Transcript S1 Measures of Dispersion

S1 Averages and Measures
of Dispersion
S1 Measures of Dispersion
Objectives:
To be able to find the median and
quartiles for discrete data
To be able to find the median and
quartiles for continuous data
using interpolation
Can you work out the rule for finding median
and quartiles from discrete data?
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LQ
1.5
Median
3
UQ
4.5
12345
123456
1234567
12345678
123456789
Can you spot any rules for n amount of
numbers in a list?
LQ  n/4
If n/4 is a whole number find the mid point
of corresponding term and the term above
If n is not a whole number, round the
number up and find the corresponding
term
UQ  3n/4
If 3n/4 is a whole number find the mid
point of corresponding term and the term
above
If n is not a whole number, round the
number up and find the corresponding
term
Median  n/2
If n/2 is a whole number find the midpoint
of the corresponding term and the term
above
If n/2 is not a whole number, round up and
find the corresponding term
Calculate the mean, median and inter quartile
range from a table of discrete data
Number of Number of
CDs(x)
students (f)
35
3
36
17
37
29
38
34
39
12
Mean = Σfx
Σf
Calculate the mean, median and inter quartile
range from a table of discrete data
Number of
CDs(x)
Number of
students (f)
Cumulative
frequency
35
3
3
36
17
20
37
29
49
38
34
83
39
12
95
Median = n/2
Median = 95/2 =
47.5 = 48th value
Median = 37 CDs
LQ = 95/4 = 23.75
LQ = 24th value
LQ (Q1) = 37 CDs
UQ (Q3) = 95/4 x 3
= 71.25
UQ = 72nd value
IQR = Q3-Q1 = 38-37=1
UQ (Q3) = 38 CDs
Calculate the mean, median and inter quartile
range from a table of continuous data
Length of
flower stem
(mm)
Number of
flowers (f)
30-31
2
2
32-33
25
27
34-36
30
57
37-39
13
70
Cumulative
frequency
Median = n/2
We do not need to
do any rounding
because we are
dealing with
continuous data
Median = 70/2 =
35th value
This lies in the 3436 class but we
don’t know the
exact value of the
term
Using interpolation to find an estimate for the
median
33.5mm
27
m
36.5mm
35
57
m – 33.5 = 35 - 27
36.5 – 33.5 = 57 - 27
m – 33.5 = 8
3
30
m – 33.5 = 0.26 x 3
m = 33.5 + 0.8 = 34.3
Using interpolation to find an estimate for the
lower quartile
LQ = 70/4 = 17.5 (in the 32-33 group)
31.5mm
2
Q1
33.5mm
17.5
27
Q1 – 31.5 = 15.5
2
25
Q1 – 31.5 = 0.62 x 2
Q1 = 31.5 + 1.24 = 32.74
Q1 – 31.5 = 17.5 - 2
33.5 – 31.5 = 27 - 2
Using interpolation to find an estimate for the
upper quartile
UQ = 70/4x3 = 52.5 (in the 34-36 group)
33.5mm
27
Q3
36.5mm
52.5
57
Q3 – 33.5 = 25.5
3
30
Q3 – 33.5 = 0.85 x 3
Q1 = 33.5 + 2.55 = 36.05
Q3 – 33.5 = 52.5 - 27
36.5 – 33.5 = 57 - 27
Summary of rules
n = total frequency
w = class width
fB = cumulative frequency below median/lq/uq
fU = cumulative frequency above median/lq/uq
Median = LB + ½n – fB x w
fU - fB
LQ
= LB + ¼n – fB x w
fU - fB
UQ
= LB + ¾n – fB x w
fU - fB
The lengths of a batch of 2000 rods were measured to the nearest
cm. The measurements are summarised below.
Length
Number of
Cumulative
Q1=74.5 + 500-250 x 5
(nearest cm) rods
frequency
60-64
65-69
70-74
75-79
80-84
85-89
90-94
95-99
11
49
190
488
632
470
137
23
738-250
11
60
250
738
1370
1840
1977
2000
Q1=77.06
Q2=79.5+1000-738 x 5
1370-738
Q2=81.57
Q3=84.5+1500-1370 x 5
1840-1370
Q3=85.88
By altering the formula slightly can you work out how to find the 3rd
decile (D3) and the 67th percentile (P67)?
Answers
D3=74.5 + 600-250 x 5
738-250
D3=78.09
P67=79.5 + 1340-738 x 5
1370-738
P67=84.26