DISCUSSES FOUR DIFFERENT METHODS OF CALCULATING MEAN DIRECT METHOD SHORT CUT METHOD STEP DEVIATION METHOD By ‘mean’ we refer to an Arithmetic mean. There are other types of ‘mean’ like geometric mean and harmonic mean. Arithmetic mean is the total of the sum of all values in a collection of numbers divided by the number of numbers in a collection.

1. MEASURES OF CENTRAL TENDENCY
MEAN / MEDIAN / MODE
condensation of a large amount of data into a single value is known as measures of central tendency.

2. MEAN
● mean is the average or the most common value in a collection of numbers.
● The ARITHMETIC MEAN and the GEOMETRIC MEAN are two types of mean that
can be calculated.
● Summing the numbers in a set and dividing by the total number gives you the
Arithmetic Mean.
● The Geometric Mean is more complicated and involves multiplication of the
numbers taking the nth root.
● It is denoted by x
̄ , (read as x bar).

3. ARITHMETIC MEAN
❖ The arithmetic mean is the simple average, or sum of a series of
numbers divided by the count of that series of numbers.
❖ It simply involves taking the sum of a group of numbers, then dividing
that sum by the count of the numbers used in the series.
❖ The arithmetic mean isn't always ideal, especially when a single outlier
can skew the mean by a large amount

4. For Ungrouped Data
CALCULATING MEAN
➔ Mean x
̄ = Sum of all observations / Number of observations
➔ X
̄ = X1 + X2 + X3 +..............+Xn n = Number of
Observations
n
➔ Eg: find arithmetic mean for given data
1, 5, 7, 9, 11, 3
➔ Ans: x
̄ = Sum of all observations / Number of observations
Sum of all observations= 1+ 5 + 7 + 9 + 11 + 3 = 36

5. For Grouped Data
● There are three methods to find the arithmetic mean for grouped data.
○ Direct method
○ Short-cut method
○ Step-deviation method
● If sum of all data inputs and sum of their frequencies are sufficiently small, the
direct method will work. But, if they are numerically large, we use the assumed
arithmetic mean method or step-deviation method.

6. I. DIRECT METHOD
❖ x
̄ = ∑xifi OR ∑fx
∑fi N
- ∑fi or N indicates the sum of all frequencies.
- ∑xifi is each value multiplied with its frequency.
Eg:
X f
3 4
6 5
8 6

7. DIRECT METHOD (contd.)
x f XiFi / fx
3 4 (3*4) = 12
6 5 (6*5) = 30
8 6 (8*6) = 48
N = 15
∑xifi / ∑fx = 90
❖ x
̄ = ∑xifi ∑fx
∑fi N
x
̄ = 90 / 15 = 6
❖ x
̄ = 6
I. FOR DISCRETE SERIES

8. II. FOR CONTINUOUS SERIES
● Here X is the mid value of the class.
● MID VALUE = LOWER LIMIT OF THE CLASS + UPPER LIMIT OF THE CLASS
2
● x
̄ = ∑xifi ∑fx
MARK No. of STUDENTS
0 - 10 5
10 - 20 8
20 - 30 2

9. II. FOR CONTINUOUS SERIES (contd.)
MARK No. of STUDENTS
( f )
MID VALUE
( X )
XiFi / fx
0 - 10 5 (0+10)/2 = 5 5 * 5 = 25
10 - 20 8 (10+20)/2 = 15 15 * 8 = 120
20 - 30 = 2 (20+30)/2 = 25 25 * 2 = 50
N= 15
∑xifi / ∑fx = 195
● x
̄ = 195 / 15 = 13

11. II. SHORT CUT METHOD
❖ The short-cut method is called as assumed mean method or change of origin method.
x
̄ = ∑fidi
∑fi
❖ The following steps describe this method.
➢ Step1: Calculate the class marks (mid-point) of each class (xi).
➢ Step2: Let A denote the assumed mean of the data.
➢ Step3: Find deviation (di) = xi – A
➢ Step4: Use the formula.
A +

12. SHORT CUT METHOD (CONTD.)
FIND THE MEAN FOR GIVEN DATA
CLASS INTERVAL FREQUENCY
0 - 10 3
10 - 20 8
20 - 30 12
30 - 40 15
40 - 50 18
50 - 60 16
60 - 70 11
70 - 80 5

13. SHORT CUT METHOD (CONTD.)
A = 35

14. III. STEP DEVIATION METHOD
❖ This is also called the change of origin or scale method.
x
̄ = ∑fiui
∑fi
❖ The following steps describe this method:
❖ Step1: Calculate the MID VALUE of each class (xi).
❖ Step2: Let A denote the ASSUMED MEAN of the data.
❖ Step3: Find d (deviation)
❖ Step4: then Find ui= d/c, where C is the CLASS INTERVAL. Or find ui directly by
ui=(xi−A)/c ( then skip step3 and step4 )
❖ Step5: Use the formula:
* C ❖ C can be denoted as h or i
A +

15. III. STEP DEVIATION METHOD (CONTD.)
● Find Mean For Given Data
CLASS FREQUENCY
0 - 10 12
10 - 20 16
20 - 30 32
30 - 40 52
40 - 50 42
50 - 60 32
60 - 70 18
70 - 80 12

16. III. STEP DEVIATION METHOD (CONTD.)
Class Frequency
( f )
Mid Value
( x )
Deviation
( d = x - A)
Step Deviation
( u = d / c )
C = 10
fiui
0 - 10 12 5 -40 -4 - 48
10 - 20 16 15 -30 -3 -48
20 - 30 32 25 -20 -2 -64
30 - 40 52 35 -10 -1 -52
40 - 50 42 45 = A 0 0 0
50 - 60 32 55 10 1 32
60 - 70 18 65 20 2 36
70 - 80 12 75 30 3 36
∑f = 216 ∑fiui = 108

17. III. STEP DEVIATION METHOD (CONTD.)
∑fiui
∑fi
A + * C
108
256
45 + * 10
x
̄ =
x
̄ =
x
̄ = 45 - 5 = 40
x
̄ = 40

18. MERITS OF MEAN
● As the formula to find the arithmetic mean is rigid, the result doesn’t change. Unlike
the median, it doesn’t get affected by the position of the value in the data set.
● It takes into consideration each value of the data set.
● Easy to Calculate
● It’s also a useful measure of central tendency, as it tends to provide useful results,
even with large groupings of numbers.
● It can be further subjected to many algebraic treatments, unlike mode and median.
For example, the mean of two or more series can be obtained from the mean of the
individual series.
● The arithmetic mean is widely used in geometry as well.

19. DEMERITS OF MEAN
● It Is Affected By Extreme Values In The Data Set.
● The Value Of The Mean Cannot Be Computed Without Making Assumptions
Regarding The Size Of The Class.
● It's Practically Impossible To Locate The Arithmetic Mean By Inspection Or
Graphically.
● It Cannot Be Used For Qualitative Types Of Data Such As Honesty, Favorite
Milkshake Flavor, Most Popular Product, Etc.
● We Can't Find The Arithmetic Mean If A Single Observation Is Missing Or Lost.