Compiled By:
Hafiza Seemab
Sadia Mazhar
Faizan Illahi
Usman Ashraf
Waleed Khalid

• 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

• 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

• 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.

• 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.

• It is the raw data which is not classified into groups or classes.
• Formula of A.M.
A.M. = 𝑥
=
𝑥 𝑛

• 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

• 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
A.M. = 𝑥
=
𝑓𝑥
𝑥
A.M. = 𝑥
= A +
𝑓𝐷
𝑥
A.M. = 𝑥
= A +
𝑓𝑢
𝑥 x h

• 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)
No. of Students
13
2
14
5
15
13
16
7
17
3

Ages (years) x
13
14
15
16
17
Total
Solution: 𝑥
=
𝑓𝑥
𝑓
=
Number of students f
13
7
3
2
5
𝑓 = 30
454
30
= 15.13 y fx
26
70
195
112
51
𝑓𝑥 = 454

𝑥
= A +
𝑓𝐷
𝑓
Where
A = Assumed Mean
f = Frequency of different groups
D = Deviation form of A
D = ( X – A )

Where
𝑥 = A + 𝑓𝑢
𝑓 x h
A = Assumed Mean
f = Frequency of different groups u = Step Deviation u = x
– A 𝒉 h = Size of Class interval

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
Marks f x fx
20 – 29
30 – 39
1 24.5
5 34.5
40 – 49 12 44.5
50 – 59 15 54.5
60 – 69
70 – 79
9 64.5
6 74.5
24.5
172.5
534.5
817.5
580.5
447.5
80 – 89 2 84.5
Total ( Σ ) 50
169.5
𝒇𝒙 = 2745
10
20
30
-30
-20
-10
0
D=x-A fD
-30
-100
-120
0
90
120
60
𝒇𝑫 = 20 u= 𝑥−𝐴 ℎ
1
2
3
-1
0
-3
-2 fu
-3
-10
-12
0
9
12
6
𝒇𝒖 = 2
Where:
A = 54.5
h = 10

𝒇 = 50 𝒇𝒙 = 2745 𝒇𝑫 = 20 𝒇𝒖 = 2
Direct Method: 𝒙 =
𝒇𝒙
𝒇
=
2750
50
= 54.9 ≅ 55
Short-Cut Method: 𝒙 = A +
𝒇𝑫
𝒇 where A = 54.5
= 54.5 +
20
50
= 54.5 + 0.4 = 54.9
Step-Deviation Method: 𝒙 = A +
𝒇𝒖 x h where A= 54.5 & h= 10
𝒇
2
= 54.5 +
50
x 10 = 54.5 + 0.4 x 10
= 54.9

• 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.

• 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).

• 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)
No. of Persons
0-10 11-20 21-30 31-40
10 20 40 30
• Marks of Stats subject obtained by student in mid exams:
Marks
Students
1-10
5
11-20
4
21-30
35
31-40
17
41-50
6

• Scores of Statistics Final Exam of BSCS Class