1. S1 Averages and Measures of Dispersion

2. 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

3. Can you work out the rule for finding median and quartiles from discrete data?
LQ Median UQ
Can you spot any rules for n amount of numbers in a list?

4. 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

5. 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

6. Calculate the mean, median and inter quartile range from a table of discrete data
Number of CDs(x)
Number of students (f) 35 3 36 17 37 29 38 34 39 12
Mean = Σfx Σf

7. Calculate the mean, median and inter quartile range from a table of discrete data
Median = n/2
Number of CDs(x)
Number of students (f) 35 3 36 17 37 29 38 34 39 12
Cumulative frequency
Median = 95/2 = 47.5 = 48th value 3 20 49 83 95
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
UQ (Q3) = 38 CDs
IQR = Q3-Q1 = 38-37=1

8. Length of flower stem (mm)
Calculate the mean, median and inter quartile range from a table of continuous data
Median = n/2
We do not need to do any rounding because we are dealing with continuous data
Length of flower stem (mm)
Number of flowers (f)
30-31 2
32-33 25
34-36 30
37-39 13
Cumulative frequency 2 27 57 70
Median = 70/2 = 35th value
This lies in the class but we don’t know the exact value of the term

9. Using interpolation to find an estimate for the median
33.5mm m
36.5mm 27 35 57
m – =
m – = 8
36.5 – =
m – = 0.26 x 3
m = = 34.3

10. Using interpolation to find an estimate for the lower quartile
LQ = 70/4 = 17.5 (in the group)
31.5mm Q1
33.5mm 2
17.5 27
Q1 – =
Q1 – = 15.5
33.5 – =
Q1 – = 0.62 x 2
Q1 = = 32.74

11. Using interpolation to find an estimate for the upper quartile
UQ = 70/4x3 = 52.5 (in the group)
33.5mm Q3
36.5mm 27
52.5 57
Q3 – =
Q3 – = 25.5
36.5 – =
Q3 – = 0.85 x 3
Q1 = = 36.05

12. 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

13. The lengths of a batch of 2000 rods were measured to the nearest cm
The lengths of a batch of 2000 rods were measured to the nearest cm. The measurements are summarised below.
Length (nearest cm)
Number of rods
60-64 11
65-69 49
70-74
190
75-79
488
80-84
632
85-89
470
90-94
137
95-99 23
Cumulative frequency
Q1= x 5
Q1=77.06 11 60
250
738
1370
1840
1977
2000
Q2= x 5
Q2=81.57
Q3= x 5
Q3=85.88
By altering the formula slightly can you work out how to find the 3rd decile (D3) and the 67th percentile (P67)?

14. Answers D3=74.5 + 600-250 x 5 738-250 D3=78.09 P67=79.5 + 1340-738 x 5
P67=84.26