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Unformatted text preview: 1  f 0
×C
Mode = l +
2f1  f 0  f 2
l =20, f1=21, f0=13, f2=16, C=5
21 − 13
Mode = 20 +
×5
2 × 21 − 13 − 16
8×5
= 20 +
42 − 29
40
= 20 +
= 20 + 3.08 = 23.08
13
Computation of Mean and Standard deviation
X
Mid – Frequen Deviations
fd
d=
cy
point
f
M
m − 22.5
5
8
4
2
2.5
05
15
3
5
7.5
510
14
2
7
12.5
1015
13
1
13
17.5
1520
0
0
21
22.5
2025
16
1
16
27.5
2530
16
2
8
32.5
3035
9
3
3
37.5
3540
N=75
∑fd =
9
178 d 2 16
9
4
1
0
1
4
9 fd’ 2 32
45
28
13
0
16
32
27
∑fd 2=
193 ∑ fd
×c
N
−9
= 22.5 +
×5
75
45
= 22.5 –
75
= 22.5 – 0.6 = 21.9 Mean = A+ ∑ fd 2 ∑ fd − ×c
N
N
2 σ= 193 − 9 − ×5
75 75 2 = 2.57 − 0.0144 × 5
= 2.5556 × 5
= 1.5986 × 5 = 7.99
Karl – Pearson’ s coefficient of skewness
Mean  Mode
=
S .D.
21.9 − 23.08
=
7.99
− 1.18
=
= −0.1477
7.99
7.10.2 Bowley’ s Coefficient of skewness:
In Karl – Pearson’ s method of measuring skewness the
whole of the series is needed. Prof. Bowley has suggested a
formula based on relative position of quartiles. In a symmetrical
distribution, the quartiles are equidistant from the value of the
median; ie.,
Median – Q1 = Q3 – Median. But in a skewed distribution,
the quartiles will not be equidistant from the median. Hence
Bowley has suggested the following formula:
Q + Q1 − 2 Median
Bowley’ s Coefficient of skewness (sk) = 3
Q 3 − Q1
= 179 Example 21:
Find the Bowley’ s coefficient of skewness for the following series.
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22
Solution:
The given data in order
2, 4, 6, 10, 12, 14, 16, 18, 20, 22 n + 1 th
Q1
= size of item
4 11 + 1 th
= size of item
4
= size of 3rd item = 6 n + 1 th
Q3
= size of 3 item
4 11 + 1 th
= size of 3 item
4
= size of 9th item
= 18 n + 1
Median
= size of th item
2 11 + 1 = size of th item
2
= size of 6th item
= 12
Q + Q1 − 2 Median
Bowley’ s coefficient skewness = 3
Q 3 − Q1
18 + 6 − 2 × 12
=0
=
18 − 6
Since sk = 0, the given series is a symmetrical data.
Example 22:
Find Bowley’ s coefficient of skewness of the following series.
Size : 4 4.5
5 5.5
6
6.5
7
7.5
8
f:
10 18 22 25
40
15
10
8
7
180 Solution:
Size
4
4.5
5
5.5
6
6.5
7
7.5
8
Q1 f
10
18
22
25
40
15
10
8
7 c.f
10
28
50
75
115
130
140
148
155 N + 1 th
= Size of item
4 155 + 1 th
= Size o...
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This note was uploaded on 01/18/2014 for the course BUS 100 taught by Professor Moshiri during the Winter '08 term at UC Riverside.
 Winter '08
 Moshiri
 Business

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