Row total + Column total = Grand total Read across AND down to find joint frequencies
Pie Charts
Angle = (frequency Γ· total) Γ 360Β° Check all angles add to 360Β°
Comparing Distributions
Average = where data centres Β· Range = how spread out Choose mean, median or mode wisely!
What you'll learn:
Read, complete and interpret two-way tables
Draw and interpret pie charts from frequency data
Use grouped frequency tables β modal class, estimating mean
Compare distributions using averages and range
Identify misleading graphs and explain why they mislead
π Learn: Interpreting Data & Two-Way Tables
Part 1: Two-Way Tables
A two-way table shows data for two categories at once. Each cell is the count for that combination.
Cats
Dogs
Row Total
Boys
12
18
30
Girls
20
10
30
Column Total
32
28
60
Row total: add cells across that row Β· Column total: add cells down that column
Grand total (bottom-right) = sum of all row totals = sum of all column totals
To find a missing cell: use the known total β the other values in that row/column
Part 2: Pie Charts
A pie chart shows proportions. Each slice angle is calculated from the frequency.
Angle = (frequency Γ· total) Γ 360Β°
Colour
Frequency
Calculation
Angle
Red
15
(15 Γ· 60) Γ 360
90Β°
Blue
24
(24 Γ· 60) Γ 360
144Β°
Green
21
(21 Γ· 60) Γ 360
126Β°
Total
60
360Β°
Always check: angles must add to exactly 360Β°
To find frequency from a pie chart: (angle Γ· 360) Γ total
Part 3: Grouped Frequency Tables
Class intervals group continuous data (e.g. 0 β€ x < 10). Key ideas:
Height (cm)
Frequency
Midpoint
f Γ midpoint
140 β€ h < 150
4
145
580
150 β€ h < 160
11
155
1705
160 β€ h < 170
9
165
1485
170 β€ h < 180
6
175
1050
Total
30
4820
Modal class = class with the highest frequency β 150 β€ h < 160
Estimated mean = Ξ£(f Γ midpoint) Γ· Ξ£f = 4820 Γ· 30 β 160.7 cm
We estimate because we don't know exact values β we assume each value is at the midpoint
Part 4: Comparing Distributions
When comparing two data sets, comment on average (centre) AND range (spread).
Average
Best whenβ¦
Avoid whenβ¦
Mean
Data is symmetrical, no extreme values
There are outliers (distorts the mean)
Median
Data has outliers or is skewed
You need to use all data values
Mode
Categorical data, most popular item
Data has no clear mode or multiple modes
Range = largest value β smallest value. A larger range means more spread out / less consistent.
Always write a comparison sentence: "On average, Group A scored higher (mean = 72) than Group B (mean = 65). Group A was also more consistent (range = 10 vs 18)."
Part 5: Misleading Graphs
Graphs can mislead if they break visual rules. Common tricks:
Truncated y-axis β axis starts above 0, making small differences look large
Unequal intervals β bars are different widths, misrepresenting frequency
No title or labels β impossible to interpret without context
3D effects β distort perceived sizes of slices/bars
Misleading scale β scale jumps are uneven, distorting the trend
Cherry-picked time range β choosing a window that hides the full picture
Always check: Does the axis start at 0? Are intervals equal? Is there a title and labels?
π‘ Worked Examples
Example 1: Completing a Two-Way Table
60 students chose a sport. 35 are in Year 7. 14 Year 7 students chose swimming. 8 Year 8 students chose swimming. Complete the table.
Swimming
Football
Total
Year 7
14
?
35
Year 8
8
?
?
Total
22
?
60
Step 1: Year 7 Football = 35 β 14 = 21
Step 2: Year 8 total = 60 β 35 = 25
Step 3: Year 8 Football = 25 β 8 = 17
Step 4: Football total = 21 + 17 = 38 β (check: 22 + 38 = 60 β)
Example 2: Drawing a Pie Chart
80 people named their favourite fruit. Apple: 20, Banana: 30, Mango: 18, Orange: 12. Find the angle for each.
Apple: (20 Γ· 80) Γ 360 = 0.25 Γ 360 = 90Β°
Banana: (30 Γ· 80) Γ 360 = 0.375 Γ 360 = 135Β°
Mango: (18 Γ· 80) Γ 360 = 0.225 Γ 360 = 81Β°
Orange: (12 Γ· 80) Γ 360 = 0.15 Γ 360 = 54Β°
β Check: 90 + 135 + 81 + 54 = 360Β° β
Example 3: Estimated Mean from Grouped Data
Time (minutes) students spend reading per day:
Time (min)
Frequency
Midpoint
f Γ m
0 β€ t < 10
6
5
30
10 β€ t < 20
14
15
210
20 β€ t < 30
8
25
200
30 β€ t < 40
2
35
70
Total
30
510
Modal class: 10 β€ t < 20 (frequency = 14, the highest)
Estimated mean: 510 Γ· 30 = 17 minutes
Example 4: Comparing Two Distributions
Two classes took a test (out of 50). Which average is more appropriate, and which class performed better?
Class A
Class B
Scores
42, 45, 44, 43, 46, 44, 2
30, 35, 34, 36, 32, 33, 34
Class A mean: (42+45+44+43+46+44+2)Γ·7 = 266Γ·7 = 38 (pulled down by outlier 2)
Class A median: Sort: 2,42,43,44,44,45,46 β median = 44 (better summary)
Class B mean: (30+35+34+36+32+33+34)Γ·7 = 234Γ·7 β 33.4
Range A = 46β2 = 44 Β· Range B = 36β30 = 6
Conclusion: Use median for Class A (outlier). Class A is better on average (median 44 > mean 33.4). Class B is far more consistent (range 6 vs 44).
Always comment on BOTH average AND spread in comparison questions.
π₯§ Live Pie Chart Builder
Enter up to 5 categories and their frequencies. Watch the pie chart draw itself!
βοΈ Exercise 1: Two-Way Tables
Complete the two-way tables by typing values into the blank cells. Hit Check when done.
βοΈ Exercise 2: Pie Chart Calculations
Calculate angles from frequencies or find frequencies from angles.
βοΈ Exercise 3: Grouped Frequency Tables
Find the modal class and estimated mean for each grouped frequency table.
βοΈ Exercise 4: Comparing Distributions
For each pair of datasets, calculate the required statistic and choose which average is most appropriate.
π Exercise 5: Spot the Misleading Graph
Click on all the problems you can spot with each graph. Collect all issues!
π Practice Questions
A two-way table shows 80 students by gender and subject. 45 are boys. 28 boys study Maths. 20 girls study Science. 35 students study Maths in total. How many girls study Maths?
In a two-way table, column total for "Sport" = 54, row total for "Year 9" = 40, and the "Year 9, Sport" cell = 22. Find "Year 9, Not Sport".
Complete: A class of 36 students. 15 like pizza. 12 boys like pizza. 20 are boys in total. Fill in the full two-way table (pizza/not pizza Γ boys/girls).
120 people voted for their favourite season. Spring: 30, Summer: 48, Autumn: 24, Winter: 18. Calculate the pie chart angle for Summer.
In a pie chart of 90 people, the "Tea" sector has an angle of 120Β°. How many people prefer tea?
A pie chart has 4 sectors. Three angles are 90Β°, 110Β°, and 85Β°. Find the fourth angle.
A sector represents 25 people out of 150. What is the sector angle?
State the modal class: Ages 10β15: 8, 15β20: 15, 20β25: 12, 25β30: 5.
Calculate the estimated mean for: Score 0β10: f=3, 10β20: f=7, 20β30: f=6, 30β40: f=4 (n=20).
A grouped table has class widths of 5. First class 20 β€ x < 25 has f=9. Second class 25 β€ x < 30 has f=6. Find the estimated mean for these two classes.
Two athletes' sprint times (seconds): Athlete A: 10.1, 10.3, 10.2, 10.8, 10.1. Athlete B: 9.9, 10.5, 10.1, 12.0, 10.2. Find the mean and range for each.
From Q11, which average is more appropriate for Athlete B? Justify your answer.
Dataset P: 4, 5, 5, 6, 7, 8, 9. Dataset Q: 2, 5, 5, 6, 7, 8, 11. Both have the same mode and median. Which has the larger range?
A shop's sales chart starts at $950 on the y-axis and goes to $1050. Sales in Jan=$980 and Feb=$1020. Why might this graph be misleading?
A bar chart compares exam results but bars have different widths. Why is this misleading?
Name TWO features every graph must have to be correctly interpreted.
A pie chart uses 3D perspective. Explain why this makes the graph misleading.
Temperatures over 12 months are shown, but only JuneβAugust is displayed. What is misleading?
In a grouped table, intervals are 0β5, 5β15, 15β20. Why are these intervals problematic?
Two classes have mean scores of 68 and 72, and ranges of 30 and 8 respectively. Write a full comparison sentence using both statistics.
A mean = (10.1+10.3+10.2+10.8+10.1)Γ·5 = 51.5Γ·5 = 10.3s; Range A = 0.7s. B mean = (9.9+10.5+10.1+12.0+10.2)Γ·5 = 52.7Γ·5 = 10.54s; Range B = 2.1s
Median is more appropriate for B β the outlier 12.0 distorts the mean. Median B = 10.2s (middle of sorted data).
Range P = 9β4 = 5. Range Q = 11β2 = 9 (Q has larger range)
The truncated y-axis makes the difference between Β£980 and Β£1020 look huge, when it is actually only Β£40 β just 4% of sales.
Wider bars represent more area even if their height is the same β visually suggesting higher frequency for that group.
A title and labelled axes (including units)
3D perspective makes some slices appear larger than their true proportion due to the viewing angle distortion.
Only showing the summer months creates a false impression of a long-term trend β the full year may show a very different pattern.
The intervals have different widths (5, 10, 5), so a standard histogram would misrepresent frequency unless frequency density is used.
Class B has a higher mean (72 vs 68) suggesting it performed better on average. However Class A has a much larger range (30 vs 8), suggesting Class B is far more consistent.
π Challenge: Real-World Data Problems
School Survey: 200 students were asked their preferred lunch option. The two-way table below is partially complete. Hot meal: Year 7=28, Year 8=35, Year 9=?. Packed lunch: Year 7=22, Year 8=?, Year 9=30. Totals: Year 7=50, Year 8=60, Year 9=?. Complete the table. What fraction of Year 8 chose hot meals?
Sports Club: 180 members play Tennis, Football or Swimming. Their pie chart angles are T=80Β°, F=160Β°, S=120Β°. How many members play each sport? What fraction play Tennis?
Weather Data: Daily maximum temperatures (Β°C) for a city over 20 days in class intervals: 15β20: 3, 20β25: 8, 25β30: 6, 30β35: 3. Find (a) the modal class, (b) the estimated mean temperature. Interpret what the modal class tells you about that city's weather.
Comparing Athletes: Three sprinters' 100m times (s): Maya: 11.2, 11.5, 11.3, 11.4, 11.1. Leo: 10.8, 11.6, 11.2, 12.9, 11.0. Sam: 11.1, 11.1, 11.1, 11.2, 11.0. (a) Find the mean and range for each. (b) Which athlete is the most consistent? (c) Which average would you use to represent Leo's data and why?
Misleading Advertising: A company shows a bar chart where the y-axis runs from 99% to 101% customer satisfaction. Their product shows 100.2% and the competitor shows 99.8%. (a) What is the actual difference in satisfaction? (b) How does the truncated axis create a misleading impression? (c) Sketch what the graph should look like with a correct axis.
Market Research: 360 shoppers were surveyed about favourite drink. Pie chart angles: Coffee=90Β°, Tea=120Β°, Juice=60Β°, Water=90Β°. (a) How many chose each drink? (b) The company claims "twice as many people prefer Tea than Juice." Is this claim accurate β show with a calculation. (c) A reporter shows only the Coffee and Tea slices and says "most people prefer hot drinks." Is this a fair interpretation?
Grouped Data Challenge: Marks scored in a test: 0β20: f=2, 20β40: f=5, 40β60: f=13, 60β80: f=8, 80β100: f=2. (a) How many students sat the test? (b) Find the modal class. (c) Estimate the mean. (d) A student says "the mean is about 60 because that's where most students scored." Explain why their reasoning is not a valid method to find the mean.
Extended Investigation: Two schools compare GCSE Maths scores (out of 100). School A: mean=72, range=20. School B: mean=68, range=6. (a) Write a full statistical comparison of the two schools. (b) A parent wants to send their child to the school with the most predictable results. Which school should they choose and why? (c) If School A's top scorer scored 90, what is School A's minimum score? (d) If School B's scores are 65, 66, 68, 69, 70, 71, 72 β verify the mean and range.
Year 9 total = 200β50β60 = 90. Year 8 packed = 60β35 = 25. Year 9 hot = 90β30 = 60. Grand total check: 28+35+60+22+25+30 = 200 β. Year 8 hot meals fraction = 35/60 = 7/12
(a) Modal class = 20β25Β°C (f=8). (b) Midpoints: 17.5, 22.5, 27.5, 32.5. Ξ£fm = 52.5+180+165+97.5 = 495. Mean = 495Γ·20 = 24.75Β°C. The modal class shows the most common temperature range β the city is typically warm but not hot.
Maya: mean=(11.2+11.5+11.3+11.4+11.1)Γ·5=56.5Γ·5=11.3s, range=0.4. Leo: mean=(10.8+11.6+11.2+12.9+11.0)Γ·5=57.5Γ·5=11.5s, range=2.1. Sam: mean=(11.1+11.1+11.1+11.2+11.0)Γ·5=55.5Γ·5=11.1s, range=0.2. (b) Sam is most consistent (range=0.2). (c) Use median for Leo (outlier 12.9 distorts mean; median = 11.2s).
(a) Difference = 100.2β99.8 = 0.4%. (b) Truncated axis makes 0.4% appear to be an enormous gap, implying ~5Γ superiority. (c) Correct graph: axis from 0% to 100%+ β bars would appear almost identical height.
(a) Coffee=(90Γ·360)Γ360=90, Tea=(120Γ·360)Γ360=120, Juice=(60Γ·360)Γ360=60, Water=(90Γ·360)Γ360=90. (b) Tea=120, Juice=60. 120Γ·60=2, so yes β Tea is exactly twice Juice. The claim is accurate. (c) Not fair β Coffee+Tea = 90+120 = 210 out of 360 = 58.3%. Technically a majority, but Juice+Water = 150 people also prefer cold drinks. The framing ignores context.
(a) n = 2+5+13+8+2 = 30. (b) Modal class = 40β60 (f=13). (c) Midpoints: 10,30,50,70,90. Ξ£fm = 20+150+650+560+180 = 1560. Mean = 1560Γ·30 = 52. (d) The student confused the modal class with the mean. The estimated mean requires calculating fΓmidpoint for each class and dividing by total frequency.
(a) School A scored higher on average (mean 72 vs 68), but School B is far more consistent (range 6 vs 20). (b) School B β small range means scores are predictable and close together. (c) Min score A = 90β20 = 70. (d) Mean B = (65+66+68+69+70+71+72)Γ·7 = 481Γ·7 = 68.7 β 68.7 β. Range B = 72β65 = 7 (note: listed as 6 β check data; actual range here = 7).