Ch 1 · 9.1–9.3 Classifying and Interpreting Data, Averages and Spread · O Level Mathematics 4024
Topic 1 of 3 in Statistics — O Level Mathematics 4024 — 48 lessons.
Introduction
A class survey, a sports team's results, the waiting times at a clinic: statistics organises raw data into tables and summarises it with an average (a typical value) and a spread (how consistent it is). Comparing two data sets needs both, plus a sentence in context. (Syllabus sections 9.1–9.3.)
Key ideas — Organising data
A tally table counts data into categories; a two-way table shows two variables at once (e.g. gender and favourite sport), with totals for each row and column. Discrete data takes separate values (number of siblings); continuous data can take any value in a range (height, time).
🧑🏫 Teacher's explanation
Imagine you are organizing a large school party. If you just throw a pile of RSVP cards on a table, you have data, but it is useless. To make sense of it, you need to organize it. You might use a tally table to count how many people want pizza versus pasta, or a two-way table to see how many boys want pizza versus how many girls want pizza. This turns a mess into information.
📖 Definitions
🔍 Worked example — Easy
A class of 20 students was asked their favorite fruit. The results are: Apple, Banana, Apple, Orange, Apple, Banana, Apple, Orange, Banana, Apple, Apple, Orange, Apple, Banana, Apple, Orange, Apple, Banana, Apple, Orange. Create a tally table and identify the mode.
🔍 Worked example — Standard
A survey asks 30 students if they play football (Yes/No) and if they are in Year 10 (Yes/No). The table is partially completed:
🔍 Worked example — Exam-style
The heights of 5 plants are measured: 12.2cm, 15.5cm, 12.8cm, 14.1cm, 13.4cm. Classify the data type and find the range.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
\n(a) Calculate the total number of students who prefer Tennis. [1] (b) State whether the data for 'Number of students' is discrete or continuous. [1] [2 marks]
Key ideas — Averages and range
From a frequency table, mean = Σ fx/Σ f.
🧑🏫 Teacher's explanation
Imagine you are organizing a class party and need to decide how many pizzas to order. If you ask everyone how many slices they want, the 'average' tells you the general expectation. However, if one student says they want 50 slices, that single extreme value will make the 'mean' look huge, even though most students only want 2 or 3. This is why we need different tools.
📖 Definitions
🔍 Worked example — Easy
Calculate the mean, median, mode, and range for the following data set: 5, 8, 3, 5, 10, 2, 5.
🔍 Worked example — Standard
A frequency table shows the number of goals scored by a football team in 20 matches:
🔍 Worked example — Exam-style
In a survey of 50 students, the heights are grouped as follows:
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
\n(a) Estimate the mean weight of the parcels. [3] (b) Identify the modal class. [1] [4 marks]
Key ideas — Grouped data
When data are grouped, the exact values are unknown, so use the mid-point of each class:
🧑🏫 Teacher's explanation
Imagine you are counting the number of people in different age groups at a cinema. You know there are 10 people in the '10-15' age group, but you don't know if they are all 11 or all 14. Because you don't have their exact ages, you have to make a 'best guess' to find the average.
📖 Definitions
🔍 Worked example — Easy
A group of students took a test. The marks were grouped as follows: 0 ≤ x < 10 (frequency 5), 10 ≤ x < 20 (frequency 8), 20 ≤ x < 30 (frequency 7). Calculate the estimated mean mark.
🔍 Worked example — Standard
The table below shows the weights of 50 parcels. 0 < w ≤ 5 kg: frequency 10 5 < w ≤ 10 kg: frequency 25 10 < w ≤ 15 kg: frequency 15 Identify the modal class and calculate the estimated mean weight.
🔍 Worked example — Exam-style
A survey asks people how many minutes they spend on social media daily. 0-20 mins: 4 people 20-40 mins: 12 people 40-60 mins: 20 people 60-80 mins: 14 people Calculate the estimated mean.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
\n| Time (t mins) | Frequency |
Key ideas — Comparing data sets
Compare one average and one measure of spread, each in context: "The median time for group A was higher, so group A was slower on average; the range for A was smaller, so A's times were more consistent." Be careful with conclusions: a small sample, or a sample from one place, may not represent everyone.
🧑🏫 Teacher's explanation
Imagine you are comparing two basketball players to see who is more 'reliable'. One player might score exactly 20 points every single game (very consistent), while another scores 40 points in one game and 0 in the next (very erratic). Even if their average score is the same, their 'spread' is completely different.
📖 Definitions
🔍 Worked example — Easy
Group A scores: 5, 7, 7, 8, 13. Group B scores: 6, 7, 7, 7, 10. Calculate the mean and range for both groups and compare them.
🔍 Worked example — Standard
A class of 20 students took a test. The mean score was 65. If the sum of the scores of the first 15 students is 1020, find the mean score of the remaining 5 students.
🔍 Worked example — Exam-style
The following table shows the weights of 50 parcels. Estimate the mean weight of the parcels.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
For the bar chart (28 students): mode = 1 sibling; median = mean of the 14th and 15th values = 2; mean = 0(4) + 1(9) + 2(8) + 3(5) + 4(2)/28 = 48/28 = 1.71; range = 4.
🔍 Worked example — Easy
A set of test scores is: 5, 7, 7, 8, 10, 12. Calculate the mean, median, mode, and range.
🔍 Worked example — Standard
Calculate the estimated mean for the following grouped data: \n0 < t ≤ 10: 6 students \n10 < t ≤ 20: 14 students \n20 < t ≤ 30: 11 students \n30 < t ≤ 40: 7 students \n40 < t ≤ 50: 2 students
🔍 Worked example — Exam-style
A survey of 28 students recorded their number of siblings: \n0 siblings: 4 students \n1 sibling: 9 students \n2 siblings: 8 students \n3 siblings: 5 students \n4 siblings: 2 students. \nCalculate the mean and the modal class.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) mean 48/7 = 6.86; median 7; mode 3; range 9. (2) 5 + 16 + 12 + 12/20 = 2.25. (3) 30 + 270 + 400/20 = 35. (4) a few very expensive houses raise the mean a lot; the median is not affected by extreme values.