Cumulative frequency - Teacher Slides.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Cumulative frequency

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Lesson 2 of 6

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

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Lesson 2 of 6

Answer each one, then check.

1

What is the median?

2

What is the range?

3

Running total of 4, 11, 20?

4

What is a grouped frequency table?

5

Write the class 20 < t ≤ 30 in words.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

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Lesson 2 of 6

1

What is the median?

The middle value when the data are in order

2

What is the range?

Largest minus smallest

3

Running total of 4, 11, 20?

4, 15, 35

4

What is a grouped frequency table?

A table showing how many values fall in each class

5

Write the class 20 < t ≤ 30 in words.

More than 20 and up to and including 30

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

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Lesson 2 of 6

1

Work out cumulative frequencies from a grouped table.

2

Plot a cumulative frequency graph.

3

Estimate the median, lower quartile and upper quartile.

4

Find the interquartile range and answer questions from the graph.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Further statistics

Lesson 2 of 6

CUMULATIVE FREQUENCY

Cumulative frequency is a running total of the frequencies. Plot each total against the upper class boundary.

Points are joined with a smooth curve that always rises.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Building the Table

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Lesson 2 of 6

Time taken (t minutes) to complete a puzzle, 60 students.

Time

Frequency

Cumulative frequency

0 < t ≤ 10

4

4

10 < t ≤ 20

11

15

20 < t ≤ 30

20

35

30 < t ≤ 40

16

51

40 < t ≤ 50

7

58

50 < t ≤ 60

2

60

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Reading the Curve

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Lesson 2 of 6

Read the quartiles at one quarter, one half and three quarters of the total.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Reading Quartiles

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Lesson 2 of 6

Use the total frequency n.

1

Median

Go across at n/2, then down to the x-axis

2

Lower quartile

Go across at n/4, then down

3

Upper quartile

Go across at 3n/4, then down

4

Interquartile range

IQR = UQ − LQ

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Median and Quartiles

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Lesson 2 of 6

For the 60 students, estimate the median, the quartiles and the interquartile range.

1

Total

n = 60

2

Median

At 30: about 27.5 minutes

3

Lower quartile

At 15: 20 minutes

4

Upper quartile

At 45: about 36 minutes

5

Interquartile range

36 − 20 = 16 minutes

ANSWER

Median ≈ 27.5 minutes, IQR ≈ 16 minutes.

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How Many More Than...?

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Lesson 2 of 6

Use the graph to estimate how many of the 60 students took more than 40 minutes.

1

Cumulative frequency at 40

51 students took 40 minutes or less

2

Subtract from the total

60 − 51 = 9

ANSWER

About 9 students took more than 40 minutes.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Common Mistakes

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Lesson 2 of 6

Watch for these.

Wrong boundary

Plot at the upper class boundary, not the middle of the class.

Straight lines

The graph is a smooth curve, not a polygon (unless the question says so).

Reading frequency, not value

Go across from the cumulative frequency, then down to the data value.

Forgetting the start

The curve starts at the lowest boundary with a cumulative frequency of 0.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

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Lesson 2 of 6

Cumulative frequency

A running total of frequencies.

Upper class boundary

The largest value in a class.

Median

The middle value; the value at n/2.

Lower quartile

The value one quarter of the way through the data.

Upper quartile

The value three quarters of the way through the data.

Interquartile range

Upper quartile minus lower quartile.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Further statistics

Lesson 2 of 6

YOUR TASK

Build a Cumulative Frequency Table

12 minutes

Heights (h cm) of 40 plants: 0 < h ≤ 10: 3, 10 < h ≤ 20: 9, 20 < h ≤ 30: 16, 30 < h ≤ 40: 8, 40 < h ≤ 50: 4. (a) Complete the cumulative frequency column. (b) At what value of n do you read the median?

1

Add each frequency to the running total.

2

The last total equals the total frequency.

WHAT A GOOD ANSWER SHOWS

(a) 3, 12, 28, 36, 40. (b) 40/2 = 20, which lies in the class 20 < h ≤ 30.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

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Lesson 2 of 6

Find cumulative frequencies.

Plot at upper class boundaries.

Draw a smooth curve.

Read the median.

Read the quartiles.

Find the IQR.

Find how many are above or below a value.

Avoid common mistakes.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Plot cumulative frequency against upper class boundaries.

Median at n/2, LQ at n/4, UQ at 3n/4.

IQR = UQ − LQ.

Values read from a graph are estimates.

EXAM FOCUS

The cumulative frequency graph shows the times taken by 80 students. Use it to estimate the median and the interquartile range. (4 marks)

Show your dashed lines on the graph and write the values down.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Cumulative frequency

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Lesson 2 of 6

Answer all questions. Show your working. · 30 minutes

Question 1 · 4 marks · Use the graph

The cumulative frequency graph shows the times, in minutes, taken by 80 students to complete a puzzle. Use the graph to find an estimate…

Question 2 · 2 marks · Use the graph

Use the same graph of the times taken by 80 students to estimate how many students took more than 45 minutes.

Question 3 · 2 marks · Complete

The table shows the times taken by 60 students. Frequencies: 0 < t ≤ 10: 4, 10 < t ≤ 20: 11, 20 < t ≤ 30: 20, 30 < t ≤ 40: 16, 40 < t ≤ 50…

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Exam Practice: Cumulative frequency (cont.)

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Lesson 2 of 6

Question 4 · 2 marks · Explain

Explain why the points on a cumulative frequency graph are plotted at the upper class boundaries.

Question 5 · 3 marks · Draw

Describe how to draw a cumulative frequency graph from a table of grouped data.

Question 6 · 2 marks · Work out

A cumulative frequency graph shows the lower quartile is 20 and the upper quartile is 36. Work out the interquartile range and explain what…

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · 4 marks · Use the graph

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Lesson 2 of 6

The cumulative frequency graph shows the times, in minutes, taken by 80 students to complete a puzzle. Use the graph to find an estimate for (a) the median (b) the interquartile range. (4 marks)

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 1 · mark scheme

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Lesson 2 of 6

4 marks available. Award a mark for each point made.

Median: reads across at 40

M1

30 (accept 29 to 31)

A1

LQ and UQ read at 20 and 60

M1

IQR about 17

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · 2 marks · Use the graph

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Lesson 2 of 6

“

Use the same graph of the times taken by 80 students to estimate how many students took more than 45 minutes.

HOW TO ANSWER IT

Command word: Use the graph. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 2 · mark scheme

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Lesson 2 of 6

2 marks available. Award a mark for each point made.

Reads about 72 at 45

M1

80 − 72 = 8

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · 2 marks · Complete

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Lesson 2 of 6

“

The table shows the times taken by 60 students. Frequencies: 0 < t ≤ 10: 4, 10 < t ≤ 20: 11, 20 < t ≤ 30: 20, 30 < t ≤ 40: 16, 40 < t ≤ 50: 7, 50 < t ≤ 60: 2. Work out the cumulative frequencies.

HOW TO ANSWER IT

Command word: Complete. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 3 · mark scheme

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Lesson 2 of 6

2 marks available. Award a mark for each point made.

At least 3 correct

M1

All correct

A1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · 2 marks · Explain

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Lesson 2 of 6

“

Explain why the points on a cumulative frequency graph are plotted at the upper class boundaries.

HOW TO ANSWER IT

Command word: Explain. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 4 · mark scheme

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Lesson 2 of 6

2 marks available. Award a mark for each point made.

Cumulative frequency includes the whole class

M1

Only true at the upper boundary

C1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · 3 marks · Draw

Further statistics

Lesson 2 of 6

“

Describe how to draw a cumulative frequency graph from a table of grouped data.

HOW TO ANSWER IT

Command word: Draw. Worth 3 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 5 · mark scheme

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Lesson 2 of 6

3 marks available. Award a mark for each point made.

Running totals

B1

Plot at upper boundaries

B1

Smooth curve

B1

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · 2 marks · Work out

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Lesson 2 of 6

“

A cumulative frequency graph shows the lower quartile is 20 and the upper quartile is 36. Work out the interquartile range and explain what it tells you.

HOW TO ANSWER IT

Command word: Work out. Worth 2 marks, so plan before writing.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Question 6 · mark scheme

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Lesson 2 of 6

2 marks available. Award a mark for each point made.

16

B1

The spread of the middle 50%

C1

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