Transcript Example of Short-Cut and Step Deviation

Arithmetic Mean
Compiled By:
Hafiza Seemab
Sadia Mazhar
Faizan Illahi
Usman Ashraf
Waleed Khalid
Contents
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Arithmetic Mean
Merits of A.M.
Demerits of A.M.
A.M. of Ungrouped Data
A.M. of Grouped Data
• Direct Method
• Short-Cut Method
• Step Deviation Method
• Properties of A.M.
• Use of A.M.
• Practice Q’s
Arithmetic Mean
• The most popular method and widely used
• Generally known as Average
• Simply Calculated as
Summing-up all the values
divided by
the Total No. of values.
• Represented by
𝒙
Merits of A.M.
•
•
•
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Arithmetic mean rigidly defined by Algebraic Formula.
It is easy to calculate and simple to understand.
It is based on all observations of the given data.
It is capable of being treated mathematically hence it is widely
used in statistical analysis.
• Arithmetic mean can be computed even if the distribution is
not known but some of the observation and number of the
observation are known.
• It is least affected by the fluctuation of sampling.
• For every kind of data mean can be calculated.
Demerits of A.M.
• It can neither be determined by inspection or by graphical
location.
• Arithmetic mean can not be computed for qualitative data.
• It is too much affected by extreme observations and hence it
is not adequately represent data consisting of some extreme
point.
• Arithmetic mean can not be computed when class intervals
have open ends.
• If any one of the data is missing then mean can not be
calculated.
A.M. of Ungrouped Data
• It is the raw data which is not classified into groups or classes.
• Formula of A.M.
A.M. =
𝑥
=
𝑥
𝑛
Example
• Sargodha’s Temperature Last week was (Celc):
38, 42, 35, 39, 42, 44, 36
find the mean temperature using A.M.
A.M. =
𝑥
𝑛
38+42+35+39+40+43+36
=
7
= 39
Hence mean temperature of the week is 39
A.M. of Grouped Data
• Grouped data is data that has been organized into groups
known as classes.
• Formula of A.M.
Method
Direct Method
Short-Cut Method
Step Deviation Method
Formula
𝑥
A.M. =
A.M. =
A.M. =
𝑥
𝑥
=
= A+
= A+
𝑓𝑥
𝑥
𝑓𝐷
𝑥
𝑓𝑢
𝑥
xh
Direct Method:
• It is the simplest method to find the A.M. value.
A.M. =
𝑥=
𝑓𝑥
𝑓
where:x = given values
• Example
f = frequency of groups
Ages (Years)
13
14
15
16
17
No. of Students
2
5
13
7
3
Ages (years)
Number of students
x
f
13
2
26
14
5
70
15
13
195
16
7
112
17
3
51
𝑓 = 30
Total
fx
𝑓𝑥 = 454
Solution:
𝑥=
𝑓𝑥
𝑓
=
454
30
=
15.13 y
Short-Cut Method
𝑥
= A+
𝑓𝐷
𝑓
Where
A = Assumed Mean
f = Frequency of different groups
D = Deviation form of A
D=(X–A)
Step-Deviation Method
𝑥 =A+
𝑓𝑢
𝑓
xh
Where
A = Assumed Mean
f = Frequency of different groups
u = Step Deviation
u=
x–A
𝒉
h = Size of Class interval
Example of Short-Cut and Step Deviation
The following frequency distribution showing the marks
obtained by 50 students in statistics at a certain college.
Find the arithmetic mean
Marks
Frequency
20-29
30-39
40-49
50-59
60-69
70-79
80-89
1
5
12
15
9
6
2
Direct
Short-Cut Method
Step-Deviation
𝑥−𝐴
ℎ
fu
-30
-3
-3
-20
-100
-2
-10
534.5
-10
-120
-1
-12
54.5
817.5
0
0
0
0
9
64.5
580.5
10
90
1
9
70 – 79
6
74.5
447.5
20
120
2
12
80 – 89
2
84.5
169.5
30
60
3
6
Total ( Σ )
50
Marks
f
x
fx
D=x-A
fD
20 – 29
1
24.5
24.5
-30
30 – 39
5
34.5
172.5
40 – 49
12
44.5
50 – 59
15
60 – 69
𝒇𝒙 =2745
𝒇𝑫 =20
Where:
A = 54.5
h = 10
u=
𝒇𝒖 =2
𝒇 = 50
𝒇𝒙 =2745
𝒇𝑫 =20
𝒇𝒖 =2
Direct Method:
𝒙 =
𝒇𝒙
𝒇
=
2750
50
= 54.9 ≅ 55
Short-Cut Method:
𝒙 = A+
= 54.5 +
𝒇𝑫
𝒇
20
50
where A = 54.5
= 54.5 + 0.4 = 54.9
Step-Deviation Method:
𝒙 = A+
= 54.5 +
= 54.9
𝒇𝒖
𝒇
2
50
xh
where A= 54.5 & h= 10
x 10 = 54.5 + 0.4 x 10
Properties of A.M.
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Every set of interval-level data has a mean.
All the values are included in computing the mean.
A set of data has a unique mean.
The mean is affected by unusually large or small data values.
The arithmetic mean is the only measure of central tendency
where the sum of the deviations of each value from the mean is
zero.
Use of A.M.
• Mean is used in fields such as business, engineering and
computer science.
• It is used in report card or in our population.
• In addition to mathematics and statistics, the arithmetic mean
is used frequently in fields such as economics, sociology, and
history, though it is used in almost every academic field to
some extent. For example, per capita GDP gives an
approximation of the arithmetic average income of a nation's
population.
• It’s used to compute the variance and SD (Standard Deviation).
Practice Q’s
• Find the A.M. of following deviations:
25, 30, 20, 63, 52, 29, 18, 8, 41
• The following data shows distance covered by persons to
perform their routine jobs.
Distance (Km)
0-10
11-20
21-30
31-40
No. of Persons
10
20
40
30
• Marks of Stats subject obtained by student in mid exams:
Marks
1-10
11-20
21-30
31-40
41-50
Students
5
4
35
17
6
• Scores of Statistics Final Exam of BSCS Class