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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