Download Statistics, contd

PROGRAMME 27
STATISTICS
(contd)
STROUD
Worked examples and exercises are in the text
Programme 27: Statistics
(REMINDER)
Mean
The arithmetic mean of a set of n observations is their average:
x
x
sum of observations

mean =
that is x 
number of observations
n
When calculating from a frequency distribution, this becomes:
xf  xf

x

n
f
[Here x now means not the individual observations, but the different values for
which frequencies are counted – J.A.B.]
STROUD
Worked examples and exercises are in the text
(Simple) Coding Method for Calculating a Mean Manually
or Mentally
[Slide added by J.A.B.]
The textbook mentions a “coding” method for calculating the mean. In class I
go through a simplified, very useful version of this. It’s easy: instead of
averaging the values themselves directly, you take a convenient number, the
“base”, that’s very roughly in the middle of or near to the values. You work out
their (positive or negative) deviations from that base value, take the average of
those deviations, and then add that average to the base. The result is the average
of the original values.
Exercise: try it with 8 values between, say, 50 and 85, using, say, 60 or 70 as the
base. Compare the difficulty of doing this with adding the values and dividing
by 8. Also check that it doesn’t matter what base you choose, leaving you free to
pick a convenenient round number.
Exercise: explain why the method works in general.
STROUD
Worked examples and exercises are in the text
Programme 27: Statistics
(NEW)
Mode of a set of data
The mode of a set of data is that value of the variable that occurs most often.
The mode of:
2, 2, 6, 7, 7, 7, 10, 13
is clearly 7. The mode may not be unique, for instance the modes of:
23, 25, 25, 25, 27, 27, 28, 28, 28
are 25 and 28.
STROUD
Worked examples and exercises are in the text
Programme 27: Statistics
Modal Class of a grouped frequency distribution
The modal class of grouped data is the class with the greatest population.
For example, the modal class of:
is the third class.
STROUD
Worked examples and exercises are in the text
Programme 27: Statistics
Mode of a grouped frequency distribution
Plotting the histogram of the data enables the mode to be found:
STROUD
Worked examples and exercises are in the text
Programme 27: Statistics
Mode of a grouped frequency distribution, contd
The mode can also be calculated algebraically:
If L = lower boundary value
l = AB = difference in frequency on the
lower boundary
u = CD = difference in frequency on the
upper boundary
c = class interval
the mode is then:
 l 
mode  L  
c
l

u


STROUD
Worked examples and exercises are in the text
Programme 27: Statistics
Mode of a grouped frequency distribution, contd
For example, the modal class of:
L = ...... l = .....
u = ...... c = .....
 l 
mode  L  
c
l

u


STROUD
Worked examples and exercises are in the text
Programme 27: Statistics
Mode of a grouped frequency distribution, contd
For example, the modal class of:
L = 15.5
u= 3
l = 16-7 = 9
c = 16-10=6
 l 
mode  L  
c
l

u


STROUD
15.5+9/(9+6)*3 = 15.5 + 1.8 = 17.3
Worked examples and exercises are in the text
Programme 27: Statistics
Median of a set of data
The median is the value of the middle datum when the data is arranged in
ascending or descending order.
If there is an even number of values the median is the average of the two
middle data.
STROUD
Worked examples and exercises are in the text
Programme 27: Statistics
Median with grouped data
In the case of grouped data the median divides the population of the largest
block of the histogram into two parts:
6  12  15  A  B  13  9  5
In this frequency distribution A + B = 20
so that A = 7:
7
The width of A 
 class interval
20
 0.35  0.3
 0.105
Therefore, Median = 30.85 + 0.105
= 30.96
STROUD
A B
Worked examples and exercises are in the text
Programme 27: Statistics
Introduction
Arrangement of data
Histograms
Measure of central tendency
Dispersion
STROUD
Worked examples and exercises are in the text
Programme 27: Statistics
Dispersion
Range
Standard deviation
Alternative formula for the standard deviation
STROUD
Worked examples and exercises are in the text
Programme 27: Statistics
Dispersion
Range
The mean, mode and median give important information about the central
tendency of data but they do not tell anything about the spread or dispersion
about the centre.
For example, the two sets of data:
26, 27, 28 ,29 30 and 5, 19, 20, 36, 60
both have a mean of 28 but one is clearly more tightly arranged about the mean
than the other. The simplest measure of dispersion is the range – the difference
between the highest and the lowest values.
STROUD
Worked examples and exercises are in the text
Programme 27: Statistics
Dispersion
Standard deviation
The standard deviation is the most widely used measure of dispersion.
The variance of a set of data is the average of the square of the difference in
value of a datum from the mean:
( x1  x )2  ( x2  x ) 2 
variance 
n
 ( xn  x ) 2
This has the disadvantage of being measured in the square of the units of the
data. The standard deviation is the square root of the variance:
n
standard deviation   
STROUD
(x  x )
i 1
2
i
n
Worked examples and exercises are in the text
Programme 27: Statistics
Dispersion
Alternative formula for the standard deviation
Since:
n

(x  x )
i
i 1

n
n

n
2
x
i 1
2
i
n
(x
i 1
2
i
n
n
 2 x  xi   x
i 1
n
 2 xi x  x 2 )
i 1
n
2

x
i 1
2
i
 2nx 2  nx 2
n
n

That is:
STROUD
x
i 1
n
2
i
 x2
  x2  x 2
Worked examples and exercises are in the text