Variance

1
Mean Deviation
Standard Deviation
Variance
2
Below is listed the test results of 2 different
classes. It is human nature to compare these
results. How can we compare the results of one
class to that of another?
Class A 52, 59, 60, 62, 64, 65, 68, 72, 80, 82,
82, 84, 86, 87, 88, 90, 91, 95, 96
Class B 44, 49, 55, 57, 61, 66, 71, 74, 81, 82,
82, 83, 85, 90, 92, 94, 98, 99, 100
Perhaps the quickest measure of central tendency
is to observe the most frequent mark for each
class. In Class A the mark of 82 occurs twice
whereas the other marks only occur once. In Class
B the exact same thing happens. For this reason
this method is unsatisfactory. This measure of
central tendency is called the MODE. It has the
advantage of being very easy and relatively quick
to determine but it is not very helpful in many
situations as it can often be very misleading.
Another way to compare classes is to compare the
middle value from each class. Since both classes
have 19 values, we are looking for the tenth
value (there are 9 values below and 9 values
above.
Class A 52, 59, 60, 62, 64, 65, 68, 72, 80,
82, 82, 84, 86, 87, 88, 90, 91, 95, 96
Class B 44, 49, 55, 57, 61, 66, 71, 74, 81,
82, 82, 83, 85, 90, 92, 94, 98, 99, 100
This is called the MEDIAN. 82 for both here as
well. This method is a little more
time-consuming especially if the numbers are not
in order as they are here. The Median is
generally a more reliable measure of central
tendency than the Mode.
3
The third and perhaps most common measure of
central tendency is the MEAN. This is also known
as the average or the arithmetic mean. It is
also the most tedious value to determine
especially with a lot of values because you have
to add them all up.
Lets assign each value in a class to variable
x. So in Class A, x1 52, x2 59, x3 60
and x19 96. When we look at the values in this
way we can express the sum of the values like so
This is a very cumbersome way to express the sum
of the data values. There is a special symbol
used in statistics to represent the sum of data
values.
We can read this symbol as follows The sum of
xi, where x ranges from i to n. n
represents the total number of data values.
52596062646568728082828486878890
919596
Even though that symbol may be weird and foreign
to you, which would you prefer to write the
weird symbol on the left side of the or the
sum of the 19 numbers on the right. One more
point sometimes it can be the sum of more than
19 numbers.
One of the reasons that we have to write it is to
express the formula for MEAN. By the way, since
we are using variable x to represent the
individual marks, we will use the following
symbol to represent the MEAN of the marks
4
The formula to determine MEAN from a set of
values is
Where n represents the total data values
Class A 52, 59, 60, 62, 64, 65, 68, 72, 80, 82,
82, 84, 86, 87, 88, 90, 91, 95, 96
Class B 44, 49, 55, 57, 61, 66, 71, 74, 81, 82,
82, 83, 85, 90, 92, 94, 98, 99, 100
We see that the mean ends up having the same
value for both classes as well. None of the 3
measures of central tendency does anything to
distinguish the results of one class from the
results of another.
There are other ways to make a distinction
between the results of both classes than using
central tendency. We can observe the VARIANCE.
This is an indication of how spread out or
dispersed the values are. It is a measure of how
much the data values deviate from the arithmetic
mean. The larger the variance, the greater the
dispersion and the smaller the variance, the more
clustered the data values.
5
The symbol for variance is s2
To calculate variance, we really should do it
step by step. 1. Calculate arithmetic mean
To calculate variance, we really should do it
step by step. 1. Calculate arithmetic mean 2.
Calculate the difference between each data value
and the mean
To calculate variance, we really should do it
step by step. 1. Calculate arithmetic mean 2.
Calculate the difference between each data value
and the mean 3. Square the result from step 2
To calculate variance, we really should do it
step by step. 1. Calculate arithmetic mean 2.
Calculate the difference between each data value
and the mean 3. Square the result from step
2 4. Calculate the sum of all of the values
from step 3 5. Divide the result of step 4 by
(n 1)
To calculate variance, we really should do it
step by step. 1. Calculate arithmetic mean 2.
Calculate the difference between each data value
and the mean 3. Square the result from step
2 4. Calculate the sum of all of the values
from step 3
6
Class A
52
59
60
62
64
65
68
72
80
82
82
84
86
87
88
90
91
95
96
52 - 77
59 - 77
60 - 77
62 - 77
64 - 77
65 - 77
68 - 77
72 - 77
80 - 77
82 - 77
82 - 77
84 - 77
86 - 77
87 - 77
88 - 77
90 - 77
91 - 77
95 - 77
96 - 77
-25
-18
-17
-15
-13
-12
-9
-5
3
5
5
7
9
10
11
13
14
18
19
625
324
289
225
169
144
81
25
9
25
25
49
81
100
121
169
196
324
361
Remember that the symbol for variance is s2 not
s. We do not take the square root of 185.67 to
determine variance. The variance is 185.67.
3342
7
Class B
44
49
55
57
61
66
71
74
81
82
82
83
85
90
92
94
98
99
100
44 - 77
49 - 77
55 - 77
57 - 77
61 - 77
66 - 77
71 - 77
74 - 77
81 - 77
82 - 77
82 - 77
83 - 77
85 - 77
90 - 77
92 - 77
94 - 77
98 - 77
99 - 77
100 - 77
-33
-28
-22
-20
-16
-11
-6
-3
4
5
5
6
8
13
15
17
21
22
23
1089
784
484
400
256
121
36
9
16
25
25
36
64
169
225
289
441
484
529
5482
8
The variance for Class A is 185.67 as compared to
the variance for Class B is 304.56. This
indicates that the values are more spread out for
Class B. You might say that the calculation we
just did is unnecessary . We can determine that
Class B is more spread out than Class A just by
observing its range (lowest to highest value).
Class A ranges from 52 to 96, a separation of 44.
Class B ranges from 44 to 100, a difference of
56. Class B is more spread out. All of that is
true but the variance is more revealing because
it takes into consideration all of the values
whereas range only takes 2 values from each class.
Actually, due to the fact that the difference
between the mean and each specific value is
squared, the variance is not the best way to
compare dispersion of the classes. To compensate
for this fact, we can square root the variance.
This allows the units measuring dispersion to be
the same as the units for the class values. When
we do this we get the STANDARD DEVIATION. The
symbol for standard deviation is s.
For Class A
For Class B
9
The owner of 2 service stations decided to record
the number of litres of gasoline needed to fill
the tank of each car that stops at one of his
service stations. One of the stations is located
along the highway and the other is located
downtown.
Highway Service Station 30, 22, 21, 28, 25, 26,
26, 24, 29, 23, 20, 27, 25, 24, 25
Downtown Service Station 25, 23, 30, 19, 35, 27,
15, 25, 17, 31, 14, 20, 33, 25, 36
Highway Service Station
30
22
21
28
25
26
26
24
29
23
20
27
25
24
25
30 - 25
22 - 25
21 - 25
28 - 25
25 - 25
26 - 25
26 - 25
24 - 25
29 - 25
23 - 25
20 - 25
27 - 25
25 - 25
24 - 25
25 - 25
5
-3
-4
3
0
1
1
-1
4
-2
-5
2
0
-1
0
25
9
16
9
0
1
1
1
16
4
25
4
0
1
0
112
10
The owner of 2 service stations decided to record
the number of litres of gasoline needed to fill
the tank of each car that stops at one of his
service stations. One of the stations is located
along the highway and the other is located
downtown.
Highway Service Station 30, 22, 21, 28, 25, 26,
26, 24, 29, 23, 20, 27, 25, 24, 25
Downtown Service Station 25, 23, 30, 19, 35, 27,
15, 25, 17, 31, 14, 20, 33, 25, 36
Downtown Service Station
25
23
30
19
35
27
15
25
17
31
14
20
33
25
36
25 - 25
23 - 25
30 - 25
19 - 25
35 - 25
27 - 25
15 - 25
25 - 25
17 - 25
31 - 25
14 - 25
20 - 25
33 - 25
25 - 25
36 - 25
0
-2
5
-6
10
2
-10
0
-8
6
-11
-5
8
0
11
0
4
25
36
100
4
100
0
64
36
121
25
64
0
121
700
11
From the box-and-whiskers plot the minimum and
maximum from the highway station are 20 and 30
respectively. The minimum and maximum from the
downtown station are 14 and 36 respectively. The
quartiles (Q1, Q2, and Q3) are as follows
Q1 23 L Q2 (Md) 25 L Q3 27 L
Q1 19 L Q2 (Md) 25 L Q3 31 L
The semi-interquartile range is basically half of
the interquartile range and it is the mean length
of a quartile.
The interquartile range (IR) is IR Q3 Q1
27 23 4 L
About half of the values will fall between Md - Q
and Md Q Q3 27 L
IR Q3 Q1 31 19 12 L
12
There is one other measure that is used to
determine the degree of dispersion of the data
values from a group. We already have defined
variance and standard deviation. The third
device is called MEAN DEVIATION. This
calculation is similar to standard deviation in
that its units are the same as the data values
given. But it is a little simpler than standard
deviation formula. Observe the 2 formulas
This is the value of the difference with the
negative sign removed if it is present.
There are 3 differences between these 2 formulas.
What are they?
No square root in the formula for mean deviation.
Mean deviation formula divides by n instead of
(n-1).
13
25
23
30
19
35
27
15
25
17
31
14
20
33
25
36
25 - 25
23 - 25
30 - 25
19 - 25
35 - 25
27 - 25
15 - 25
25 - 25
17 - 25
31 - 25
14 - 25
20 - 25
33 - 25
25 - 25
36 - 25
0
-2
5
-6
10
2
-10
0
-8
6
-11
-5
8
0
11
0
2
5
6
10
2
10
0
8
6
11
5
8
0
11
84
14
Mr. White had carpeting installed in each of the
10 units in his 2 apartment buildings located on
Stone Street. The following table shows the
surface area covered in each unit.
a) Calculate the standard deviation of the data
collected for each building. Building 1
__________ Building 2 __________
b) Using standard deviation for each building,
determine for which building the carpet area
differs the most from one unit to the next.
Explain. _________________________________ _______
__________________________ _______________________
__________
c) Would the variance have allowed you to draw
the same conclusions? Why? ______________________
___________ _________________________________
15
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