Worked example · Easy
Read a bar chart
A bar chart shows frequencies 6, 9, 5 and 10 for four clubs. Find the total and the most popular club.
Total \(=6+9+5+10=\boxed{30}\). The fourth club is most popular, with \(\boxed{10}\) members.
Chapter 46
Statistical Diagrams
Learn how to draw and interpret the statistical diagrams, estimate results correctly, and communicate conclusions accurately.
By the end of this chapter, you should be able to
- read and construct bar charts, dual/composite bars, pictograms and pie charts;
- interpret strong, weak, positive, negative and zero correlation in scatter diagrams;
- draw a sensible line of best fit and recognise unreliable extrapolation;
- construct and use cumulative-frequency tables and curves;
- estimate medians, quartiles, percentiles and interquartile ranges;
- construct and interpret histograms of unequal-width intervals using frequency density;
- calculate an estimated mean where grouped data or histogram areas are supplied.
1
Subtopic 1
Bar charts and pictograms
Bar charts display frequencies for categories or discrete groups. The horizontal axis shows categories and the vertical axis shows frequency. Bars usually have equal widths and clear gaps because each category is separate.
With dual (comparative) bar charts, two adjacent bars represent groups in the same category. Composite (stacked) bar charts stack the groups, showing the total and the composition simultaneously. Always use a labelled scale and a key.
A pictogram uses symbols to represent frequencies. First read the key: a half-symbol represents half the stated frequency. Never assume that one drawing means one observation.
Reading accurately: For a stacked bar, the height of a coloured segment is the difference between its top and bottom values; the full height is the category total.
Worked example · Medium
Interpret a pictogram
One complete symbol represents 4 people. A category contains 6 full symbols and a half-symbol. How many people are in it?
\[(6\times4)+\tfrac12(4)=24+2=\boxed{26}\]
Worked example · Medium
Compare two groups
For three weeks, a shop sells 12, 17 and 11 laptops and 8, 13 and 16 tablets. How many more laptops than tablets were sold in total?
\[\text{Laptops}=12+17+11=40\]\[\text{Tablets}=8+13+16=37\]\[\text{Difference}=40-37=\boxed3\]
Worked example · Exam
Read a stacked chart
A composite bar chart for a survey has two stacked groups. One bar reaches 42 altogether, with the lower segment ending at 18. Find the upper-group frequency and the fraction of the total.
\[\text{Upper group}=42-18=24\]\[\text{Fraction}=\frac{24}{42}=\boxed{\frac47}\]
2
Subtopic 2
Pie charts
A pie chart divides a circle into sectors. Each sector angle represents its share of the total. A whole circle is \(360^\circ\), so the angles must add to \(360^\circ\).
From frequency to sector angle
\[\text{Sector angle}=\frac{\text{category frequency}}{\text{total frequency}}\times360^\circ\]To recover a frequency, divide the sector angle by \(360^\circ\), then multiply by the total number of observations. If the total is unknown, a pie chart shows proportions but does not reveal absolute frequencies.
For accurate construction, calculate each angle, draw a circle, measure sectors using a protractor, label each category and check the total. Use a sensible legend for narrow sectors.
Worked example · Easy
Find a sector angle
9 out of 36 pupils choose football. Find the sector angle.
\[\frac9{36}\times360^\circ=\boxed{90^\circ}\]
Worked example · Medium
Find the frequency from a chart
A sector measures \(54^\circ\) in a pie chart representing 120 customers. How many customers are in this category?
\[\frac{54}{360}\times120=\boxed{18}\]
Worked example · Medium
Complete a pie chart
Four activities attract 14, 21, 7 and 28 pupils. Calculate the four angles.
Total \(=14+21+7+28=70\).
\[\frac{14}{70}\times360=72^\circ,\quad\frac{21}{70}\times360=108^\circ\]\[\frac7{70}\times360=36^\circ,\quad\frac{28}{70}\times360=144^\circ\]Check: \(72+108+36+144=360^\circ\).
Worked example · Exam
Reverse a percentage
A sector of \(144^\circ\) represents pupils who travel by bus. There are 48 such pupils. Find the total number surveyed.
The sector is \(144/360=2/5\) of the whole.
\[\frac25N=48\implies N=48\times\frac52=\boxed{120}\]3
Subtopic 3
Scatter diagrams
A scatter diagram plots paired numerical observations as points \((x,y)\). First label both axes with units and sensible scales. The pattern of dots suggests a relationship called correlation.
- Positive correlation: as one variable increases, the other tends to increase.
- Negative correlation: as one variable increases, the other tends to decrease.
- Zero or no correlation: there is no clear increasing or decreasing trend.
Patterns may be strong or weak depending on how closely points follow the overall trend. A straight line of best fit is drawn by eye through the middle of correlated data, with points reasonably balanced on each side. It need not pass through all points or through the origin.
Use the line to estimate unknown values within the data range (interpolation). An estimate beyond the data range is an extrapolation and is less reliable. Correlation alone does not prove one quantity causes changes in the other.
Worked example · Easy
Name the correlation
On a scatter diagram the points slope downwards from left to right. Identify the correlation.
As \(x\) increases, \(y\) generally decreases. This is negative correlation.
Worked example · Medium
Interpret the line of best fit
A line of best fit passes through approximately \((2,15)\) and \((8,39)\). Estimate \(y\) when \(x=5\).
Estimated gradient \(=(39-15)/(8-2)=4\). Equation \(y=4x+7\).
\[y(5)=4(5)+7=\boxed{27}\]Worked example · Medium
Recognise an unreliable prediction
Data were collected for ages from 10 to 16 years. Explain why using the best-fit line to predict a value for a 65-year-old would be unreliable.
65 is far outside the 10–16 age range. This would be extrapolation, and the observed linear trend may not continue that far.
Worked example · Exam
Draw conclusions cautiously
A scatter plot of ice-cream sales and beach attendance shows a strong positive correlation. A report claims buying ice cream causes more people to visit the beach. Explain whether this follows.
No. Positive correlation means the two quantities tend to increase together, but it does not prove causation. Hot weather could make both ice-cream sales and beach attendance rise.
4
Subtopic 4
Cumulative frequency diagrams
Cumulative frequency is a running total: the number of observations up to a given value or upper class boundary. First make a cumulative-frequency table by adding each new class frequency to the previous total.
Cumulative frequency
\[CF_k=f_1+f_2+\cdots+f_k\]For continuous grouped data, plot cumulative frequency against the upper class boundary, and include a starting point at the lower boundary of the first group with cumulative frequency 0. Draw a smooth, increasing curve (an ogive).
To estimate how many items are below a value, read upwards from the measurement to the curve, then across to the cumulative-frequency axis. To find the number above a value, subtract that cumulative frequency from the total. Values read from grouped-data curves are estimates, not exact counts.
Worked example · Easy
Find the running totals
Four consecutive groups have frequencies 5, 8, 12 and 5. List the cumulative frequencies.
\[5,\quad5+8=13,\quad13+12=25,\quad25+5=\boxed{30}\]
Cumulative frequency row: \(\boxed{5,13,25,30}\).
Worked example · Medium
Interpret a cumulative curve
A cumulative frequency graph for 80 parcels shows 29 parcels below 3 kg. Estimate how many weigh 3 kg or more.
\[80-29=\boxed{51\text{ parcels}}\]
Worked example · Exam
Estimate from a grouped distribution
At the upper boundaries 10, 20, 30 and 40, the cumulative frequencies are 4, 20, 44 and 60. Assuming an even spread within the 20–30 class, estimate how many observations are below 25.
The 20–30 class has \(44-20=24\) observations. 25 is halfway through this class.
\[CF(25)\approx20+\frac{25-20}{30-20}\times24=20+12=\boxed{32}\]5
Subtopic 5
Median, quartiles and percentiles
The median divides ordered data into two equal halves and corresponds to the 50th percentile. The lower quartile \(Q_1\) is the 25th percentile, and the upper quartile \(Q_3\) is the 75th percentile. A general percentile describes the value at or below which a given percentage of observations lie.
Positions on a cumulative-frequency graph
\[\begin{aligned}\text{Median position}&=\tfrac12N\\Q_1\text{ position}&=\tfrac14N\\Q_3\text{ position}&=\tfrac34N\\P_k\text{ position}&=\tfrac{k}{100}N\\\mathrm{IQR}&=Q_3-Q_1\end{aligned}\]Read the estimated median and quartiles by starting at \(N/2\), \(N/4\) and \(3N/4\) on the cumulative-frequency axis, moving across to the curve and then down to the measured-value axis. These graphical estimates differ slightly depending on the curve drawn.
The interquartile range (IQR) measures the spread of the central half of the data and is less affected by extreme values than the overall range. When comparing distributions, consider both the typical value and the spread.
Worked example · Easy
Identify percentile ranks
There are 120 measurements. At which cumulative-frequency levels should you read the lower quartile, median and upper quartile?
\[Q_1:120/4=\boxed{30},\quad\mathrm{Median}:120/2=\boxed{60},\quad Q_3:3(120)/4=\boxed{90}\]
Worked example · Medium
Find the interquartile range
A cumulative-frequency graph gives \(Q_1=22.5\) cm and \(Q_3=37.8\) cm. Find the IQR.
\[\mathrm{IQR}=37.8-22.5=\boxed{15.3\text{ cm}}\]
Worked example · Medium
Find a percentile
A distribution has 150 readings. Which cumulative-frequency level corresponds to the 80th percentile?
\[\frac{80}{100}\times150=\boxed{120}\]
Read across from cumulative frequency 120 to the curve, then down to the horizontal axis.
Worked example · Exam
Compare two distributions
For Group A, median \(=42\) and quartiles \(31,55\). For Group B, median \(=39\) and quartiles \(35,46\). Which group has the higher typical value, and which is more consistent?
Group A has the greater median: \(42>39\). Its IQR is \(55-31=24\). Group B has IQR \(46-35=11\). Therefore Group A has the higher typical value, while Group B is more consistent (smaller central spread).
6
Subtopic 6
Histograms
A histogram displays grouped continuous data. The bars touch because the class intervals are adjacent. Unlike an ordinary bar chart, the area of a histogram bar, not just its height, is proportional to the number of observations.
Histogram calculations
\[\boxed{\text{Frequency density}=\frac{\text{frequency}}{\text{class width}}}\qquad\boxed{\text{Frequency}=\text{density}\times\text{width}}\]On the horizontal axis use continuous class boundaries, and on the vertical axis use frequency density. If class widths are unequal, the tallest bar is not necessarily the class with the greatest frequency. The modal class is the class represented by the highest density bar.
To estimate the mean from a grouped frequency table or a histogram, recover the frequencies (using bar area if necessary), multiply each frequency by its class midpoint and divide by the total frequency.
Estimating the mean from grouped data
\[\text{Estimated mean}=\frac{\sum(\text{class midpoint}\times f)}{\sum f}\]Common exam errors: Do not use frequencies directly as histogram heights when class widths differ. Do not read frequency from a histogram bar without multiplying its height by its width.
Worked example · Easy
Calculate frequency density
A class has frequency 18 and width 6. Find its frequency density.
\[\frac{18}{6}=\boxed3\]
Worked example · Medium
Recover frequency from a histogram
A histogram bar extends from 20 to 35 on the horizontal axis, and has a frequency density of 2.4. Find the frequency.
\[\text{Class width}=35-20=15\]\[\text{Frequency}=15\times2.4=\boxed{36}\]
Worked example · Medium
Compare unequal-width groups
Class A covers 0–5 with frequency 18. Class B covers 5–20 with frequency 30. Calculate both histogram heights and identify the taller bar.
\[\text{A density}=18/5=3.6\]\[\text{B density}=30/15=2\]
Class A is taller although B has greater frequency.
Worked example · Exam
Estimate the mean from unequal classes
Three grouped intervals 0–10, 10–20 and 20–40 have frequency densities 1.2, 1.8 and 0.6. Estimate the mean.
Frequencies are \(10(1.2)=12\), \(10(1.8)=18\) and \(20(0.6)=12\). Midpoints are \(5,15,30\).
\[\bar x\approx\frac{5(12)+15(18)+30(12)}{12+18+12}=\frac{690}{42}=\boxed{16.4}\quad(3\text{ s.f.})\]✓
Independent practice
Chapter 46 exercises
40 new exercises progress from basic questions to multi-step exam-style problems. Try each question before opening Show Answer to see the method and answer.
Build confidence — Questions 1–18
1Read a bar chartPractice
Five bars have frequencies 4, 7, 9, 6 and 4. How many observations are represented?
Show Answer
\[4+7+9+6+4=\boxed{30}\]
2Find the highest barPractice
A bar chart shows 11, 8, 14 and 7 people in categories A, B, C and D. State the modal category.
Show Answer
C has the largest frequency (14), so category C is modal.
3Interpret pictogram symbolsPractice
A full symbol represents 6 books. How many books do 4 complete symbols represent?
Show Answer
\[4\times6=\boxed{24\text{ books}}\]
4Interpret part symbolsPractice
A full symbol represents 8 students. How many students do 3 full symbols and one quarter-symbol represent?
Show Answer
\[3(8)+\tfrac14(8)=24+2=\boxed{26}\]
5Compare two bar groupsPractice
A chart shows 18 girls and 14 boys took a subject. How many more girls took it?
Show Answer
\[18-14=\boxed4\]
6Read a stacked barPractice
A stacked bar reaches frequency 28. Its lower section reaches 17. What is the frequency in the top section?
Show Answer
\[28-17=\boxed{11}\]
7Pie chart anglePractice
5 of 20 respondents chose swimming. Find the sector angle.
Show Answer
\[\frac5{20}\times360^\circ=\boxed{90^\circ}\]
8Pie chart percentagePractice
A sector measures \(72^\circ\). What percentage of a circle does it represent?
Show Answer
\[\frac{72}{360}\times100\%=\boxed{20\%}\]
9Pie chart frequencyPractice
A pie chart represents 90 voters. One sector measures \(120^\circ\). Find its frequency.
Show Answer
\[\frac{120}{360}\times90=\boxed{30}\]
10Describe positive correlationPractice
The plotted points rise from left to right. What correlation is shown?
Show Answer
This is positive correlation: the quantities generally increase together.
11Describe negative correlationPractice
The plotted points fall from left to right. What correlation is shown?
Show Answer
This is negative correlation: one quantity generally decreases as the other increases.
12No relationshipPractice
Points in a scatter diagram have no clear pattern. What conclusion about correlation can you draw?
Show Answer
There is no clear correlation between the variables.
13Cumulative frequencyPractice
A table has frequencies 3, 7, 5, 9. Give the running totals.
Show Answer
\[3,\quad3+7=10,\quad10+5=15,\quad15+9=\boxed{24}\]
The cumulative frequencies are \(\boxed{3,10,15,24}\).
14Read the upper boundaryPractice
A grouped class is \(20\le x<30\). At which horizontal-axis value should its cumulative total be plotted?
Show Answer
Plot the cumulative total at the upper class boundary 30.
15Median CF levelPractice
A cumulative-frequency graph represents 84 objects. At which cumulative-frequency value do you locate the median?
Show Answer
\[84/2=\boxed{42}\]
16Quartile levelsPractice
A cumulative-frequency graph represents 200 results. Give the CF levels for \(Q_1\) and \(Q_3\).
Show Answer
\[Q_1:200/4=\boxed{50},\quad Q_3:3(200)/4=\boxed{150}\]
17Find the IQRPractice
The lower and upper quartiles are 16 and 29. Find the interquartile range.
Show Answer
\[29-16=\boxed{13}\]
18Find densityPractice
A histogram group has width 5 and frequency 35. Calculate its frequency density.
Show Answer
\[35/5=\boxed7\]
Develop your skills — Questions 19–27
19Construct pie anglesIntermediate
A survey of 60 students gives categories with frequencies 15, 20 and 25. Calculate all sector angles.
Show Answer
\[15/60\times360=90^\circ\]\[20/60\times360=120^\circ\]\[25/60\times360=150^\circ\]
Check: \(90+120+150=360^\circ\).
20Missing categoryIntermediate
A pie chart has sectors \(45^\circ\), \(90^\circ\), \(135^\circ\) and one unknown sector. Find its angle and its percentage of the total.
Show Answer
\[360-(45+90+135)=\boxed{90^\circ}\]\[90/360\times100=\boxed{25\%}\]
21Reverse the pie chartIntermediate
A \(54^\circ\) sector represents 18 pupils. Find the total number of pupils.
Show Answer
\[N=18\times\frac{360}{54}=\boxed{120}\]
22Best-fit estimateIntermediate
A straight best-fit line passes through (1, 7) and (5, 23). Estimate \(y\) when \(x=4\).
Show Answer
\[m=(23-7)/(5-1)=4\]\[y=4x+3\implies y(4)=16+3=\boxed{19}\]
23Extrapolation checkIntermediate
Measurements only cover 30–60 minutes. Explain why predicting a value at 200 minutes using their trend line is questionable.
Show Answer
200 minutes is outside the range of the original observations. The prediction is extrapolation, and the trend may not continue.
24Cumulative-frequency tableIntermediate
Class frequencies are 8, 12, 21, 9 and 10. Write a cumulative-frequency row and find the total.
Show Answer
\[8,\quad20,\quad41,\quad50,\quad\boxed{60}\]
The total is 60 observations.
25Reading counts above a valueIntermediate
In a sample of 150, the cumulative frequency below 26 is 92. Estimate how many values are at least 26.
Show Answer
\[150-92=\boxed{58}\]
26Read the 90th percentileIntermediate
For 240 measurements, find the cumulative frequency at which to read the 90th percentile.
Show Answer
\[0.90\times240=\boxed{216}\]
27Find histogram frequencyIntermediate
A density bar from 12 to 20 has height 4.5. Find the frequency.
Show Answer
\[\text{Width}=20-12=8\]\[\text{Frequency}=8(4.5)=\boxed{36}\]
Original exam-style practice — Questions 28–40
28Dual bar chart totalsExam-style
A shop sells drinks A and B in four periods. A sells 14, 20, 18, 16; B sells 11, 17, 22, 14. Find each total and the percentage of all sales that were B.
Show Answer
\[A=14+20+18+16=68\]\[B=11+17+22+14=64\]\[\text{All}=68+64=132\]\[\frac{64}{132}\times100=\boxed{48.5\%}\ (3\text{ s.f.})\]
29Pie chart with unknown frequencyExam-style
A pie chart of 150 people has sector angles 96°, 120° and 72°, plus a final sector. Find the final angle and its frequency.
Show Answer
\[\text{Missing angle}=360-(96+120+72)=72^\circ\]\[\text{Frequency}=\frac{72}{360}(150)=\boxed{30}\]
30Find pie sector from ratiosExam-style
Three groups occur in the ratio 2:3:4. Calculate the sector angle for the largest group.
Show Answer
Total parts \(=2+3+4=9\); largest group has 4 parts.
\[\frac49\times360^\circ=\boxed{160^\circ}\]31Compare distributions with quartilesExam-style
A set of exam scores has median 68, lower quartile 55 and upper quartile 81. Another has median 66, lower quartile 61 and upper quartile 76. Which has the larger median and which has the smaller IQR?
Show Answer
First median: \(68>66\), so the first is higher.
\[\mathrm{IQR}_1=81-55=26\]\[\mathrm{IQR}_2=76-61=15\]The second group has the smaller IQR and is more consistent.
32Estimate a missing cumulative countExam-style
Cumulative frequencies at upper boundaries 20 and 30 are 34 and 58. Estimate the cumulative frequency at 27, assuming the data are uniformly spread in this class.
Show Answer
Class frequency \(=58-34=24\). 27 is \(7/10\) of the way through.
\[34+\frac7{10}(24)=34+16.8=\boxed{50.8}\]About 51 observations.
33Use frequency densitiesExam-style
Three histogram classes of widths 4, 6 and 10 have densities 5, 3 and 1.2. Calculate the total frequency and identify the modal class.
Show Answer
\[f_1=4(5)=20,\quad f_2=6(3)=18,\quad f_3=10(1.2)=12\]\[\text{Total}=20+18+12=\boxed{50}\]
The first class is modal (highest density 5).
34Find a missing histogram heightExam-style
A class has frequency 24 and extends from 35 to 47. Find its histogram height.
Show Answer
\[\text{Class width}=47-35=12\]\[\text{Density}=24/12=\boxed2\]
35Estimated mean from histogramExam-style
A histogram has groups 0–5 (density 2), 5–15 (density 3) and 15–25 (density 1). Estimate the mean value.
Show Answer
Widths \(5,10,10\); frequencies \(10,30,10\); midpoints \(2.5,10,20\).
\[\bar x\approx\frac{10(2.5)+30(10)+10(20)}{10+30+10}=\frac{525}{50}=\boxed{10.5}\]36Cumulative frequency IQRExam-style
There are 160 race times in a cumulative-frequency diagram. Give the cumulative-frequency values for the median, lower quartile, upper quartile and 75th percentile.
Show Answer
\[\text{Median}=160/2=80\]\[Q_1=160/4=40,\quad Q_3=3(160)/4=120\]
The 75th percentile is also at 120. Use these levels to read the corresponding times from the curve.
37Interpret histogram areaExam-style
A histogram has one class 10–20 with density 2 and another class 20–50 with density 0.8. Which class contains more observations?
Show Answer
\[f_1=10(2)=20\]\[f_2=30(0.8)=24\]
The 20–50 class contains more observations, despite its shorter bar.
38Interpret a scatter diagram cautiouslyExam-style
A scatter diagram shows a strong negative correlation between daily rainfall and sunshine hours. State the relationship and explain why it does not prove rainfall alone determines sunshine.
Show Answer
Higher rainfall tends to be associated with fewer sunshine hours: strong negative correlation. The relationship is observational, and both may be affected by other weather conditions. Correlation does not establish a sole cause.
39Compare two cumulative curvesExam-style
Group A has 100 times with median 28 min, \(Q_1=21\) and \(Q_3=40\). Group B has median 31 min, \(Q_1=28\) and \(Q_3=35\). Which is usually faster, and which has less variation?
Show Answer
A has the lower median time (28 rather than 31), so A is usually faster.
\[\mathrm{IQR}_A=40-21=19\]\[\mathrm{IQR}_B=35-28=7\]Group B has less variation in its central half.
40Use a frequency-density ratioExam-style
A histogram group has width 12 and density 2.5. The next group has width 5 and frequency 20. Find the density of the next group and the combined frequency.
Show Answer
\[f_1=12(2.5)=30\]\[\text{Density}_2=20/5=\boxed4\]\[\text{Total}=30+20=\boxed{50}\]
✓
Exam checklist
Chapter 46 key points
Check your graph before answering: Does it show frequency, cumulative frequency, or frequency density? Each uses a different reading method.
Remember these key rules
\[\text{Pie angle}=\frac{f}{N}\times360^\circ\] \[\text{Histogram frequency}=\text{class width}\times\text{frequency density}\] \[\mathrm{IQR}=Q_3-Q_1\]- Bar chart: separate category bars; use a key for multiple groups.
- Pie chart: sector angles add to 360° and express proportions.
- Scatter: discuss direction, strength and limitations of a line of best fit.
- Cumulative frequency: plot upper class boundaries and begin at zero frequency.
- Quartiles: estimate at 25%, 50% and 75% of the cumulative total.
- Histograms: bar area is proportional to frequency, so use density when intervals differ.