Week 5 - ftms.edu.my
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Transcript Week 5 - ftms.edu.my
Prepared by: Nurazrin Jupri
differences
will be
large
differences
will be
small
MATH0102|Nurazrin Jupri
Two
groups of three students
Group 1
4
7
10
Group 2
7
7
7
Mean
mark
Group 1
Group 2
4 + 7 + 10 = 21/3 = 7
7 + 7 + 7 = 21/3 = 7
Same
mean mark, but Group 1’s marks are
widely spread, Group 2’s are all the same
The following diagram reinforces this point
MATH0102|Nurazrin Jupri
MATH0102|Nurazrin Jupri
A
measure of the average amount by
which the values in a distribution (x)
differ from the arithmetic mean
Average of the absolute deviations from
the arithmetic mean (ignoring the sign)
MATH0102|Nurazrin Jupri
grouped
Vertical
bars = all
differences
are taken as
positive
MATH0102|Nurazrin Jupri
ungroupe
d
X1
= 2, X2 = 4, X3 = 3
MD
=
X1 X X2 X X3 X
n
MD
=
=
2 3 4 3
=⅔
3 3
1 1 0
3
3
MATH0102|Nurazrin Jupri
Mean Deviation of grouped data
MATH0102|Nurazrin Jupri
X
MATH0102|Nurazrin Jupri
MATH0102|Nurazrin Jupri
If
we square all the deviations from the
arithmetic mean, then we no longer need
to bother with dropping the signs since
all the values will be positive.
Variance
is the average of the squared
deviations from the arithmetic mean
MATH0102|Nurazrin Jupri
Variance =
X
n
i
X
2
i 1
n
1.
2.
3.
To calculate the variance
Calculate the mean value X
Subtract the mean from each value in turn,
that is, find X i X
Square each answer to get X X 2
i
MATH0102|Nurazrin Jupri
4. Add up all these squared values to get
X
n
i
X
2
i 1
5. Divide the result by n to get
X
n
1
X
2
i 1
n
6. You now have the average of the squared deviations
from the mean (in square units)
MATH0102|Nurazrin Jupri
This
is simply the square root of the
variance
An advantage is that we avoid the square
units of the variance
Larger SD, larger the average dispersion
of data from the mean
Smaller SD, smaller the average
dispersion of data from the mean
MATH0102|Nurazrin Jupri
xi
x1 - x
(x1 – x)2
4
7
10
Total
4–7=-3
7–7= 0
10 – 7 = 3
(-32) = 9
02 = 0
32 = 9
18
MATH0102|Nurazrin Jupri
X
n
Variance = i 1
i
X
2
n
18
6square units
3
Standard deviation is square root of 6
= 2.449 units
MATH0102|Nurazrin Jupri
xi
xi - x
7
7
7
Total
7–7=0
7–7=0
7–7=0
MATH0102|Nurazrin Jupri
(xi – x)2
02 = 0
02 = 0
02 = 0
0
X
n
Variance =
i
i 1
X
2
n
0
0 square units
3
Standard deviation is square root of 0 = 0
i.e. there is no spread of values
MATH0102|Nurazrin Jupri
j
F
S
2
i
Xi
i 1
j
F
i 1
i
2
Fi X i
i 1
j
Fi
i 1
j
2
where Fi = Frequency of ith class interval
Xi = mid point of ith class interval
j = number of class intervals
MATH0102|Nurazrin Jupri
LCB
UCB
F
X
FX
FX^2
5.5
10.5
8
8
64
512
10.5
15.5
4
13
52
676
15.5
20.5
6
18
108
1944
224
3132
18
MATH0102|Nurazrin Jupri
S
2
3132
18
224
18
2
S2 = 174 – 12.442
S2 = 174 – 154.86
S2 = 19.14
S = √ 19.14 = 4.375
MATH0102|Nurazrin Jupri