GCSE Statistics · Higher · Chapter 6: Averages · Lesson 2

Averages from grouped data

About 55 minutesCalculator allowed44 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
The 27 boxes in van A have a mean mass of $53$ kg. The 18 boxes in van B have a mean mass of $30$ kg. Work out the mean mass of all the boxes in the two groups together.
[2 marks]
Answer
A2
From chapter 4
Town R has 15\,000 people and 14% of them are aged 65 and over. Town S has 50\,000 people and 6% of them are aged 65 and over. Kemi says, "The bar for people aged 65 and over is longer on Town R's pyramid, so more people aged 65 and over live in Town R." Is Kemi correct?
[2 marks]
A3
Maths skill
A car is worth £ $1900$ . Its value falls by $30%$ each year. Find its value after 2 years.
[2 marks]
Answer
A4
Maths skill
A bag holds only red, blue and green counters. $P (red) = 0.15$ and $P (blue) = 0.55$ . There are $140$ counters in the bag. How many are green?
[2 marks]
Answer

Example, then your turn

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

Example 1

The table shows the times, $t$ minutes, that 40 people waited at a clinic. Work out an estimate of the mean waiting time. Explain why your answer is only an estimate.

Hint Use the midpoint of each class.

B1
The table shows the masses, $m$ grams, of 50 apples. Work out an estimate of the mean mass.
[3 marks]
Answer

Example 2

The table shows the heights, $h$ cm, of 35 plants. Write down the modal class. Find the class that contains the median.

Hint Half of 35 is 17.5. Add up the frequencies until you pass it.

B2
The table shows the ages, $a$ years, of 60 people at a gym class. Write down the modal class and the class that contains the median.
[2 marks]
Answer

Example 3

The table shows the distances, $d$ km, that 60 people travel to work. Estimate the median distance.

Hint Find how far into the median class the 30th value is.

B3
The table shows the lengths, $t$ minutes, of 80 phone calls. Estimate the median length.
[3 marks]
Answer

Example 4

The table shows the heights, $h$ cm, of 20 seedlings. Their actual heights add up to 171 cm. Compare the estimate of the mean with the true mean. Explain why they are different.

Hint Work out both means first.

B4
The table shows the times, $t$ minutes, of 40 commutes. Work out an estimate of the mean. The actual mean of the 40 times is 26.4 minutes. Give a reason why your estimate is different.
[4 marks]

Practice

C1
The table shows the ages, $a$ years, of 50 people at a café. The classes are not all the same width. Work out an estimate of the mean age.
[3 marks]
Answer
C2
Ana says, "There are five classes, so the median is in the middle one, $20 < x \leq 30$ ."
The table shows the number of minutes, $x$ , that 50 students spent on homework. Is Ana right? Give a reason.
[2 marks]

C3
The table shows the number of hours, $h$ , that some pupils spent reading last week. An estimate of the mean is 7.6 hours. Work out $x$ .
[3 marks]
Answer
C4
A table of 30 bus journey times has a last class $60 < t \leq 120$ minutes. Only two journeys are in it, and they took 61 and 65 minutes. Will the estimate of the mean from the table be too high or too low? Explain why.
[2 marks]

Exam-style questions

D1
Sam says, "The estimate of the mean is $(5 + 15 + 25 + 35 + 45) \div 5 = 25$ seconds."
The table shows the times, $t$ seconds, that 50 web pages took to load. Is Sam correct? You must show your working.
[3 marks]

D2
The table shows the waiting times, $t$ minutes, of 40 customers at Shop A.
(a)
Work out an estimate of the mean waiting time.
[3 marks]
Answer
(b)
Estimate the median waiting time. Give your answer to 2 decimal places.
[3 marks]
Answer
(c)
At Shop B, the median waiting time was 4.9 minutes. Compare the waiting times at the two shops.
[2 marks]
Total for D2: 8 marks

Extension

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

E1
The table shows the lengths, $l$ cm, of some leaves. One frequency is $x$ . An estimate of the median, found by interpolation, is 23 cm. Work out $x$ .
[3 marks]
Hint 1 · What to try
There are $40 + x$ leaves, so the median is the $\frac{40 + x}{2}$ th value. It is in $20 < l \leq 30$ .
Hint 2 · The first line
$23$ is $3$ cm into a class of width $10$ , so the median is $\frac{3}{10} \times 20 = 6$ values into the class.
Hint 3 · The full method
$\frac{40 + x}{2} = 8 + x + 6$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can estimate the mean from a grouped table and say why it is an estimate.
I can find the modal class, the median class and the median by interpolation.
I can say why an estimate differs from the true value.

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