Ch 1 · 5.1 Representation of Data · A Level Mathematics 9709
Topic 1 of 5 in Probability & Statistics 1 — A Level Mathematics 9709 — 71 lessons.
Introduction
A school compares its exam results with last year's; a doctor compares waiting times at two clinics. Statistics starts with turning raw numbers into a clear picture and two summary numbers: a centre and a spread. Paper 5 expects you to draw the diagrams, calculate the summaries, and compare data sets in context. (Syllabus section 5.1, Paper 5.)
Key ideas — Choosing a diagram
A stem-and-leaf diagram needs a key (e.g. 3 | 7 means 37) and leaves in order.
🧑🏫 Teacher's explanation
Imagine you are a librarian trying to organize a pile of books. If you have 10 books, you might just list their titles. If you have 100, you might group them by genre. If you have 10,000, you need a complex filing system. In statistics, the 'diagram' you choose is your filing system. It depends entirely on how much data you have and what you want to show.
📖 Definitions
🔍 Worked example — Easy
The test scores for 7 students are: 45, 52, 48, 55, 45, 61, 52. Represent this data using a stem-and-leaf diagram, including a key.
🔍 Worked example — Standard
A set of data is represented by a histogram. The class intervals are [10, 20), [20, 30), and [30, 50). The frequencies are 5, 10, and 15 respectively. Calculate the frequency density for the third class interval.
🔍 Worked example — Exam-style
A cumulative frequency graph is drawn for the weights of 80 students. The median weight is found to be 65 kg. Use this information to estimate the number of students weighing between 60 kg and 70 kg, given that the cumulative frequency at 60 kg is 25 and at 70 kg is 55.
⚠️ Common errors
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🎯 Exam question
Key ideas — Histograms
Plot frequency density on the vertical axis:
🧑🏫 Teacher's explanation
Imagine you are looking at a crowd of people. If you count them by age, you might find 10 people are aged 10-14 and 50 people are aged 15-19. If you just draw bars of equal width, the 50 people will look much taller than the 10 people, even though the age gap is the same. But what if the age groups aren't the same size? What if one group covers 5 years and another covers 10 years?
📖 Definitions
🔍 Worked example — Easy
A dataset of test scores is grouped into intervals. The interval 40 ≤ x < 50 has a frequency of 15. Calculate the frequency density for this class.
🔍 Worked example — Standard
In a histogram, a bar represents the interval 10 ≤ x < 15 with a frequency of 20. Another bar represents 15 ≤ x < 25 with a frequency of 30. Calculate the frequency density for both bars.
🔍 Worked example — Exam-style
A histogram has a bar for the class 20 ≤ x < 30 with a frequency density of 5. Calculate the frequency of this class.
⚠️ Common errors
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🎯 Exam question
Key ideas — Centre and spread
σ = √(Σ x^2/n - bar x^2) (grouped: √(Σ fx^2/Σ f - bar x^2))
🧑🏫 Teacher's explanation
When we look at a collection of numbers, we want to know two things: where the 'middle' is and how 'spread out' the numbers are. Imagine you are comparing the heights of two different groups of students. If the average height is similar, the groups are centered at the same place. But if one group has heights ranging from 150cm to 160cm and the other from 140cm to 170cm, the second group is much more 'spread out' or variable.
📖 Definitions
🔍 Worked example — Standard
A set of data has Σ x = 150 and Σ x^2 = 2500 for n=10 observations. Find the mean and standard deviation.
🔍 Worked example — Exam-style
Compare two sets of data. Set A: 12, 14, 15, 16, 18. Set B: 10, 15, 15, 15, 25. Comment on the center and spread.
⚠️ Common errors
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🎯 Exam question
Key ideas — Cumulative frequency graphs
Plot cumulative frequency against the upper class boundary and join with a smooth curve (or straight lines). For n values, read the median at n/2, Q_1 at n/4, Q_3 at 3n/4, and the 90th percentile at 0.9n. To find how many are above a value, read up from that value and subtract from n.
🧑🏫 Teacher's explanation
Imagine you are standing at the entrance of a cinema. As people walk in, you count them. A cumulative frequency graph is like a running tally of how many people have entered the building at any given time. Instead of seeing how many people entered *during* a specific minute, you see how many people are *already inside*.
📖 Definitions
🔍 Worked example — Easy
A set of 80 test scores is recorded. A cumulative frequency graph is drawn. Use the graph to find the score below which 20 students fall.
🔍 Worked example — Standard
Given a cumulative frequency graph for n=120 observations, find the position of the Upper Quartile (Q_3).
🔍 Worked example — Exam-style
A cumulative frequency graph shows that for a sample of 200 people, the 90^th percentile is 65 kg. How many people weigh more than 65 kg?
⚠️ Common errors
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🎯 Exam question
(a) Estimate the median weight. (b) Find the weight below which 25\% of the parcels fall. (c) Find the number of parcels that weigh more than 15 kg. [5 marks]
Key ideas — Coded data and combining sets
If y = x - a, then bar x = a + Σ(x - a)/n and the standard deviation is unchanged:
🧑🏫 Teacher's explanation
Imagine you are measuring the heights of students in a classroom. If everyone grows exactly 5 cm taller, the average height increases by 5 cm, but the 'spread' or variation between students stays exactly the same. This is what 'coding' data does; we subtract a constant a from every value to make the numbers smaller and easier to work with. This new set of numbers is called 'coded data'.
📖 Definitions
🔍 Worked example — Easy
A set of n=10 observations has a mean barx = 50 and a standard deviation σ = 5. If each observation is transformed by y = x - 40, find the new mean bary and the new standard deviation σ_y.
🔍 Worked example — Standard
A group of n=20 students has a mean score of 65 and a standard deviation of 8. A second group of n=30 students has a mean score of 72 and a standard deviation of 10. Calculate the mean of the combined group.
🔍 Worked example — Exam-style
For a set of n=50 values, the coded data y = x - 100 has Σ y = 250 and Σ y^2 = 15000. Find the standard deviation σ of the original data x.
⚠️ Common errors
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🎯 Exam question
Worked examples — Example 1: from totals
For 20 values, Σ x = 360 and Σ x^2 = 6880. bar x = 18; σ = √(6880/20 - 18^2) = √(20) = 4.47.
🔍 Worked example — Easy
A set of 10 numbers has a sum of 50 and a sum of squares Σ x^2 = 300. Calculate the mean and standard deviation.
🔍 Worked example — Standard
For a sample of 20 data points, Σ x = 360 and Σ x^2 = 6880. Calculate the mean and standard deviation to 2 decimal places.
🔍 Worked example — Exam-style
A dataset has n=5, Σ x = 15, and Σ x^2 = 55. Calculate the standard deviation.
⚠️ Common errors
Worked examples — Example 2: coded totals
For 12 values, Σ(x - 50) = -24 and Σ(x - 50)^2 = 390. bar x = 50 + -24/12 = 48; σ = √(390/12 - (-2)^2) = √(28.5) = 5.34.
🔍 Worked example — Easy
A set of 10 numbers is coded by subtracting 10 from each value. Given Σ(x - 10) = 50, find the mean barx of the original data.
🔍 Worked example — Standard
For a sample of 12 values, Σ(x - 50) = -24 and Σ(x - 50)^2 = 390. Calculate the standard deviation σ of the original data.
🔍 Worked example — Exam-style
A data set of 20 values is coded by y = x - 100. Given Σ y = 200 and Σ y^2 = 5000, find the original mean barx and original standard deviation σ.
⚠️ Common errors
Worked examples — Example 3: grouped data and a histogram
Mid-points 5, 15, 30, 50; Σ f = 30, Σ fx = 695, Σ fx^2 = 22525. Mean = 23.2 min; σ = √(22525/30 - 23.17^2) = 14.6 min. Frequency densities: 0.5,\ 1.2,\ 0.4,\ 0.25.
🔍 Worked example — Easy
A set of data is grouped into intervals of width 10. The frequencies are 5, 12, 8, and 5. Calculate the frequency density for each interval.
🔍 Worked example — Standard
Given grouped data with mid-points x = 5, 15, 30, 50, frequencies f = 5, 12, 8, 5, and Σ fx = 695, calculate the mean barx.
⚠️ Common errors
Worked examples — Example 4: combining two groups
Group A: n = 10, bar x = 12, Σ x^2 = 1500. Group B: n = 15, bar x = 14, σ = 3. B: Σ x = 210, Σ x^2 = 15(9 + 196) = 3075. Combined: n = 25, Σ x = 330, Σ x^2 = 4575. Mean = 13.2, σ = √(183 - 13.2^2) = √(8.76) = 2.96.
🔍 Worked example — Easy
Group A has n=5, barx=10. Group B has n=5, barx=20. Calculate the combined mean.
🔍 Worked example — Standard
Group A: n=10, barx=12, Σ x^2=1500. Group B: n=15, barx=14, σ=3. Find the combined mean and combined Σ x^2.
🔍 Worked example — Exam-style
Using the results from the previous example (Combined n=25, Σ x=330, Σ x^2=4575), calculate the combined standard deviation σ.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) median 104; Q_1 = 98, Q_3 = 112, IQR = 14 g. (2) bar x = 12; σ = √(154 - 144) = 3.16. (3) bar x = 20 + 3 = 23; σ = √(45 - 9) = 6. (4) boundaries 15 and 25, width 10: density 3. (5) Box plot: medians and spreads (IQR) can be compared at a glance. Stem-and-leaf: every original mark is kept, so the shape and the exact values can be seen.