Standard deviation :grouped data/Continuous data
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May 02, 2020
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About This Presentation
Study material for S.E.E Students
Secondary Education Board, Nepal
Optional Mathematics: Statistics
Slide Content
5/2/2020 Janak Singh Saud 1
0- 10
10 –20
20 –30
30 –40
40 –50
50 –60
60 –70
0
5
10
15
20
NO. of
students
5
12
15
20
10
42
0- 1010 –2020 –3030 –4040 –5050 –60 60 –70
0- 10
10 –20
20 –30
30 –40
40 –50
50 –60
60 –70
0
5
10
15
20
NO. of
students
5
12
15
20
10
42
0- 1010 –2020 –3030 –4040 –5050 –60 60 –70
OBJECTIVES
5/2/2020 Janak Singh Saud 2
The learners are expected to
•Calculate the Standard Deviation of a given set of
grouped data.
•Compare the consistency and variability of two given
set of data by Coefficient of variance.
5/2/2020 Janak Singh Saud 2
The learners are expected to
•Calculate the Standard Deviation of a given set of
grouped data.
•Compare the consistency and variability of two given
set of data by Coefficient of variance.
STANDARDDEVIATION
5/2/2020 JanakSingh Saud 3
It is considered as the most reliable
measure of variability because
Its value is based on all the observations
Deviation of each item is taken from the central
value
All algebraic sign are also considered
5/2/2020 JanakSingh Saud 3
It is considered as the most reliable
measure of variability because
Its value is based on all the observations
Deviation of each item is taken from the central
value
All algebraic sign are also considered
5/2/2020 Janak Singh Saud 4
STANDARDDEVIATION
S.D is a special form of average deviation from the
mean
S.D is affected by the individual values or items in
the distribution
STANDARDDEVIATION
S.D is a special form of average deviation from the
mean
S.D is affected by the individual values or items in
the distribution
•The concept of standard deviation was first introduced
by Karl pearsonin 1893
•The standard deviation is the most useful and the most
popular measure of dispersion.
•It is always calculated from the arithmetic mean;
median and mode is not considered.
5/2/2020 Janak Singh Saud 5
Standard
by Karl pearsonin 1893
•The standard deviation is the most useful and the most
popular measure of dispersion.
•It is always calculated from the arithmetic mean;
median and mode is not considered.
5/2/2020 Janak Singh Saud 5
Standard
Definition:
•Standard Deviation is the positive square root of the average (i.e.
mean) of squared deviation taken about mean.
•The standard deviation is represented by the Greek letter ??????(sigma).
•Formula.
•Standard deviation ( ??????)=
��−??????
2
??????
•Alternatively ??????=
��
2
??????
−
��
??????
2
, mean (??????)=
��
??????
5/2/2020 JanakSingh Saud 6
•Standard Deviation is the positive square root of the average (i.e.
mean) of squared deviation taken about mean.
•The standard deviation is represented by the Greek letter ??????(sigma).
•Formula.
•Standard deviation ( ??????)=
��−??????
2
??????
•Alternatively ??????=
��
2
??????
−
��
??????
2
, mean (??????)=
��
??????
5/2/2020 JanakSingh Saud 6
5/2/2020 Janak Singh Saud 7
Comparing Standard Deviation
10 20 30 40 50 60 70 80 90 100
Data: A
Mean = 57
S.D = 28.302
Data: B
Mean = 63
S.D = 11
10 20 30 40 50 60 70 80 90 100
Data: C
Mean = 55
S.D = 40.410
10 20 30 40 50 60 70 80 90 100
Comparing Standard Deviation
10 20 30 40 50 60 70 80 90 100
Data: A
Mean = 57
S.D = 28.302
Data: B
Mean = 63
S.D = 11
10 20 30 40 50 60 70 80 90 100
Data: C
Mean = 55
S.D = 40.410
10 20 30 40 50 60 70 80 90 100
Three Methods
a)Actual Mean Method or Direct Method
b)Assumed Mean Method or Short-cut Method
c)Step-Deviation Method
5/2/2020 Janak Singh Saud 8
a)Actual Mean Method or Direct Method
b)Assumed Mean Method or Short-cut Method
c)Step-Deviation Method
5/2/2020 Janak Singh Saud 8
•The S.D. for the Grouped data is given by the formula:
??????=
��−??????
2
??????
, Where ??????is the arithmetic mean,
m is the mid-point of classes,
??????is the corresponding frequency
and N = ??????
5/2/2020 Janak Singh Saud 9
??????=
��−??????
2
??????
, Where ??????is the arithmetic mean,
m is the mid-point of classes,
??????is the corresponding frequency
and N = ??????
5/2/2020 Janak Singh Saud 9
Mean
(??????)
30.59
30.59
30.59
30.59
30.59
30.59
30.59
I.Q 0-10 10 –20 20 –30 30 –40 40 –50 50 –60 60 –70
NO. of
students
5 12 15 20 10 4 2
f. (�−??????)
2
3274.25
2916.6
468.75
389
2076.5
2383.4
2368.1
??????.(�−??????)
2
=
13876.6
Mean (??????) =
��
??????
=
����
��
= 30. 59
S.D (??????)=
��−??????
�
??????
=
�����.�
��
= 14.29
C. S.D =
??????.??????
��??????�
=
14.28
3�.��
= 0.47
I.Q No.of
students
(f)
0-10
10-20
20-30
30-40
40-50
50-60
60-70
5
12
15
20
10
4
2
N= ??????=
68
Mid-
value
(m)
5
15
25
35
45
55
65
f.M
25
180
375
700
450
220
130
????????????
= 2080
�−??????
-25.59
-15.59
-5.59
4.41
14.41
24.41
34.41
(�−??????)
2
654.85
243.05
31.25
19.45
207.65
595.85
1184.05
5/2/2020 JanakSingh Saud
10
(??????)
30.59
30.59
30.59
30.59
30.59
30.59
30.59
I.Q 0-10 10 –20 20 –30 30 –40 40 –50 50 –60 60 –70
NO. of
students
5 12 15 20 10 4 2
f. (�−??????)
2
3274.25
2916.6
468.75
389
2076.5
2383.4
2368.1
??????.(�−??????)
2
=
13876.6
Mean (??????) =
��
??????
=
����
��
= 30. 59
S.D (??????)=
��−??????
�
??????
=
�����.�
��
= 14.29
C. S.D =
??????.??????
��??????�
=
14.28
3�.��
= 0.47
I.Q No.of
students
(f)
0-10
10-20
20-30
30-40
40-50
50-60
60-70
5
12
15
20
10
4
2
N= ??????=
68
Mid-
value
(m)
5
15
25
35
45
55
65
f.M
25
180
375
700
450
220
130
????????????
= 2080
�−??????
-25.59
-15.59
-5.59
4.41
14.41
24.41
34.41
(�−??????)
2
654.85
243.05
31.25
19.45
207.65
595.85
1184.05
5/2/2020 JanakSingh Saud
10
•The S.D. for the Grouped data is given by the formula:
??????=
��
2
??????
−
��
??????
2
,
•Where ??????= ??????− A, ??????is the assumed mean
•??????is the corresponding frequency
and N = ??????
5/2/2020 JanakSingh Saud 11
??????=
��
2
??????
−
��
??????
2
,
•Where ??????= ??????− A, ??????is the assumed mean
•??????is the corresponding frequency
and N = ??????
5/2/2020 JanakSingh Saud 11
I.Q 0-10 10-20 20-30 30-40 40-50 50-60 60-70
NO. of students5 12 15 20 10 4 2
fd
2
4500
4800
1500
0
1000
1600
1800
????????????
2
=
15200
Assume mean (A) = 35
S.D (??????)=
��
�
??????
−
��
??????
�
=
�����
��
−
−���
��
�
= 14.29
Mean (??????)=A +
��
??????
= 35 +
−���
��
= 30.59
I.Q No.of
students (f)
0-10
10-20
20-30
30-40
40-50
50-60
60-70
5
12
15
20
10
4
2
N= ??????=
68
Mid-value
(m)
5
15
25
35
45
55
65
d=m-A
(A = 35)
-30
-20
-10
0
10
20
30
fd
-150
-240
-150
0
100
80
60
????????????= -300
5/2/2020 JanakSingh Saud 12
C. S.D = ?
NO. of students5 12 15 20 10 4 2
fd
2
4500
4800
1500
0
1000
1600
1800
????????????
2
=
15200
Assume mean (A) = 35
S.D (??????)=
��
�
??????
−
��
??????
�
=
�����
��
−
−���
��
�
= 14.29
Mean (??????)=A +
��
??????
= 35 +
−���
��
= 30.59
I.Q No.of
students (f)
0-10
10-20
20-30
30-40
40-50
50-60
60-70
5
12
15
20
10
4
2
N= ??????=
68
Mid-value
(m)
5
15
25
35
45
55
65
d=m-A
(A = 35)
-30
-20
-10
0
10
20
30
fd
-150
-240
-150
0
100
80
60
????????????= -300
5/2/2020 JanakSingh Saud 12
C. S.D = ?
5/2/2020 Janak Singh Saud 13
•The S.D. for the Grouped data is given by the formula
S.D (??????) = i×
��
′
2
??????
−
��
′
??????
2
, where ??????
′
=
�−??????
??????
, A is assumed mean,
i= common class interval and
??????is the corresponding frequency and N = ??????
•The S.D. for the Grouped data is given by the formula
S.D (??????) = i×
��
′
2
??????
−
��
′
??????
2
, where ??????
′
=
�−??????
??????
, A is assumed mean,
i= common class interval and
??????is the corresponding frequency and N = ??????
5/2/2020 Janak Singh Saud 14
I.Q 0-10 10-20 20-30 30-40 40-50 50-60 60-70
No. of
students
5 12 15 20 10 4 2
Assume mean (A) = 45
f�
′
�
45
48
15
0
10
16
18
f�
′
�
=152
I.Q f
0-10
10-20
20-30
30-40
40-50
50-60
60-70
5
12
15
20
10
4
2
N= 68
m
5
15
25
35
45
55
65
A
35
35
35
35
35
35
35
�
′
=
�−??????
��
-3
-2
-1
0
1
2
3
f.�
�
-15
-24
-15
0
10
8
6
-30
S.D (??????) = i×
��
′
2
??????
−
��
′
??????
2
S.D (??????) = 10×
152
68
−
−30
68
2
= 10×1.4285
= 14.29
Mean (??????)= A +
��
′
??????
×??????
= 35 +
−30
68
×10
= 30.59
C.S.D=
??????
??????
=
14.29
30.59
= 0.47
I.Q 0-10 10-20 20-30 30-40 40-50 50-60 60-70
No. of
students
5 12 15 20 10 4 2
Assume mean (A) = 45
f�
′
�
45
48
15
0
10
16
18
f�
′
�
=152
I.Q f
0-10
10-20
20-30
30-40
40-50
50-60
60-70
5
12
15
20
10
4
2
N= 68
m
5
15
25
35
45
55
65
A
35
35
35
35
35
35
35
�
′
=
�−??????
��
-3
-2
-1
0
1
2
3
f.�
�
-15
-24
-15
0
10
8
6
-30
S.D (??????) = i×
��
′
2
??????
−
��
′
??????
2
S.D (??????) = 10×
152
68
−
−30
68
2
= 10×1.4285
= 14.29
Mean (??????)= A +
��
′
??????
×??????
= 35 +
−30
68
×10
= 30.59
C.S.D=
??????
??????
=
14.29
30.59
= 0.47
•The S.D. for the Grouped data is given by the formula.
??????=
��
2
??????
−
��
??????
2
,
•Where m= mid point of corresponding class,
•??????is the corresponding frequency
and N = ??????
5/2/2020 JanakSingh Saud 15
??????=
��
2
??????
−
��
??????
2
,
•Where m= mid point of corresponding class,
•??????is the corresponding frequency
and N = ??????
5/2/2020 JanakSingh Saud 15
I.Q 0-10 10-20 20-30 30-40 40-50 50-60 60-70
NO. of
students
5 12 15 20 10 4 2
fm
2
125
2700
9375
24500
20250
12100
8450
????????????
2
= 77500
S.D (??????)=
��
�
??????
−
��
??????
�
Mean (??????) =
��
??????
=
����
��
= 30.59
Coefficient of S.D =
??????.??????
��??????�
=
14.29
30.59
= 0.4
=
�����
��
−
����
��
�
= ����.��−���.��
= ���.��
= 14. 29
I.Q No.of
students
(f)
Mid-
value (m)
0-10
10-20
20-30
30-40
40-50
50-60
60-70
5
12
15
20
10
4
2
5
15
25
35
45
55
65
N= ??????=
68
f.m
25
180
375
700
450
220
130
????????????
= 2080
m
2
25
225
625
1225
2025
3025
4225
5/2/2020 JanakSingh Saud
NO. of
students
5 12 15 20 10 4 2
fm
2
125
2700
9375
24500
20250
12100
8450
????????????
2
= 77500
S.D (??????)=
��
�
??????
−
��
??????
�
Mean (??????) =
��
??????
=
����
��
= 30.59
Coefficient of S.D =
??????.??????
��??????�
=
14.29
30.59
= 0.4
=
�����
��
−
����
��
�
= ����.��−���.��
= ���.��
= 14. 29
I.Q No.of
students
(f)
Mid-
value (m)
0-10
10-20
20-30
30-40
40-50
50-60
60-70
5
12
15
20
10
4
2
5
15
25
35
45
55
65
N= ??????=
68
f.m
25
180
375
700
450
220
130
????????????
= 2080
m
2
25
225
625
1225
2025
3025
4225
5/2/2020 JanakSingh Saud
5/2/2020 Janak Singh Saud 17
From the following information, examine company A or B has greater variability in wage
distribution.
Company A Company B
Average monthly wage Rs. 182 Rs. 178.50
S.D of distribution of wageRs. 8.8 R.S. 9.9
Company B:
??????= 178.50 S.D (??????) = 9.9 C.V =
?
∴C.V. =
??????.??????.
��??????�
×100%
=
9.9
178.50
×100%
= 5.55 %
Since C.V
A< C.V
B
So, the variability of wage distribution in company B is greater.
Company A:
??????= 182 S.D (??????) = 8.8 C.V = ?
∴C.V. =
??????.??????.
��??????�
×100%
=
8.8
182
×100%
= 4.835 %
5/2/2020 Janak Singh Saud 18
distribution.
Company A Company B
Average monthly wage Rs. 182 Rs. 178.50
S.D of distribution of wageRs. 8.8 R.S. 9.9
Company B:
??????= 178.50 S.D (??????) = 9.9 C.V =
?
∴C.V. =
??????.??????.
��??????�
×100%
=
9.9
178.50
×100%
= 5.55 %
Since C.V
A< C.V
B
So, the variability of wage distribution in company B is greater.
Company A:
??????= 182 S.D (??????) = 8.8 C.V = ?
∴C.V. =
??????.??????.
��??????�
×100%
=
8.8
182
×100%
= 4.835 %
5/2/2020 Janak Singh Saud 18
Email: [email protected]
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5/2/2020 JanakSingh Saud 19
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w_as=subscriber
https://www.blogger.com/blogger.g?blogID=5544241353531082206#al
lposts
https://www.facebook.com/janaksinghsaud
https://www.slideshare.net/janaksinghsaud
5/2/2020 JanakSingh Saud 19
THANK YOU
5/2/2020 Janak Singh Saud 20
5/2/2020 Janak Singh Saud 20
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