Mode, Median and Mean Great Marlow School Mathematics Department

Mode, Median and Mean

Great Marlow School Mathematics Department

Calculating the mode, median and mean for a grouped frequency distribution of discrete data

Number of phones x

Frequency f

0 - 4 5

5 - 9 8

10 - 14 4

15 - 19 9

20 - 24 3

• The modal group is 15 – 19 phones.

• It is impossible to find the exact value of the median when the data has been grouped.

It can be estimated by using interpolation.

The total frequency Σf = 29

The position of the median is

29 + 1 = 15th value

2

To find this value, use a table with running values.

A local shop sells mobile phones and keeps a record of the daily

sales. The table shows these sales.


Estimating the median using interpolationNumber of phones x

Frequency f Running total

0 - 4 5 55 - 9 8 13

10 - 14 4 1715 - 19 9 2620 - 24 3 29

Look at the group 10 – 14.

The group is 5 numbers wide: 10, 11, 12, 13 and 14.

The frequency is 4, so the group has 4 values in it.

The 15th value is the 2nd of these 4 values.

So the estimate would be 2 of the way through this group.

4

An estimate of the median = 10 + 2 x 5 = 12.5

4 Great Marlow School Mathematics Department

Estimating the meanNumber of phones x

Frequency f Mid-values x f x x

0 - 4 5 2 5 x 2 = 105 - 9 8 7 8 x 7 = 56

10 - 14 4 12 4 x 12 = 4815 - 19 9 17 9 x 17 = 15320 - 24 3 22 3 x 22 = 66

It is impossible to find the exact value of the mean when the data has

been grouped.

It can be estimated by using the mid-values of each group.

The mid-values are ½(0 + 4) = 2, ½(5 + 9) = 7 and so on.

To work out the mean add two extra columns to your table.

Use the mid-values as the x values.

Estimate of the mean:

Σf = 29 Σfx = 333

f

fx__X = 5.11

29

333


Calculating the mode, median and mean for a grouped frequency distribution of continuous data

• You cannot give an exact value to continuous data because it is impossible to measure it exactly. Continuous data always has to be given to a chosen degree of accuracy. The groups for continuous data need some adjusting when working out means and medians.

• Measurements of time, weight height and speed are often given to the nearest unit. The groups for these are usually written like this:


This means 30.5 up to, but not including 40.5

You use 30.5 and 40.5 to work out the mid-value

You use 30.5 as the beginning of the median calculations.

This means 200 up to, but not including 300

You use 200 and 300 to work out the mid-value

You use 200 as the beginning of the median calculation

21 – 30

31 – 40

41 – 50

100 –

200 –

300 –

Special care is needed with age, if it is counted in completed years.

21 – 30

31 – 40

41 – 50

This means 31 up to, but not including 41

You use 31 and 41 to work out the mid-value

You use 31 as the beginning of the median calculation Great Marlow School Mathematics Department

Sometimes algebra is used in a table to show the size of the group.

10 < x 20

10 is not included in this group but 20 is included

You use 10 and 20 to work out the mid-values

You use 10 as the beginning of the median calculations

10 X < 20

10 is included in the group but 20 is not included

You still use 10 and 20 to work out the mid-value

You still use 10 as the beginning of the median calculations

Example: The table shows the time in minutes, to the nearest minute, spent by people travelling to work.

Time in minutes x

Number of People f

Frequency

0 - 14 10

15 - 29 14

30 - 44 12

45 - 59 7

Estimate the median time spent travelling to work.

The median is 44 / 2 = 22nd value

An estimate for the median = 14.5 + 12/14 x (29.5 – 14.5)

= 27.4 minutes to 1 d.p.


Exercise 3:4 Question 1Dillon collects apples from the tree in his garden. He weighed each one

and recorded the weight, to the nearest gm, in a table.Weight in grams x

Number of apples f

121 - 140 8

141 - 160 6

161 - 180 9

181 - 200 5

201 - 220 4

Weight in grams x

Number of apples f

Mid-values x

f x x

121 - 140 8141 - 160 6161 - 180 9181 - 200 5

(a) Write down the modal group.(b) Estimate the mean weight, to the nearest gram, of

Dillon’s apples.(c) Estimate the median weight, to the nearest gram, of an

apple in Dillon’s garden.

Σf = Σfx =


Exercise 3:4 Question 1Dillon collects apples from the tree in his garden. He weighed each one

and recorded the weight, to the nearest gm, in a table.Weight in grams x

Number of apples f

121 - 140 8

141 - 160 6

161 - 180 9

181 - 200 5

201 - 220 4

Weight in grams x

Number of apples f

Mid-values x

f x x

121 - 140 8 130.5 1044141 - 160 6 150.5 903161 - 180 9 170.5 1534.5181 - 200 5 190.5 952.5201 - 220 4 210.5 842

(a) Modal group = 161 – 180 gm

Σf = 32 Σfx = 5276

__X =

f

fx(b) An estimate of the mean = 5276 = 164.875 = 165

gm

(c) The median is the = 33/2 = 16.5 = 17th value

An estimate of the median =160.5 + 3/9 x (180 – 161) = 166.8 gms


Question 2: Carla measures the height of pupils in her class in cm. The table gives her measurements.

Height in cm x Number of students f

110 ≤ x < 120 10

120 ≤ x < 130 6

130 ≤ x < 140 5

140 ≤ x < 150 4

150 ≤ x < 160 3


Mid-values x

f x x

110 ≤ x < 120 10

120 ≤ x < 130 6

130 ≤ x < 140 5

140 ≤ x < 150 4

For the pupils in Carla’s class:

(a) Write down the modal group.

(b) Estimate the mean height to the nearest centimetre.

(c) Estimate the median height to the nearest centimetre.

Σf = Σfx =


Question 2: Carla measures the height of pupils in her class in cm. The table gives her measurements.


110 ≤ x < 120 10

120 ≤ x < 130 6

130 ≤ x < 140 5

140 ≤ x < 150 4

150 ≤ x < 160 3


Mid-values x

f x x

110 ≤ x < 120 10 115 1150

120 ≤ x < 130 6 125 750

130 ≤ x < 140 5 135 675

140 ≤ x < 150 4 145 580

150 ≤ x < 160 3 155 465

(a) The modal group =

(b) An estimate of the mean = 3620/28 = 129.28 cm = 129.3 cm

(c) The median is the 29/2 = 14.5 = 15th value

An estimate of the median = 120 + 5/6 x (130 – 120) = 128.3 cm

Σf = 28 Σfx = 3620

110 ≤ x < 120


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Mode, Median and Mean Great Marlow School Mathematics Department