 1 cm. In Year 2 it raised £6,000 (three times as much), shown by a coin of radius 3 cm (three times the radius).
(a)Using π = 3.14, work out the area of the Year 1 coin (radius 1 cm).(2)
(b)Using π = 3.14, work out the area of the Year 2 coin (radius 3 cm).(2)
(c)Compare the ratio of the two coins' areas to the ratio of the two amounts raised, and explain why scaling the radius like this could mislead a reader.(2)
(d)Suggest a fairer way the charity could redraw the pictogram so that the visual impression matches the data, and explain why your method is fairer.(2)
(Total for Question 5 is 8 marks)
6
A survey of 200 cafe customers found their favourite drink: Tea (80 customers), Coffee (60 customers), Juice (40 customers), Water (20 customers). A magazine draws a 3D pie chart of this data, printing the percentages 45%, 35%, 25% and 15% next to the Tea, Coffee, Juice and Water sectors.
(a)Work out the correct percentage of customers whose favourite drink was Coffee.(2)
(b)The magazine's pie chart shows the percentages 45%, 35%, 25% and 15% for Tea, Coffee, Juice and Water. Show that these percentages do not sum to 100%, and state the total they do sum to.(2)
(c)The magazine's chart is also drawn in 3D with a tilt. Referring to both faults, explain why this pie chart could give a misleading impression of the survey results.(2)
(d)Suggest two specific corrections the magazine should make before it republishes this pie chart.(2)
(Total for Question 6 is 8 marks)
7
A record shop compares sales last month: Pop albums sold 40 copies, Jazz albums sold 20 copies. On a promotional poster, both bars are drawn to the correct heights, but the Pop bar is also drawn twice as wide as the Jazz bar.
(a)Write down the ratio of albums sold, Pop to Jazz, in the form n : 1.(1)
(b)The Pop bar is drawn twice as wide as the Jazz bar, as well as twice as tall. Work out the ratio of the areas of the two bars.(2)
(c)Explain why making the bars different widths, as well as different heights, could mislead a reader even though the heights themselves are drawn correctly.(2)
(Total for Question 7 is 5 marks)
8
60 runners' finishing times (minutes) in a fun run were grouped into classes with different widths.
Time (minutes)
20-30
30-40
40-60
60-100
Frequency
12
18
20
10
A poster for the event plots these frequencies directly as bar heights on a chart, with time on the horizontal axis, so wider classes are also drawn as wider bars, without adjusting for the different class widths.
(a)Show that the frequency densities for the 20-30 class and the 60-100 class are 1.2 and 0.25 respectively.(2)
(b)Explain why plotting frequency (rather than frequency density) as the bar height on this chart, given the classes have different widths, gives a misleading impression of how finishing times are distributed.(2)
(c)Work out the correct frequency density for the 40-60 class, and use frequency densities to make a fair comparison between the 40-60 and 60-100 classes.(2)
(d)State what should be plotted on the vertical axis instead, to draw this histogram correctly.(1)
(Total for Question 8 is 7 marks)
9
A small business tracks its weekly profit (£1000s) over six weeks: 10, 11, 12, 10, 13, 25. Graph U plots this data using a tall, narrow set of axes. Graph V plots the identical data and values using a short, wide set of axes.
(a)Work out the percentage increase in profit from Week 5 to Week 6.(2)
(b)Explain how stretching the vertical axis of a graph, while keeping the horizontal axis the same, can change the apparent steepness of a trend without changing any of the underlying data.(2)
(c)The two graphs use the same data but different aspect ratios (height compared with width). Suggest one way a reader could check whether a line graph's apparent trend is a fair reflection of the data, rather than an effect of the aspect ratio chosen.(2)
(Total for Question 9 is 6 marks)
10
Priya is investigating whether adverts for phone data plans use misleading bar charts to make their own network look the best value.
(a)Plan: Suggest one thing Priya should decide before she starts collecting her sample of adverts, and give a reason for your suggestion.(2)
(b)Collect: Priya records, for each of her 15 sampled adverts, whether the bar chart's vertical axis starts at zero. Her results: Yes - 6 adverts, No - 9 adverts. Write down the fraction of adverts in her sample where the axis did not start at zero, in its simplest form.(1)
(c)Process: Work out this fraction as a percentage.(2)
(d)Interpret: Priya concludes 'Most phone data plan adverts are misleading, because their bar chart's vertical axis doesn't start at zero.' Comment on whether her sample is likely to be reliable evidence for this general conclusion.(3)
(Total for Question 10 is 8 marks)
11
A wellness blog plots two data sets over five hours: Temperature (degrees C) - 10, 20, 30, 40, 50; Cold drinks sold - 20, 40, 60, 80, 100. The blog uses a left-hand vertical axis from 0 to 50 for temperature and a separate right-hand vertical axis from 0 to 100 for drinks sold. Drawn this way, the two lines lie exactly on top of each other.
(a)Using the table, work out how many times greater the number of drinks sold is than the temperature value, at the 10 degree reading.(2)
(b)Explain why drawing the two data sets so their lines lie exactly on top of each other, using two different vertical axes, could mislead a reader.(2)
(c)Suggest a fairer way to present this data so a reader could judge the relationship between temperature and drinks sold without being misled by the matching lines.(2)
(Total for Question 11 is 6 marks)
12
A gym advertises its membership growth using a bar chart of new members joining each quarter: Q1 = 15, Q2 = 18, Q3 = 20, Q4 = 45. The chart's vertical axis starts at 12 (not zero), and the Q4 bar is also drawn twice as wide as the other three bars 'to fit the label'.
(a)Work out the percentage increase in new members from Q3 to Q4.(2)
(b)Identify two separate faults in how this chart is drawn that could exaggerate the growth shown in Q4, referring to specific features of the chart.(2)
(c)Describe exactly how you would redraw this chart to give a fair impression of new member growth, referring to both the vertical axis and the bar widths.(3)
(Total for Question 12 is 7 marks)
13
A company shows profit for two branches using 3D cube icons on a poster. Branch A made £8,000 profit, shown as a cube of side length 2 cm. Branch B made £16,000 profit (twice as much), shown as a cube of side length 4 cm (twice the side length).
(a)Work out the volume of the Branch A cube (side length 2 cm).(1)
(b)Work out the volume of the Branch B cube (side length 4 cm).(1)
(c)Compare the ratio of the two cubes' volumes to the ratio of the two branches' profits, and explain why scaling every side of the cube like this could give a very misleading impression.(2)
(d)Suggest a fairer way to show this data using icons, and justify your suggestion using a calculation.(3)
(Total for Question 13 is 7 marks)
14
A property website groups the prices (£1000s) of 80 houses sold in a town using unequal class widths.
Price (£1000s)
100-150
150-200
200-300
300-500
Frequency
20
24
24
12
The website plots these frequencies directly as bar heights, with price on the horizontal axis, without adjusting for the different class widths.
(a)Work out the frequency density for the 200-300 class and for the 300-500 class.(2)
(b)Show that, when frequency (not frequency density) is plotted as bar height, the 200-300 bar and the 300-500 bar would have the same visual area (height x width), even though they represent very different numbers of houses.(2)
(c)Use frequency density to explain which of these two classes truly has the greater concentration of house prices per £1000, and by what factor.(2)
(d)State what a correctly drawn histogram should show for these two classes, referring to bar height and bar area.(1)
(Total for Question 14 is 7 marks)
15
A gym's full sign-up data for six weeks is: Week 1 = 20, Week 2 = 21, Week 3 = 19, Week 4 = 22, Week 5 = 21, Week 6 = 25. An advert for the gym shows only Weeks 5 and 6, with a vertical axis starting at 20, and the Week 6 bar drawn one and a half times as wide as the Week 5 bar.
(a)Work out the percentage increase in sign-ups the advert's graph covers, from Week 5 to Week 6.(2)
(b)Work out the percentage increase in sign-ups over the full six weeks shown in the underlying data, from Week 1 to Week 6.(2)
(c)Identify three separate features of how the advert's graph is presented that could make the increase look more dramatic than the roughly 19-25% figures above suggest, and explain the effect of each. Then give an overall conclusion.(4)
(Total for Question 15 is 8 marks)
16
A campaign group claims: 'Our graph proves that switching to reusable bottles has cut plastic bottle use in half.' Their bar chart shows 'Plastic bottles used per household per week': Before = 6, After = 3, with the vertical axis starting at 2, and no source or sample size stated anywhere on the chart.
(a)Work out the true percentage decrease in plastic bottles used, from Before to After.(2)
(b)The claim states the graph 'proves' the reduction. Evaluate this claim, referring to (i) whether the chart's own presentation is fair, and (ii) whether a bar chart like this could ever be enough evidence on its own to 'prove' a cause-and-effect claim about an entire campaign's impact.(3)
(Total for Question 16 is 5 marks)
Mark scheme · H25 Recognising Misleading Graphs
Question 1
(i) B1 True cao
(i) Answer: True
(ii) B1 False cao
(ii) Answer: False
(iii) B1 False cao
(iii) Answer: False
(iv) B1 True cao
(iv) Answer: True
Question 2
B1 Is frequency density (frequency divided by class width), rather than frequency, plotted on the vertical axis for classes of unequal width?
B1 Does the area of each bar (not just its height) correctly represent the frequency for that class?
B1 Does the vertical axis start at zero, with equally spaced numbers marked on it?
B1 Is the chart clearly labelled with a title, units and axis labels, so the class widths cannot be misread? (oe - accept any four valid checks, e.g. whether bars are drawn to a consistent scale, or whether the sample size/source is stated)
Answer: e.g. (1) Is frequency density plotted, not frequency, for unequal-width classes? (2) Does bar area represent frequency? (3) Does the vertical axis start at 0 with equal intervals? (4) Is the chart fully labelled?
Question 3
(a) B1 Oakdale = 64 and Ferrytown = 70, both cao
(a) Answer: Oakdale = 64, Ferrytown = 70
(b) B1 identifies that the vertical axis starts at 60, not 0 (a truncated/non-zero axis)
(b) B1 explains this makes the real difference of 6 plants look much bigger than it really is, since the Ferrytown bar is drawn far taller even though it is only a small amount more
(b) Answer: The vertical axis starts at 60 instead of 0, so the small true difference of 6 plants (64 compared with 70) is stretched to fill most of the chart, making Ferrytown's sales look far higher than Oakdale's.
(c) B1 suggests starting the vertical axis at 0 (instead of 60)
(c) B1 explains that on a 0 to 72 scale the two bars would then be almost the same height (64 out of 72, and 70 out of 72), giving a fair impression that matches the true difference of about 9%, instead of looking two and a half times as big
(c) Answer: Redraw the chart with the vertical axis starting at 0. On a 0 to 72 scale, Oakdale's bar would reach 64/72 of the height and Ferrytown's would reach 70/72, so the bars would look almost the same height, fairly reflecting the true 6-plant (about 9%) difference.
Question 4
(a) M1 (29-26)/26 x 100, oe
(a) A1 awrt 11.5%
(a) Answer: 11.5%
(b) B1 identifies that squeezing the horizontal distance between April and May, while the vertical rise stays the same size, makes that part of the line much steeper on the page than the earlier, equally-spaced months
(b) B1 explains a reader could interpret this steep, near-vertical section as a sudden dramatic jump, when the true increase (about 11.5%) is only slightly bigger than the rate of increase in the earlier, fairly spaced months (each about 1 percentage point higher than the last)
(b) Answer: Compressing the April-May gap keeps the same vertical rise but squeezes it into a much shorter horizontal distance, so the line becomes much steeper there than in the rest of the graph; a reader could easily read this as a sudden, dramatic surge, when the true increase of about 11.5% is only a little larger than the roughly steady rate of increase shown in the earlier months.
(c) B1 suggests spacing all five months equally along the horizontal axis (in proportion to the equal time gap between each pair of months)
(c) B1 explains that equal time gaps mean the slope drawn between each pair of points fairly reflects the true rate of change, so no part of the trend is exaggerated relative to another
(c) Answer: Space Jan, Feb, Mar, Apr and May equally along the horizontal axis, since each represents the same one-month time gap; this way, the steepness of the line between any two months fairly reflects the true rate of change, and no single month-to-month rise is exaggerated compared with the others.
Question 5
(a) M1 3.14 x 12, oe
(a) A1 3.14 cm2 cao
(a) Answer: 3.14 cm2
(b) M1 3.14 x 32, oe
(b) A1 28.26 cm2 cao
(b) Answer: 28.26 cm2
(c) B1 correctly finds the area ratio 28.26 : 3.14 = 9 : 1, compared with the money ratio of 6000 : 2000 = 3 : 1
(c) B1 explains the Year 2 coin looks nine times as big as the Year 1 coin, even though only three times as much money was raised, exaggerating how much more successful Year 2 was
(c) Answer: The area ratio is 28.26 : 3.14 = 9 : 1, but the true amount-raised ratio is only 3 : 1; tripling the radius makes the picture look nine times bigger, so the poster exaggerates how much Year 2's fundraising improved on Year 1's.
(d) B1 valid method, e.g. keep every coin the same fixed size and draw three identical coins for Year 2 for every one coin used for Year 1, or scale the AREA of a single coin directly in proportion to the amount raised (so the area ratio is exactly 3 : 1) rather than scaling the radius
(d) B1 explains why the suggested method is fairer, e.g. repeating identical, equally-sized coins means the number of coins (not their size) shows the multiple, so the visual impression matches the data exactly
(d) Answer: Keep the coin the same fixed size throughout, and show Year 2 with three identical coins for every one coin used for Year 1; since every coin is now the same size, the number of coins shown is directly proportional to the amount raised, giving a fair visual impression.
Question 6
(a) M1 60/200 x 100, oe
(a) A1 30% cao
(a) Answer: 30%
(b) M1 45 + 35 + 25 + 15, oe
(b) A1 120%, correctly identified as not equal to 100%
(b) Answer: 120%
(c) B1 explains that because the percentages sum to 120% rather than 100%, every sector has effectively been labelled with too large a share of the whole, so the chart no longer fairly represents parts of a single complete set of 200 customers
(c) B1 explains that the added 3D tilt makes the sector drawn nearest the front of the chart look even bigger than its (already inflated) printed percentage, distorting the comparison between drinks still further
(c) Answer: Because the printed percentages add up to 120% instead of 100%, every sector's labelled share is inflated beyond its true proportion of the 200 customers, so the parts no longer fairly represent the whole; on top of this, the 3D tilt makes whichever sector faces the viewer look even larger than its stated percentage, distorting the comparison between drinks further.
(d) B1 recalculate the sector angles/percentages directly from the true frequencies (80, 60, 40, 20 out of 200) so they sum to exactly 100% (360 degrees)
(d) B1 redraw the chart as a flat 2D pie chart instead of a 3D one
(d) Answer: (1) Recalculate each sector from the true frequencies out of 200, so the percentages sum to exactly 100%. (2) Redraw the pie chart as a flat 2D chart instead of a 3D one.
Question 7
(a) B1 2 : 1 cao
(a) Answer: 2 : 1
(b) M1 (2 x 2) : (1 x 1), oe
(b) A1 4 : 1 cao
(b) Answer: 4 : 1
(c) B1 identifies that a reader's eye naturally compares the overall size (area) of bars, not just their height
(c) B1 explains the extra width makes the Pop bar look four times as big as the Jazz bar, when truly only twice as many albums were sold; every bar in a fair bar chart should be drawn the same width, with only the height varying
(c) Answer: A reader naturally judges bars by their overall size, not just their height, so making the Pop bar twice as wide as well as twice as tall makes it look four times as big as the Jazz bar, when really only twice as many albums were sold. In a fair bar chart every bar should be the same width, with only the height showing the value.
Question 8
(a) M1 12/10 and 10/40, oe
(a) A1 1.2 and 0.25 both shown correctly
(a) Answer: 1.2 and 0.25
(b) B1 identifies that because bar width also varies here, the area of each bar (not just its height) is what a reader visually judges as the amount of runners, and area = frequency x width once widths differ, not just frequency
(b) B1 explains this makes the wide 60-100 class look like it takes up a similar visual area to the 40-60 class, even though it contains only half as many runners (10, compared with 20)
(b) Answer: Because the class widths are unequal, plotting raw frequency as the bar height means the visual area of each bar is frequency multiplied by width, not frequency alone; this makes the wide 60-100 bar (10 runners, width 40) occupy a visual area similar to the 40-60 bar (20 runners, width 20), even though the 60-100 class actually has only half as many runners.
(c) M1 20/20, oe
(c) A1 1 (frequency density), correctly compared with the 60-100 class's 0.25 to show 40-60 is four times as densely packed with runners per minute
(c) Answer: The 40-60 class has frequency density 1, compared with 0.25 for the 60-100 class, so 40-60 has four times the concentration of runners per minute - confirming it is the more populous interval per minute, despite the misleading chart making the two classes look similarly sized.
(d) B1 frequency density (frequency divided by class width)
(d) Answer: Frequency density (frequency divided by class width)
Question 9
(a) M1 (25-13)/13 x 100, oe
(a) A1 awrt 92.3%
(a) Answer: 92.3%
(b) B1 explains that a taller, more stretched vertical scale makes the same rise cover a bigger visual angle, so the line looks steeper
(b) B1 explains that none of the numbers or data have changed at all; only the proportions of the drawn axes have changed, so the 'steepness' a reader sees is a drawing choice, not a fact about the data
(b) Answer: Stretching the vertical axis spreads the same numerical rise over a much bigger vertical distance on the page, so the line covers a steeper angle for the same true change; but the actual profit figures have not changed at all, so how steep the trend looks depends on the axes chosen, not on the underlying data.
(c) B1 valid method, e.g. calculate the actual percentage or numerical change between points directly from the data, rather than judging steepness by eye
(c) B1 correct supporting reason, e.g. a calculated percentage change does not depend on how the axes are drawn, so it gives a fair comparison whatever aspect ratio the graph uses
(c) Answer: Calculate the actual percentage change between the points directly from the figures, rather than judging the trend by how steep the line looks; a calculated percentage does not depend on how tall, narrow, short or wide the axes are drawn, so it gives a fair, consistent comparison.
Question 10
(a) B1 valid decision suggested, e.g. which networks or sources to include, how many adverts to sample, or what counts as a 'bar chart' for this investigation
(a) B1 valid reason linked to getting a fair, representative sample or a workable, consistent definition
(a) Answer: e.g. Priya should decide in advance exactly how many adverts and from how many different networks she will look at, so that her sample is a consistent size and is not chosen to fit whatever result she expects to find.
(b) B1 3/5 cao
(b) Answer: 3/5
(c) M1 9/15 x 100, oe
(c) A1 60% cao
(c) Answer: 60%
(d) B1 identifies a possible limitation of the sample, e.g. it may only cover a small number of networks, a short time period, or adverts from a single source, so it may not be representative of all data plan adverts
(d) B1 suggests a valid improvement, e.g. a bigger and more varied sample (more networks, more adverts, over a longer time period) would give more reliable evidence
(d) B1 explains that 60% is a statistic about Priya's sample only, and does not automatically prove a fact about every phone data plan advert that exists
(d) Answer: A sample of only 15 adverts, possibly from a small number of networks or a short time window, may not represent all phone data plan advertising; a bigger, more varied sample (more networks, more adverts, over a longer period) would give more reliable evidence. Even a correctly worked out 60% is a fact about Priya's own sample, not automatic proof about every advert that has ever been produced.
Question 11
(a) M1 20/10, oe
(a) A1 2 (times) cao
(a) Answer: 2 times
(b) B1 identifies that the two vertical axes use very different scales and units (temperature in degrees C, drinks sold as a count), chosen deliberately so the lines coincide
(b) B1 explains a reader could wrongly think the two quantities are directly linked, matched, or of similar size, without noticing the mismatched scales - drinks sold are actually always twice the temperature value, not literally equal to it
(b) Answer: The left axis (0 to 50) and right axis (0 to 100) use completely different scales and units, chosen so the two lines exactly coincide; a reader glancing at the graph could think temperature and drinks sold are closely matched or directly linked in size, without noticing that drinks sold are actually always double the temperature reading in this data set.
(c) B1 valid method, e.g. plot the paired values as a scatter graph of temperature against drinks sold, or clearly label both axes with their true scales so a reader can see they are different
(c) B1 explains this fairer method lets the reader judge the true relationship from the paired values, rather than being misled by a coincidental visual match caused by the choice of axes
(c) Answer: Plot the data as a scatter graph of temperature against drinks sold (one point per hour), rather than as two separate lines on mismatched axes; this way a reader can judge the true relationship directly from the paired values, instead of being misled by a visual overlap that was only created by choosing two very different axis scales.
Question 12
(a) M1 (45-20)/20 x 100, oe
(a) A1 125% cao
(a) Answer: 125%
(b) B1 the vertical axis starts at 12, not 0 (a truncated axis)
(b) B1 the Q4 bar is drawn wider than the other three bars (inconsistent bar width), adding extra visual area beyond the true height difference
(b) Answer: (1) The vertical axis starts at 12, not 0. (2) The Q4 bar is drawn twice as wide as the Q1, Q2 and Q3 bars, giving it extra visual area on top of its already-taller height.
(c) B1 start the vertical axis at 0
(c) B1 make every bar (including Q4) the same width
(c) B1 explains that a chart redrawn this way would show Q4's bar as taller than the others only in proportion to the true 125% increase, without being exaggerated by an extra-wide bar or a squeezed axis
(c) Answer: Redraw the chart with the vertical axis starting at 0, and make the Q4 bar the same width as Q1, Q2 and Q3. With both changes, Q4's bar would be taller than the others in proportion to the true 125% increase only, without any extra visual exaggeration from the axis or the bar width.
Question 13
(a) B1 8 cm3 cao
(a) Answer: 8 cm3
(b) B1 64 cm3 cao
(b) Answer: 64 cm3
(c) B1 correctly finds the volume ratio 64 : 8 = 8 : 1, compared with the true profit ratio of 16000 : 8000 = 2 : 1
(c) B1 explains that scaling all three dimensions of the cube by the same factor cubes the visual effect, so Branch B's cube looks eight times as big by volume even though it made only twice the profit - a bigger distortion than scaling a flat shape's area
(c) Answer: The volume ratio is 64 : 8 = 8 : 1, but the true profit ratio is only 2 : 1; doubling every side of the cube makes its volume eight times as big (since 23 = 8), so Branch B's cube looks eight times bigger even though it made only twice as much profit - a far bigger distortion than resizing a flat 2D shape would give.
(d) M1 attempts the cube root of 2 (the side-length scale factor that would double the volume), oe, or an alternative method (e.g. repeating equally-sized cubes in a ratio of 2 : 1 for Branch B against Branch A)
(d) A1 cube root of 2 awrt 1.26
(d) B1 correctly applies this to state the fairer side length, awrt 2.52 cm (= 2 x 1.26), or gives a fully justified equal-icon alternative (e.g. keep the cube a fixed size and draw two identical cubes for Branch B for every one used for Branch A, so the icon count is exactly proportional to profit)
(d) Answer: To make the volume exactly double (matching the true 2 : 1 profit ratio), the side length should scale by the cube root of 2, awrt 1.26, giving a fairer side length of about 2.52 cm (2 x 1.26) rather than 4 cm. Alternatively, keep every cube a fixed size and draw two identical cubes for Branch B for every one cube for Branch A, so the number of icons is exactly proportional to the profit.
Question 14
(a) M1 24/100 and 12/200, oe
(a) A1 0.24 and 0.06, both cao
(a) Answer: 0.24 and 0.06
(b) M1 24 x 100 and 12 x 200, oe
(b) A1 both equal 2400, correctly concluding the visual areas are equal despite frequencies of 24 and 12 (i.e. the 300-500 class represents exactly half as many houses)
(b) Answer: 24 x 100 = 2400 and 12 x 200 = 2400 - the two bars have the same visual area even though the 300-500 class contains only half as many houses (12, compared with 24) as the 200-300 class.
(c) B1 identifies the 200-300 class has frequency density 0.24, compared with 0.06 for the 300-500 class
(c) B1 0.24 / 0.06 = 4, so the 200-300 class is four times as densely packed with houses per £1000 as the 300-500 class, even though the misleading chart makes them look similarly common
(c) Answer: The 200-300 class has frequency density 0.24, four times the 300-500 class's 0.06 (0.24/0.06 = 4), so house prices are four times as concentrated per £1000 in the 200-300 range, despite the chart making the two classes look like they cover a similar area.
(d) B1 bar heights should be the frequency densities (0.24 and 0.06), so the 300-500 bar would be drawn much shorter than the 200-300 bar, and the area of each bar (not raw frequency) would then correctly represent the class's frequency
(d) Answer: The bar heights should be 0.24 and 0.06 (the frequency densities), making the 300-500 bar much shorter than the 200-300 bar; the area of each bar would then correctly represent its true frequency.
Question 15
(a) M1 (25-21)/21 x 100, oe
(a) A1 awrt 19.0%
(a) Answer: 19.0%
(b) M1 (25-20)/20 x 100, oe
(b) A1 25% cao
(b) Answer: 25%
(c) B1 axis truncation: the vertical axis starts at 20, not 0, stretching the modest gap between 21 and 25 to fill most of the height of the chart
(c) B1 unequal bar width: the Week 6 bar is drawn one and a half times as wide as the Week 5 bar, adding extra visual area beyond the true height difference
(c) B1 cherry-picked window: showing only Weeks 5 and 6 hides that sign-ups were fairly flat over the first four weeks (20, 21, 19, 22), presenting the final small uptick in isolation as if it were the whole story
(c) B1 overall conclusion that combining all three features turns a genuinely modest increase of about one-fifth to one-quarter into something that looks like a dramatic surge
(c) Answer: (1) The vertical axis starts at 20, not 0, stretching the true gap between 21 and 25 to fill most of the chart. (2) The Week 6 bar is drawn 1.5 times as wide as the Week 5 bar, adding extra visual area on top of the height difference. (3) Showing only Weeks 5 and 6 hides that sign-ups were roughly flat for the first four weeks (20, 21, 19, 22), so the small final rise is shown in isolation as if it were a sudden breakthrough. Together, these three choices turn a genuinely modest 19-25% increase into what looks like a dramatic surge in growth.
Question 16
(a) M1 (6-3)/6 x 100, oe
(a) A1 50% cao
(a) Answer: 50%
(b) B1 comments on the presentation: the axis starting at 2 stretches the true (correct) 50% difference further than necessary, and with no source or sample size given, a reader cannot judge how many households, or which households, this data came from
(b) B1 comments on proof: even a perfectly fair chart of a small or unspecified sample cannot 'prove' cause and effect for an entire campaign, since other explanations (e.g. fewer people buying bottled drinks generally, seasonal changes, or an unrepresentative sample) could also explain the fall
(b) B1 gives a balanced overall conclusion: the 50% figure may well be accurate, but both the chart's presentation and the word 'proves' overstate what a single, unsourced bar chart can honestly establish
(b) Answer: The true decrease is a genuine 50%, but the chart's truncated axis (starting at 2) exaggerates this beyond what is needed, and without a stated source or sample size a reader cannot judge how reliable or representative the data is. Even if the chart were drawn perfectly fairly, a single before-and-after bar chart cannot 'prove' that the campaign caused the fall, since other factors could also explain it. The 50% figure may well be true, but the claim of 'proof' overstates what this chart can honestly demonstrate.