Prepared by: Nurazrin Jupri. differences will be large differences will be small MATH0102|Nurazrin...
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Transcript of Prepared by: Nurazrin Jupri. differences will be large differences will be small MATH0102|Nurazrin...
Prepared by: Nurazrin Jupri
differences will be
large
differences will be
small
MATH0102|Nurazrin Jupri
Two groups of three studentsGroup 1 4 7 10Group 2 7 7 7
Mean markGroup 1 4 + 7 + 10 = 21/3 = 7Group 2 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
groupedungroup
ed
Vertical bars = all difference
s are taken as positive
MATH0102|Nurazrin Jupri
X1 = 2, X2 = 4, X3 = 3
MD =
MD = =
= ⅔
X X X X X X
n
1 2 3
2 3 4 3 3 3
3
1 1 0
3
MATH0102|Nurazrin Jupri
Mean Deviation of grouped data
XMATH0102|Nurazrin Jupri
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 =
To calculate the variance1. Calculate the mean value 2. Subtract the mean from each value in
turn, that is, find 3. Square each answer to get
n
XXn
ii
1
2
X
XX i
2XX i
MATH0102|Nurazrin Jupri
4. Add up all these squared values to get
5. Divide the result by n to get
6. You now have the average of the squared deviations from the mean (in square units)
n
ii XX
1
2
n
XXn
i
1
2
1
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
Variance = square units
Standard deviation is square root of 6 = 2.449 units
6
3
181
2
n
XXn
ii
MATH0102|Nurazrin Jupri
xi xi - x (xi – x)2
7
7
7
Total
7 – 7 = 0
7 – 7 = 0
7 – 7 = 0
02 = 0
02 = 0
02 = 0
0
MATH0102|Nurazrin Jupri
Variance = square units
Standard deviation is square root of 0 = 0 i.e. there is no spread of values
0
3
01
2
n
XXn
ii
MATH0102|Nurazrin Jupri
where Fi = Frequency of ith class interval Xi = mid point of ith class interval
j = number of class intervals
2
1
1
1
1
2
2
j
i
j
iii
j
ii
j
iii
iF
XF
F
XFS
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
18 224 3132
MATH0102|Nurazrin Jupri
S2 = 174 – 12.442 S2 = 174 – 154.86 S2 = 19.14 S = √ 19.14 = 4.375
22
18
224
18
3132
S
MATH0102|Nurazrin Jupri