Organising data: linear

interpolation

OBJECTIVES:

To find the median and quartiles

from grouped data using linear

interpolation.

Interpolation

When you estimate a value between two known values.

Example: Suppose a small jar of sweets contains 50 sweets,

and a large jar contains 100. We can estimate how many

sweets will go in the medium jar.

We interpolate (estimate) roughly between

these 2 values to get the medium jar

containing 75 sweets.

Proper method = more accurate

Finding the median from a grouped

frequency table

Example: Parcels

Calculate an estimate of the median weight, shown to the nearest gram,

in the following grouped frequency table:

Weight (g) 1-10 11-20 21-30 31-40 41-50

Frequency 10 13 28 15 9

Finding the median from a

grouped frequency table

Weight (g) 1-10 11-20 21-30 31-40 41-50

Frequency 10 13 28 15 9

Cumulative

Frequency

10 23 51 66 75

The median lies in the ½ (75) = 37.5 therefore:

38th position

Therefore the median lies in the 21 – 30 class.

Remember it was rounded:

ACTUAL CLASS BOUNDARY: 20.5 – 30.5

Weight (g) 11-20 21-30

Frequency 13 28

Cumulative

Frequency

23 51

Assumption:

data are evenly spread over

each class interval

Useful diagram:

20.5 30.5 Class width = 10

23rd item 51st item 38th item

28 items in class

(51 – 23)

15 items

(38 - 23)

Lower class boundary + number of items up to median x class width

Number of items in the class

.).1(9.25

....85714.25

1028

155.20

pd

Estimating upper quartile

Weight (g) 1-10 11-20 21-30 31-40 41-50

Frequency 10 13 28 15 9

Cumulative

Frequency

10 23 51 66 75

Q3 = ¾ of 75 = 56.25 ≈ 57th value

57th is in the interval 30.5 – 40.5

The group below this account for 51 parcels, leaving 6 (57 – 51)

to get to the 57th.

So the 57th parcel is 6 out of 15in the interval 30.5 – 40.5.

Assuming equal spacing: UQ = 1015

65.30

NOW lets do this:

Find the median, LQ and UQ

The table shows the distributions of the weights, to the

nearest 0.1kg, of the babies born in a hospital during a

14-day period.

Weight

(kg)

2.0 – 2.9 3.0 – 3.1 3.2 – 3.3 3.4 – 3.5 3.6 – 3.9 4.0 – 4.4

Frequency 3 7 10 8 4 2

Cum Freq

Actual class

width

Exam question:

The time taken for 55 pupils to eat their lunch, to the nearest minute, are given

below. Work out the median time taken.

Time

(min)

3-4 5-9 10-19 20-29 30-44 45-60

Freq 2 7 16 21 9 0

Exercise C page 17

Numbers 1, 2, 3