S3 Mathematics Practice Book · Unit 13: Interpretation of statistical data

Unit 13 Interpretation of statistical data

Key unit competence: Interpret statistical graphs from ungrouped and grouped data to inform conclusions or decisions.

Key ideas

Ungrouped data are individual values; grouped data are organised in classes. Choose a graph that suits the data.
Bar chart: categories or discrete values; read the frequency from the bar height. Pie chart: each sector is a fraction of the whole; frequency $= \frac{angle}{36 0 ^{\circ}} \times total$ .
Line graph: change over time; look for trends (increase, decrease), highest and lowest points.
Histogram: grouped continuous data; bars touch. Frequency polygon: joins midpoints of bar tops; useful to compare two distributions.
Cumulative frequency curve (ogive): plot cumulative frequency against upper class boundary. Read the median at $\frac{N}{2}$ , $Q_{1}$ at $\frac{N}{4}$ , $Q_{3}$ at $\frac{3 N}{4}$ ; IQR $= Q_{3} - Q_{1}$ .
Interpreting: compare averages (mean, median, mode) and spread (range, IQR); watch out for misleading graphs (axes not starting at zero, unequal scales, 3-D effects).

Formulas to know

f = \frac{θ}{36 0 ^{\circ}} \times N

median at \frac{N}{2}, Q_{1} at \frac{N}{4}, Q_{3} at \frac{3 N}{4}

I QR = Q_{3} - Q_{1}

Common misconceptions

"In a pie chart, a bigger sector means a bigger number in every chart." Only within the same chart; two pie charts with different totals cannot be compared by sector size alone.
"An ogive is plotted at class midpoints." No: cumulative frequencies are plotted at upper class boundaries.
"A bar starting high on a cut axis shows a big difference." Check the scale: a truncated axis exaggerates differences.

Worked examples

Example 1. A pie chart of 180 students shows "walking" with an angle of $14 0^{\circ}$ . How many walk?

Solution

$\frac{140}{360} \times 180 = 70$ students.

Example 2. An ogive for 80 students' marks gives 52 at the 40th student, 41 at the 20th and 63 at the 60th. Find the median and IQR.

Solution

Median $= 52$ ; $Q_{1} = 41$ , $Q_{3} = 63$ ; IQR $= 22$ .

Practice questions

Easier and harder questions are mixed, just like in a real exam.

A pie chart of 240 students' favourite subjects has Maths $10 5^{\circ}$ , Biology $7 5^{\circ}$ , English $9 0^{\circ}$ , Physics the rest. Find the number for each.
Show answer
Physics $9 0^{\circ}$ . Maths 70, Biology 50, English 60, Physics 60.
Common mistake: Forgetting to compute the remaining angle first.
A pie chart's "Other" sector is $5 4^{\circ}$ and represents 27 people. Find the total surveyed.
Show answer
$\frac{54}{360} N = 27$ , $N = 180$ .
Common mistake: Using $\frac{27}{360}$ .
Explain three ways a graph can be misleading.
Show answer
A vertical axis not starting at 0; unequal intervals on an axis; 3-D or pictures whose areas grow faster than the values.
Common mistake: Saying a graph is misleading just because it has many bars.
A bar chart shows maize harvests 2019–2023: 4, 5, 3, 6, 7 tonnes. Find the mean harvest and the year with the largest increase.
Show answer
Mean $= 5$ t. Largest increase: 2021 to 2022 (+3 t).
Common mistake: Confusing the largest value with the largest increase.
From cumulative frequencies 6, 20, 45, 64, 72, 80 at upper boundaries 10, 20, 30, 40, 50, 60, find the class containing the median.
Show answer
$N = 80$ , $\frac{N}{2} = 40$ ; it lies between cf 20 and 45: the class 20–30.
Common mistake: Taking the class with the highest cumulative frequency.
An ogive for 120 workers' wages gives $Q_{1} = 45000$ , median $60000$ , $Q_{3} = 72000$ FRW. Interpret these values.
Show answer
A quarter of workers earn under 45 000 FRW, half under 60 000 FRW, and three quarters under 72 000 FRW; the middle half earn between 45 000 and 72 000 FRW (IQR 27 000 FRW).
Common mistake: Treating $Q_{3}$ as the maximum wage.
A line graph shows temperatures 18, 21, 25, 24, 20 (°C) for Monday to Friday. Describe the trend.
Show answer
Temperatures rose from Monday to a peak of 25 °C on Wednesday, then fell to 20 °C by Friday.
Common mistake: Describing only the highest value.
Draw (describe) a frequency polygon from the table: 0–10: 3, 10–20: 7, 20–30: 12, 30–40: 5.
Show answer
Plot the points (5, 3), (15, 7), (25, 12), (35, 5); join them with straight lines and close at (−5, 0) and (45, 0) if required.
Common mistake: Plotting at class limits instead of midpoints.

Matching and drag-and-drop

Matching

Match each item in column A with the correct answer in column B. Write the letter next to each number.

Column A	Column B
1. A pie chart of 72 people shows a $5 0^{\circ}$ sector for "bus". How many take the bus?	a. 60
2. In a histogram with equal class widths, the modal class is the one with:	b. 15%
3. In a pie chart, tea has $12 0^{\circ}$ , coffee $15 0^{\circ}$ and juice the rest. 36 people chose juice. How many chose coffee?	c. the tallest bar
4. A pie-chart sector has angle $5 4^{\circ}$ . What percentage does it represent?	d. 10

Show answer

1 → d (10) · 2 → c (the tallest bar) · 3 → a (60) · 4 → b (15%)

$\frac{50}{360} \times 72$ .
Highest frequency.
Step 1. Juice angle $= 36 0^{\circ} - 27 0^{\circ} = 9 0^{\circ}$ , so 36 people $= \frac{1}{4}$ of the total: 144 people.
Step 2. Coffee: $\frac{150}{360} \times 144 = 60$ .
Step 1. $\frac{54}{360} = 0.15$ .
Step 2. $0.15 \times 100% = 15%$ .

Drag and drop
Choose the correct word or phrase from the box for each gap. One word in the box is not needed.
upper class boundaries · 60 · 30 · the frequency is highest
1. On an ogive, cumulative frequency is plotted against .
2. An ogive is steepest where .
3. For $N = 60$ , the median is read on an ogive at cumulative frequency .
Show answer
1: upper class boundaries · 2: the frequency is highest · 3: 30
- All values up to the upper boundary are counted.
- Steep rise means many values in that range.
- $\frac{N}{2}$ .

Matching

Match each item in column A with the correct answer in column B. Write the letter next to each number.

Column A	Column B
1. A pie chart sector of $9 0^{\circ}$ represents what fraction of the total?	a. 90
2. A histogram has class 10–20 with frequency 30 and class 20–40 with frequency 40. Which bar is taller?	b. 10–20 (frequency density 3 against 2)
3. The vertical axis of a bar chart starts at 90 instead of 0. This can:	c. $\frac{1}{4}$
4. In a pictogram, one symbol = 20 cars and a row shows $4 \frac{1}{2}$ symbols. How many cars?	d. exaggerate differences between bars

Show answer

1 → c ( $\frac{1}{4}$ ) · 2 → b (10–20 (frequency density 3 against 2)) · 3 → d (exaggerate differences between bars) · 4 → a (90)

$\frac{90}{360}$ .
Step 1. Bar height is frequency density: $\frac{30}{10} = 3$ and $\frac{40}{20} = 2$ .
Step 2. So the 10–20 bar is taller even though its frequency is smaller.
A truncated axis is misleading.
$4.5 \times 20$ .

Drag and drop
Choose the correct word or phrase from the box for each gap. One word in the box is not needed.
$9 0^{\circ}$ · 30 · 45 · 14
1. An ogive gives $Q_{1} = 24$ and $Q_{3} = 38$ . The IQR is .
2. A pie chart shows 25% for "football". The angle is .
3. For $N = 60$ , $Q_{3}$ is read at cumulative frequency .
Show answer
1: 14 · 2: $9 0^{\circ}$ · 3: 45
- $38 - 24$ .
- $0.25 \times 36 0^{\circ}$ .
- $\frac{3 N}{4}$ .

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