3. Arithmetic Mean
• Arithmetic mean is one of the methods of calculating central
tendency.
• It is an average.
• It is calculated to reach a single value which represents the
entire data.
• Calculating an Average
• For example (1, 2, 2, 2, 3, 9). The arithmetic mean is 1 + 2 + 2 + 2
+ 3 + 9
6
= 19/6 = 3.17
Arithmetic means are used in situations such as working out cricket
averages.
• Arithmetic means are used in calculating average incomes.
4. Mean = (Sum of all the observations/Total number of
observations)
5. Different Series and different methods
• Individual Series
» Direct Method
» Shortcut Method
• Discrete Series
» Direct Method
» Shortcut Method
• Continuous Series
» Direct Method
» Shortcut Method
» Step Deviation Method
6. Individual Series - Direct Method
The Problem – Calculate the arithmetic mean for the data given below
1. Write the sum in rows and column format.
2. Formula to find the arithmetic mean=
= 2+7+10+8+6+3+5+4+5+0
10
= 50
10
= 5
Ans : The arithmetic mean
is 5
Student A B C D E F G H I J
Marks 2 7 10 8 6 3 5 4 5 0
∑x
N is the number of
observations in our
e.g. there are 10
students so N =10
Student X
A 2
B 7
C 10
D 8
E 6
F 3
G 5
H 4
I 5
J 0
50
7. 1. Draw 2 columns ;
a. Under X write the marks.
b. dx = X-A
2. Assume a Mean (A) = 2
= 2 + 0+5+8+6+4+1+3+2+3(-2)
10
= 2 + 32-2
10
=
2+3
3. Formula to find the arithmetic mean =
=2 + 30
10
= 5
Ans : The arithmetic mean is 5
30
= A + ∑dx / N
X A = 2
dx = X - A
2 2 -2 = 0
7 7- 2 = 5
10 10 - 2 = 8
8 8 - 2 = 6
6 6 - 2 = 4
3 3 -2 = 1
5 5 -2 = 3
4 4 -2 = 2
5 5 - 2 = 3
0 0 - 2= -2
Student A B C D E F G H I J
Marks 2 7 10 8 6 3 5 4 5 0
8. DISCRETE SERIES- DIRECT METHOD
Step 1. Column ‘x’ and put all the data of
marks received by students in this
column.
Step 2. Column ‘’f and put all the data
of frequency under this.
Step 3. Third column ‘fx’ ; fx = f x X
i.e multiply data in col. f with data in col.
X.
Step 4. Find ∑fx add all the data in
column ‘ fx’ = 20+ 12 +24 +24 = 80
Step 5. Find ∑f Add all data in column
‘f’
= 4+6+8+6 = 24
Step 6 .Formula to find mean =
(∑fx)
(∑f)
Marks Received by students 5 2 3 4
Frequency 4 6 8 6
x f fx
Marks Received by students Frequency
5 4 4 x 5 = 20
2 6 6 x 2 = 12
3 8 8 x 3 = 24
4 6 6 x 4 = 24
24 80
9. DISCRETE SERIES- SHORTCUT METHOD
Step 1. Make a column ‘x’ and put all the data
of marks received by students in this column.
Step 2. Column ‘’f and put all the data of
frequency under this.
Step 3. Column Assume a no. A in our
e.g. we have assumed A = 2 ‘dx’ ; dx = x -
A
Step 4. Find ∑fdx Multiply data in column
‘f’ with data in column ‘dx’.
∑fdx = sum of the column fdx = 32
Step 5. Find ∑f Add all data in column ‘f’
= 4+6+8+6 = 24
Step 6 . Formula = A + (∑fdx)/(∑f)
= 2 + 32/24 = 3.33 ; ANS = 3.33
Marks Received by students 5 2 3 4
Frequency 4 6 8 6
x f A = 2
Marks Received by
students
Frequency dx = x - A fdx
5 4 5 - 2= 3 4 x 3 = 12
2 6 2 - 2= 0 6 x 0 = 0
3 8 3 - 2= 1 8 x 1 = 8
4 6 4 - 2= 2 6 x 2 = 12
24 32
10. CONTINUOUS SERIES - DIRECT METHOD
The following table shows wages of workers. Calculate the arithmetic mean.
The wages are given in a group series
e.g. how many workers get wages between 10 to 20 How
many workers get wages between 20 and 30
Ans. 8.
Ans. 9.
Here ‘X’ is the wages, but we cannot identify one number for ‘X’. So we need to
find the ‘Mid Value’ of each group.
e.g. To find the Mid value of the first group ‘10 – 20’
Formula to find the Mid Value L1 + L2/ 2
L1 = the first number in the group in our example it is ‘10’
L2 = the second number in the group, in our example it is ‘20 ‘
Divide by 2, because we need to divide into two halves so we know which
number is in the middle.
So the mid value (M.V) for the first group ‘10 – 20’ =
10 + 20 / 2 = 30 /2 = 15
Wages 10 - 20 20-30 30 - 40 40 - 50 50 - 60
No. of Workers 8 9 12 11 6
11. Continuous Series - Direct Method.. Cont.….
Arrange the numbers in the following manner
Add a column “MV”(Mid Value) which will be your new ‘X’, and find the Mid Value
Add another column ‘fx’ i.e multiple value in column ‘f’ and the value in column ‘X’.
Find ∑f i.e (find the total of column ‘f’) Find ∑fx i.e (find the total of column ‘fx’)
Formula = ∑fx/ ∑f
Answer = 34.56
Wages f
No. of Workers
X
MV (Mid Value)
fx
10 - 20 8 10+20/2 = 15 8 x 15 = 120
20-30 9 20+30/2 = 25 9 x 25 = 225
30 - 40 12 30+40/2= 35 12 x 35 = 420
40 - 50 11 40+50/2 = 45 11 x 45 = 495
50 - 60 6 50+60/2 = 55 6 x 55 = 330
∑f = 46 ∑fx 1590
= 1590
46
12. CONTINUOUS SERIES – SHORTCUT METHOD
1.Draw the 3 columns similar to the direct method i.e Wages, f (no.of workers and x-Midvalue)
2. Next assume a no. A (preferably from column’x’). In our eg we have assumed A = 15.
3. Find dx, dx = x – A.
4. Find fdx ; Multiply the answer in dx column with the 2nd column ‘f’.
5. Formula to find the arithmetic mean = A + ∑fdx
∑f
= A + ∑fdx
∑f
= 15 + 900
46
Answer = 34.56
f x Assume A = 15
Wages No.of Workers Mid Value dx = x - A fdx
10 - 20 8 10 + 20/2 = 15 15 - 15 = 0 0 x 8 = 0
20 - 30 9 20 + 30/2= 25 25 - 15= 10 10 x 9 = 90
30 - 40 12 30 + 40 /2 = 35 35 - 15= 20 20 x 12 = 240
40 - 50 11 40 + 50/2 = 45 45 - 15= 30 30 x 11 = 330
50 - 60 6 50 + 60/2 = 55 55 - 15= 40 40 x 6 = 240
46 900
13. CONTINUOUS SERIES – STEP DEVIATION
METHOD
1.Draw the columns similar to the shortcut method i.e Wages, f, x and dx
2. Next assume a no. C. In our eg we have assumed c = 10.
3. Find dx’, dx’= dx /c.
4. Find fdx’ ; Multiply the answer in dx’ column with the 2nd column ‘f’.
5. Formula to find the arithmethic mean = A + ∑fdx’ x
c
= A + ∑fdx’ x c
∑f
Answer = 34.56
f x Assume A = 15 Assume C= 10
Wages No.of Workers Mid Value dx = x - A dx' = dx/c
10 - 20 8 10 + 20/2 = 15 15 - 15 = 0 0/10 = 0
20 - 30 9 20 + 30/2= 25 25 - 15= 10 10/10= 1
30 - 40 12 30 + 40 /2 = 35 35 - 15= 20 20/10= 2
40 - 50 11 40 + 50/2 = 45 45 - 15= 30 30/10= 3
50 - 60 6 50 + 60/2 = 55 55 - 15= 40 40/10= 4
46
14. Individual Series
Direct Method :
Shortcut Method :
Discrete Series
Direct Method : = ∑fx
∑f
Shortcut Method : = A +
∑fdx
∑f
Continuous Series
Direct Method
Shortcut Method
Step Deviation
X
X dx
No. of Boxes
X f dx fdx
X f fx
X f x(M.V) fx
x f x (M.V) dx fdx
x f x (M.V) dx dx' fdx'
15. EXAMPLES : ─ MEAN (GROUPED
DATA)
Example 1: There are 40 students in Grade 8. The
marks obtained by the students in mathematics are
tabulated below. Calculate the mean marks.
Marks Obtained (xi) Number of
Students (fi)
fixi
100 6 100×6 =600
95 8 95×8 = 760
88 10 88×10=880
76 9 76×9= 684
69 7 69×7= 483
Total Σ fi= 40 Σfixi = N=3407
16. SOLUTION : ─
We know ,
Mean =Σfixi / Σfi
x =3407/40
Mean = 85.175
Therefore ,the mean marks = 85.175
17. EXAMPLE 2 :
The percentage of marks of 50 students in a test is
given in the following table. find the mean of the
percentage.
Class
(percentage
of marks)
0−20 20−40 40−60 60−80 80−100
No of
students
3 7 15 20 5
18. SOLUTION
Class Class marks(xi) Frequency (fi) fixi
0−20 10 3 10×3=30
20−40 30 7 30×7=210
40−60 50 15 50×15=750
60−80 70 20 70×20=140
80−100 90 5 90×5=450
Total N=Σ fi =50 Σ fixi =2840
We now,
Mean = Σfixi / Σfi
Mean = 2840/ 50
Mean =56..80
Therefore the mean percentage is 56.80
19. EXAMPLE 3
The maximum temperatures in degree Celsius of
30 towns, in the last summer, is shown in the
following table.Find the mean of the maximum
temperatures.
maximum
(temp)
24−28 28−32 32−36 36−40 40−44
No. of towns 4 5 7 8 6
20. SOLUTION
Class (temp in
celcius)
Class mark(xi) Frequency (fi) fixi
24−28 26 4 26×4=104
28−32 30 5 30×5=150
32−36 34 7 34×7=238
36−40 38 8 38×8=304
40−44 42 6 42×6=252
Total N=Σ fi = 30 Σ fixi = 1048
We know,
Mean = Σfixi / Σfi
Mean =1048 /30
Mean = 34.9 degree Celsius
Therefore the mean of maximum temperature is 34.9 degree Celcius
