GCSE Statistics · Higher · Chapter 6: Averages · Lesson 1

Mean, median and mode

About 55 minutesCalculator allowed49 marks

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Date

Do Now

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
A class has 10 boys and 20 girls. The boys' mean test mark is $80$ and the girls' mean test mark is $74$ . Find the mean mark for the whole class.
[2 marks]
Answer
A2
Last lesson
A cumulative frequency graph shows the marks of $200$ students. Reading up from $40$ gives a cumulative frequency of $71$ . How many of the students had $40$ or less, and what percentage of the students is that?
[2 marks]
Answer
A3
Maths skill
Write $480$ g : $0.4$ kg as a ratio in its simplest form.
[2 marks]
Answer
A4
Maths skill
$\frac{3}{8}$ of $40$ is equal to $\frac{1}{4}$ of $B$ . Find $B$ .
[2 marks]
Answer

Example, then your turn

Your teacher works through each example with you. Then try the one beside it.

Example 1

The prices, in £1000s, of seven houses in a street are $185$ , $192$ , $198$ , $204$ , $210$ , $215$ and $960$ . Work out the mean and the median. Which average better represents a typical price? Give a reason.

Hint Look for a value far from the rest.

B1
Eight students did a puzzle. Their times, in minutes, were $12$ , $14$ , $15$ , $15$ , $17$ , $18$ , $19$ and $58$ . Work out the mean and the median. Which average better represents a typical time? Give a reason.
[4 marks]

Example 2

The table shows the number of goals scored in 30 football matches. Work out the mean, the median and the mode.

Hint Mean

= Σ f x \div Σ f

. The median is the middle of the 30 matches.

B2
The table shows the number of pets in each of 40 homes. Work out the mean, the median and the mode.
[4 marks]
Answer

Example 3

In a test, 24 students in Group A had a mean mark of 62, and 26 students in Group B had a mean mark of 70. Work out the mean mark of all 50 students.

Hint Find each group's total mark first.

B3
A running club has 18 members in the morning group, with a mean 5 km time of 31.5 minutes, and 12 in the evening group, with a mean of 28 minutes. Work out the mean time of all 30 members.
[2 marks]
Answer

Example 4

A job applicant is scored on an interview (weight 50%), a test (30%) and a reference (20%). She scores $72$ , $85$ and $60$ . Work out her weighted mean score.

Hint Multiply each score by its weight, then add.

B4
A café sold 120 teas at £1.80, 200 coffees at £2.40 and 80 hot chocolates at £2.75 in a day. Work out the mean price of a drink sold.
[2 marks]
Answer

Example 5

A shop's sales rose by 10%, rose by 25%, then fell by 4% over three years. Use the geometric mean to find the average yearly percentage change. Explain why the mean of 10, 25 and $- 4$ is not used.

Hint The multipliers are

1.10

1.25

and

0.96

B5
The number of members of a gym changed by $+ 2%$ , $+ 15%$ and $- 6%$ in three years. Work out the geometric mean of the multipliers, and interpret it. Give the percentage to 1 decimal place.
[3 marks]

Practice

C1
The mean mass of 9 parcels is 4.2 kg. One of them, of 11.4 kg, was weighed wrongly and is removed. Work out the mean mass of the other 8 parcels.
[2 marks]
Answer
C2
Leo says, "The shop should use the mean shoe size to decide which size to order most of."
A shoe shop records the size of every pair it sells. Is Leo right? Give a reason.
[2 marks]

C3
The mean of six test marks is 15. A seventh mark is added and the mean becomes 16. Work out the seventh mark.
[2 marks]
Answer
C4
A firm has 20 workers who earn between £21 000 and £34 000 a year, and a director who earns £240 000. Explain why the median is a better average than the mean for the salaries at this firm.
[2 marks]
C5
In a survey, 4 students read no books, $x$ students read 1 book, 10 read 2 books and 6 read 3 books last month. The mean number of books read was 1.6. Work out $x$ .
[3 marks]
Answer

Exam-style questions

D1
The manager says, "The mean output of all 40 workers is $(46 + 38) \div 2 = 42$ units."
At a factory, 30 day-shift workers made a mean of 46 units each, and 10 night-shift workers made a mean of 38 units each. Is the manager correct? You must show your working.
[3 marks]

D2
The table shows the number of days that each of 40 pupils was absent in a term.
(a)
Work out the mean number of days absent.
[2 marks]
Answer
(b)
The head teacher wants one number to describe a typical pupil's absence. Should she use the mean or the median? Give a reason.
[2 marks]
Total for D2: 4 marks
D3
A bus ticket's price changed over three years. Its chain base index numbers were $112$ , $95$ and $108$ .
(a)
Work out the geometric mean of the index numbers, to 1 decimal place. Interpret your answer in context.
[3 marks]
(b)
Mia says, "The mean of $112$ , $95$ and $108$ gives a 5% rise a year, so it is just as good." Explain why the geometric mean is better.
[2 marks]
Total for D3: 5 marks

Extension

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
The value of a painting rose by the same percentage in each of two years, then fell by 20% in the third year. The geometric mean yearly change over the three years was $+ 2%$ . Work out the percentage rise in each of the first two years, to 1 decimal place.
[3 marks]
Hint 1 · What to try
Write the three multipliers: $m$ , $m$ and $0.8$ .
Hint 2 · The first line
$m \times m \times 0.8 = 1.0 2^{3}$ .
Hint 3 · The full method
$m^{2} = 1.061208 \div 0.8 = 1.32651$ , so $m = 1.1517$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can choose the best average and say why.
I can find averages from a frequency table, and combined and weighted means.
I can find and interpret a geometric mean.

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