Download 2. Measures of Location and measures of spread

A.MEASURES OF LOCATION
&
B.MEASURES OF SPREAD
Central tendency and measures of dispersion
Measures of
Location
Central tendency
Spread
Dispersion tendency
A Measures of Location (Central tendency)
Common measures of location are
1.
Mean
2.
Median
3.
Mode
1. Mean
Mean is of 3 types such as
a. Arithmetic Mean/Average
b. Harmonic Mean
c. Geometric Mean
Arithmetic Mean
The most widely utilized measure of central
tendency is the arithmetic mean or average.
The population mean is the sum of the values of the
variables under study divided by the total
number of observations in the population. It is
denoted by μ (‘mu’). Each value is algebraically
denoted by an X with a subscript denotation ‘i’.
For example, a small theoretical population
whose objects had values 1,6,4,5,6,3,8,7 would
be denoted X1 =1, X2 = 6, X3 = 4……. X8=7 …….1.1
Mean….
We would denote the population size with a
capital N. In our theoretical population N=8.
The pop. mean μ would be
1 6  4  5  6  3  8  7
 5
8
Formula 1.1: The algebraic shorthand formula
N
for a pop. mean is
Xi

i 1
μ=
N
Mean…..
• The Greek letter  (sigma) indicates summation,
the subscript i=1 means to start with the first
observation, and the superscript N means to
continue until and including the Nth observation.
For the example above,  Xi would indicate the
sum of X2+X3+X4+X5 or 6+4+5+6 = 21. To reduce
clutter, if the summation sign is not indexed, for
example  Xi, it is implied that the operation of
addition begins with the first observation and
continues through the last observation in a
X

population, that is,
=
5
i2
N
i 1
Xi
i
Mean…
N
X
i
The sample mean is defined by X =
n
Where n is the sample size. The sample mean is
usually reported to one more decimal place
than the data and always has appropriate
units associated with it.
The symbol X (X bar) indicates that the
observations of a subset of size n from a
population have been averaged.
i 1
Mean….
X is fundamentally different from μ because
samples from a population can have different
values for their sample mean, that is, they
can vary from sample to sample within the
population. The population mean, however, is
constant for a given population.
Mean…..
Again consider the small theoretical population
1,6,4,5,6,3,8,7. A sample size of 3 may consists
of 5,3,4 with X = 4 or 6,8,4 with X = 6.
Actually there are 56 possible samples of size 3
that could be drawn from the population 1.1.
Only four samples have a sample mean the
same as the population mean ie X = μ.
Mean…
Sample
X3, X6, X7
X2, X3, X4
X5, X3, X4
X8, X6, X4
Sum
4+3+8
6+4+5
6+4+5
7+3+5
X
5
5
5
5
Mean…
Each sample mean X is an unbiased estimate of
μ but depends on the values included in the
sample size for its actual value. We would
expect the average of all possible X ‘s to be equal
to the population parameter, μ . This is in fact,
the definition of an unbiased
of the pop. mean.
estimator
Mean…
If you calculate the sample mean for each of the
56 possible samples with n=3 and then average
these sample means, they will give an average
value of 5 , that is, the pop. mean, μ. Remember
that most real populations are too large or too
difficult to census completely, so we must rely
on using a single sample to estimate or
approximate the population characteristics.
Harmonic mean
Geometric mean
n= no of obs., X1, X2, X3……..Xn are individual obs.
Median
The second measure of central tendency is the
MEDIAN. The median is the middle most value
of an ordered list of observations. Though the
idea is simple enough, it will prove useful to
define in terms of an even simple notion. The
depth of a value is its position relative to the
nearest extreme (end) when the data are
listed in order from smallest to largest.
Median: Example 2.1
Table below gives the circumferences at chest
height (CCH) in cm and their corresponding
depths for 15 sugar maples measured in a
forest in Ohio.
CCH
Depth
18
1
21 22 29 29 36 37 38 56 59 66 70 88 93 120
2
3
4
5
6
7
8
7
6
5 4
3
No. of obs. = 15 (odd)
The population median M is the observation whose
depth is d = N 2 1 , where N is the population size.
2
1
Median…
A sample median M is the statistic used to
approximate or estimate the population
median. M is defined as the observation
n 1
whose depth is d = 2 where n is the
sample size. In example 2.1 the sample size is
n=15 so the depth of the sample median is
d=8. the sample median X n  1 = X8 = 38 cm.
2
Median: Example 2.2
The table below gives CCH (cm) for 12 cypress
pines measured near Brown lake on North
Stradebroke Island
CCH
17 19 31 39 48 56 68 73 73 75 80 122
Depth 1 2 3 4 5 6 6
5
4 3 2
1
No. of observation = 12 (even)
12  1
Since n=12, the depth of the median is
= 6.5. Obviously no observation has
2
depth 6.5 , so this is the interpretation as the average of both observations whose
depth is 6 in the list above. So M = 56  68 = 62 cm.
2
Mode
The mode is defined as the most
frequently occurring value in a data
set. The mode in example 2.2 would
be 73 cm while example 2.1 would
have a mode of 29 cm.
Mean, median and mode concide
• In symmetrical
distributions (NORMAL
DISTRIBUTION), the
MEAN, MEDIAN and
MODE coincide.
Exercise
Hen egg sizes(ES,g) on 12 wks of lay were
randomly measured in a layer flock as follows.
Determine mean, median and mode of eggs.
size.
Hen 01
No.
02
03
04
05
06
07
08
09
10
11
12
ES
41
47
50
49
44
46
41
39
38
45
40
44
B Measures of Spread (dispersion)
It measures variability of data. There are 4
measures in common.
1. Range
2. Variance
3. Standard Deviation (SD)
4. Standard Error (SE)
Range
Range: The simplest measure of dispersion or
spread of data is the RANGE
Formula: The difference between the largest
and smallest observations (two extremes) in a
group of data is called the RANGE.
Sample range= Xn – X1 ; Population range=XN-X1
The values Xn and X1 are called ‘sample range
limits’.
Range: Example
Marks of Biometry of 10 students are as follows
(Full marks 100)
Student ID
Marks Obtained
Marks ordered
01
35
80
02
40
75
03
30
70
04
25
60
05
75
40
06
80
40
07
39
39
08
40
35
09
60
30
10
70
25
Here, Range =
X1-X10=80-25
= 55
Range…
The range is a crude estimator of dispersion
because it uses only two of the data points and is
somewhat dependent on sample size. As sample
size increases, we expect largest and smallest
observations to become more extreme. Therefore,
sample size to increase even though population
range remains unchanged. It is unlikely that
sample will include the largest and smallest values
from the population, so the sample range usually
underestimates the population range and is
,therefore, a biased estimator.
Variance
Suppose we express each observation as a
distance from the mean xi = Xi - X . These
differences are called deviates and will be
sometimes positive (Xi is above the mean) and
sometimes negative (Xi is below the mean). If
we try to average the deviates, they always sum
to zero. Because the mean is the central
tendency or location, the negative deviates will
exactly cancel out the positive deviates.
Variance…
X
Mean
2
3
1
8
6
Sum
-2
-1
-3
4
2
0
4
=
Deviates
(X  X )
i
0
Example
Variance…
• Algebraically one can demonstrate the same result more generally,
n
n
n
 ( Xi  X )   X   X
i 1
Since
X
n
(X
i 1
i 1
i 1
is a constant for any sample,
 X )  i 1 X i  n X ,
n
i
i
Variance…
Since
X

X 
n
n
(X
i 1
i
then n X   X i ,
n
i
 X )  X i  i 1 X i  0
i 1
n
so
Variance…
• To circumvent the unfortunate property , the
widely used measure of dispersion called the
sample variance utilizes the square of the
deviates. The quantity  ( X  X ) is the sum of
these squared deviates and is referred to as
the corrected sum of squares (CSS). Each
observation is corrected or adjusted for its
distance from the mean.
n
2
i 1
i
Variance…
• Formula: The CSS is utilized in the formula for
the sample variance

s
2

2
(
X

X
)
 i
n
The sample variance is usually reported to two
more decimal places than the data and has
units that are the square of the measurement
units.
Variance…
Or
s
2
X


2
i
 ( X i ) / n
2
n 1
With a similar deviation the population variance
computational formula can be shown to be
 22 
22
X
X


(
(
X
X
)
)



 ii // Nn
22
ii
N
Variance…Example(unit Kg)
• Data set 3.1, 17.0, 9.9, 5.1, 18.0, 3.8, 10.0, 2.9,
21.2
X
i
 91
X
2
i
 1318.92
n=9
1318.92  (91) 2 / 9 1318.92  920.11 398.81
s 


 49.851Kg 2
9 1
8
8
2
Variance…
Remember, the numerator must
always be a positive number because
it is sum of squared deviations.
Population variance formula is rarely
used since most populations are too
large to census directly.
Standard deviation (SD)
• Standard deviation is the positive square root
of the variance
 X i  ( X i ) / N
2

And
s
X
2
N
2
i
 ( X i ) 2 / n
n 1
Standard Error (SE)
SE 
SD
n= no. of observation
n
Exercise 2
Daily milk yield (L) of 12 cows are tabulated
below. Calculate mean, median, mode,
variance and standard error.
Cow no
1
Milk yield Cow no
23.7
7
Milk yield
21.5
2
12.8
8
25.2
3
4
5
6
28.9
21.4
14.5
28.3
9
10
11
12
21.4
25.2
19.5
19.6
Problem 1
• Two herds of cows located apart in Malaysia
gave the following amount of milk/day (L).
Compute arithmetic mean, median, mode,
range, variance, SD and SE of daily milk yield
in cows of the two herds. Put your comments
on what have been reflected from two sets of
milk records as regards to their differences.
Table
Herd 1
Herd 2
• Cow no. 1 18.25
•
2 12.60
•
3 15.25
•
4 16.10
•
5 18.25
•
6 15.25
•
7 12.80
•
8 15.65
•
9 14.20
•
10 10.20
•
11 10.90
•
12 12.60
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Cow no. 1 7.50
2 6.95
3 4.20
4 5.10
5 4.50
6 6.15
7
6.90
8
7.50
9
7.80
10 10.20
11 6.30
12
7.50
13 5.75
14 4.75
Problem 2
• Sex adjusted weaning weight of lambs in two
different breeds of sheep were recorded as
follows. Compute mean, median, range,
variance and SE in weaning weight of lambs in
two breed groups. Put your comments on
various differences between the two groups.
Weaning wt. (Kg) of lambs
Breed 1
Breed 2
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
7.5
6.9
8.1
5.8
5.9
5.8
6.2
7.5
9.1
8.7
8.1
8.5
5.6
4.7
9.8
4.5
6.1
3.6
5.7
4.9
5.1
5.1
5.9
4.0
9.8
10.2
Problem No 3
In a market study data on the price (RM) of 10
kg rice were collected from 2 different
markets in Malaysia. Using descriptive
statistics show the differences relating to price
of rice in the two markets.
Pasar 1: 20, 25, 22, 23, 22, 24, 23, 21, 25,
25,23,22,25,24,24
Pasar 2: 25, 24, 26, 23, 26, 25, 25, 26, 24, 26, 24,
23,22, 25, 26, 26, 24