Unit 16 Section 2 : Grouped Data

When dealing with grouped data it is important to think about the type of data that is being processed. You also have to decide the range of values that each group contains.

When calculating the mean of grouped data, we assume that all the values lie at the midpoint of the group.

These ideas are illustrated in the following examples.

Example 1

The table below shows the times taken by a group of walkers to complete a 15-mile walk. Their times have been recorded to the nearest hour.

Illustrate the data using a bar chart and a frequency polygon.

Time (hours)	3	4	5	6	7	8
Frequency	2	5	12	11	4	3

A time of 5 hours actually means a time that is greater than or equal to 4 hours but is less than 5 hours, so the bar representing this time on the bar chart will begin at 4.5 and end at 5.5.

Similarly, the bar for a time of 3 will begin at 2.5 and end at 3.5.

The bar chart is shown below:

The frequency polygon is shown below. We obtain it by joining the midpoints of the tops of the bars from the previous graph.

Example 2

At a school fair, visitors enter a 'Guess the weight of the cake' competition. Their guesses, rounded to the nearest 100 grams, were recorded in the following table:

Guess (kg)	0.5 - 0.7	0.8 - 1.0	1.1 - 1.3	1.4 - 1.6	1.7 - 1.9
Frequency	5	32	26	11	6

(a)

Illustrate the data using a bar chart.

The guesses have been recorded to one decimal place, in other words to the nearest 100 grams. This means that the first category, nominally described as '0.5 - 0.7 kg' actually includes guesses greater than or equal to 0.45 kg but less than 0.75 kg. The precise description of the first category is therefore

0.45 kg ≤ guess < 0.75 kg

The nominal descriptions of the other classes must also be interpreted precisely if we are to represent the data accurately.

Guess (kg)	0.45 ≤ G < 0.75	0.75 ≤ G < 1.05	1.05 ≤ G < 1.35	1.35 ≤ G < 1.65	1.65 ≤ G < 1.95
Frequency	5	32	26	11	6

The precise descriptions of the classes indicate how the bars should be drawn on the bar chart.

(b)

Estimate the mean of the data.

The mean can be estimated by assuming that all the values in a class are equal to the midpoint of the class.

Class	Midpoint	Frequency	Frequency × Midpoint
0.45 ≤ G < 0.75	0.6	5	5 × 0.6 = 3
0.75 ≤ G < 1.05	0.9	32	32 × 0.9 = 28.8
1.05 ≤ G < 1.35	1.2	26	26 × 1.2 = 31.2
1.35 ≤ G < 1.65	1.5	11	11 × 1.5 = 16.5
1.65 ≤ G < 1.95	1.8	6	6 × 1.8 = 10.8
TOTALS		80	90.3

Estimate of mean = = 1.12875 kg = 1.1 kg to 2 significant figures

(c)

State the modal class.

The modal class is the one with the highest frequency. In this case, the modal class has nominal description '0.8 - 1.0 kg', which means guesses in the interval 0.75 kg ≤ G < 1.05 kg, i.e. 750 grams ≤ G < 1050 grams.

Exercises

Question 1

The following table lists the results of a survey that recorded the heights of pupils in one year group. The heights have been given to the nearest 10 cm.

Height (cm)	140	150	160	170	180	190
Frequency	3	5	57	63	30	2

(a)

Illustrate the data on a bar chart.

(b)

Estimate the mean height of the pupils.

cm (to the nearest cm)

Question 2

The following table lists the masses of a group of students, recorded to the nearest kg:

Mass (kg)	60	61	62	63	64	65	66	67	68	69	70
Frequency	3	7	9	11	10	22	17	23	11	9	5

(a)

Illustrate the data using a frequency polygon.

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(b)

Estimate the mean mass for these students.

kg (to 1 d.p.)

Question 3

An English class looked at the number of words per sentence for an essay that one of them had written. Their results are summarised in the following table:

Number of Words	6 - 8	9 - 11	12 - 14	15 - 17	18 - 20
Frequency	13	10	8	4	3

(a)

Estimate the mean number of words per sentence.

(to the nearest integer)

(b)

What is the modal class?

Question 4

The time taken for people to solve a puzzle is recorded, to the nearest minute, in the following table:

Time (mins)	2 - 5	6 - 9	10 - 13	14 - 17	18 - 21
Frequency	3	19	20	12	6

Estimate the mean time taken to solve the puzzle.

minutes (to 1 d.p.)

Question 6

The heights of some plants grown in a laboratory were recorded after 4 weeks. The results are listed in the following table:

Height (cms)	11 - 15	16 - 20	21 - 25	26 - 30	31 - 35	36 - 40
Frequency	3	7	19	20	11	2

(a)

Draw a frequency polygon for the data.

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(b)

State the modal class.

(c)

Calculate an estimate of the mean height.

cm (to 1 d.p.)

Question 8

Children were asked to sell tickets for a school play. A record was kept of how many tickets each child sold.

Tickets Sold	0 - 10	11 - 20	21 - 50	51 - 100
Frequency	7	42	8	3

(a)

Estimate the mean number of tickets sold.

(to the nearest integer)

(b)

Estimate the total number of tickets sold.

tickets

Question 9

A company owns a fleet of 20 vans. The mileage on each van is recorded.
The results are given in the following table:

Mileage	0 ≤ M < 5000	5000 ≤ M < 10 000	10 000 ≤ M < 15000	15000 ≤ M < 20 000
Frequency	1	4	8	7

(a)

Illustrate the data with a bar chart.

(b)

Estimate the mean mileage.

miles

Question 10

Joshua is given the data below and asked to estimate the mean.

Value	100 - 104	105 - 109	110 - 114	115 - 119
Frequency	5	16	32	7

(a)

Calculate an estimate of the mean.

(to 1 d.p.)

(b)

Determine a value that the mean must be greater than.

(to 1 d.p.)

The lower bound is obtained by using the lower class boundaries, i.e.

Mean > = = 107.9 (to 1 d.p.)

(c)

Determine a value that the mean must be less than.

(to 1 d.p.)

Question 11

Lyn recorded the temperature at lunch time every day for a week.
She started to draw a bar chart to show her results.

(a)

The temperature on Friday was 25°C.
The temperature on Saturday was 19°C.
On a copy of Lyn's bar chart, draw the bars for Friday and Saturday.

What was the temperature on Monday?

°C

(b)

Five more pupils recorded the temperature every day for different weeks in the year.

Match the pupils' comments to their bar charts. The first is done for you.

PUPILS' COMMENTS

Pupil A:	"It was very warm at first, then it suddenly got much colder."
Pupil B:	"Each day was colder than the day before."
Pupil C:	"The temperature was about the same all week."
Pupil D:	"Each day was hotter than the day before."
Pupil E:	"There were 5 warm days and 2 cold days."

Pupil A:	Bar Chart 2
Pupil B:	Bar Chart
Pupil C:	Bar Chart
Pupil D:	Bar Chart
Pupil E:	Bar Chart