Measures of central tendency and dispersion

1. Measures of central tendency and dispersion

Indramani Tripathi
M.Sc-Biochemistry

2. .
:
MEASURE OF CENTRAL VALUE
1-CROXTON AND COWDEN-
“Average is single value within the the range of data that used to
represented all the value in series””
2-CLARK &SCHAKADA-
“Average is an attempt to find one single to describe whole group of
figure”
3-Dr B0WELY
“Statistics is the science of average”
Since an average is somewhere within the range of data it is also called a
Measure of central value

3. Central tendency = means – average
Characteristics of Average
• ClearDefination
The average should be properly defined
• Easy to calculation
It should have easy to calculation
• Representative of series
The should be based on each and every item of the seris
• Unaffected
The average should not be affected to the extreme value
• Algebraic and mathematical calculation
It should be easily subjected to further mathematical calculation

4. • Absolut number
The average should be in Absolut no and the not percentage
• Sampling
The average should be least affect by the function of
sampling
Limitation of average
• The value of central tendency or average does not
completely describe the data
• Average is obtain very judiciously and is ideal for a
particular investigation
the best average has its own series
• Other limitation of average

5. • Types of average
1. Mathematical Average
2. Positional average
3. Partition average
Mathematical average or (mean)
Average represent mathematically is termed as mean
Mean
1. Arithmetic mean
2. Geometric mean
3. Harmonic mean

6. Arithmetic mean
Arithmetic average mean is the quantity obtained by the sum of all the
The value of the items in a series by their number
Formula
Where
AM Arethemathic mean =m
n=no of item

7. Formula: Arithmetic Mean = sum of elements / number of elements
= a1 + a2 + a3 + ... + an / n
Example to find the Arithmetic Mean of 3, 5, 7.
Step 1: The sum of the numbers are:
3 + 5 + 7 = 15
Step 2: The total numbers are:
There are 3 numbers.
Step 3: The Arithmetic Mean is:
15 / 3 = 5

8. Geometric Mean
The geometric mean is defined as the nth root of the
Product of nth observation
It is denoted the GM
Formula
Geometric Mean :
Geometric Mean = ((X1)(X2)(X3)........(XN))1/N
where
X = Individual score
N = Sample size (Number of scores)

9. Formula: Geometric Mean = ((x1)(x2)(x3) ... (xn))1/n
where x = Individual score and n = Sample size
(Number of scores)
Example to find the Geometric Mean of 1, 2 ,3 ,4 ,5.
Step 1: n = 5 is the total number of values. Find 1
/ n.
1 / n = 0.2
Step 2: Find Geometric Mean using the formula:
[(1)(2)(3)(4)(5)]0.2 = 1200.2
Geometric Mean = 2.60517

10. Harmonic Mean
The harmonic mean is defined as the
“The reciprocal of the arithmetic mean of the value reciprocal of given value”
Formula
Harmonic Mean Formula :
Harmonic Mean = N/(1/a1+1/a2+1/a3+1/a4+.......+1/aN)
Where,
X = Individual score
N = Sample size (Number of scores)

11. Harmonic Mean
Example:
To find the Harmonic Mean of 1,2,3,4,5.
Step 1
Calculate the total number of values. N = 5
Step 2:
Now find Harmonic Mean using the above formula.
N/(1/a1+1/a2+1/a3+1/a4+.......+1/aN)
= 5/(1/1+1/2+1/3+1/4+1/5)
= 5/(1+0.5+0.33+0.25+0.2)
= 5/2.28
So
Harmonic Mean = 2.19

12. Average of position
Averages of the postion indicate the postion of the an average in series of
observation arranged in inccrasing of the magnitude
Average of the position
1. Median
2. Mode
Median=
Median means value of the middle most observation when data
Arranged ascending or desending order of magnitude
.
.
s.

13. Example 2:
To find the median of 4,5,7,2,1,8 [Even]
Step 1:
Count the total numbers given.
There are 6 elements or numbers in the distribution.
Step 2:
Arrange the numbers in ascending order. 1,2,4,5,7,8
Step 3:
The total elements in the distribution (6) is even.
As the total is even, we have to take average of number at n/2 and (n/2)+1
So the position are n/2= 6/2 = 3 and 4 The number at 3rd and 4th position are 4,5
Step 4:
Find the median. The average is (4+5)/2 = Median = 4.5

14. Mode
That value of the variable for which the frequency is maximum
It is denoted by the MO
• Unimodal frequency distribution
Data having one mode
• Biomodal frequency distribution
Data having two mode
• Multimodal frequency distribution
Data having two or more mode
• Antimode
In u shaped distribution the law point at the middle of the distribution

15. Example
The following is the number of
problems that Ms. Matty assigned for
homework on 10 different days.
What is the mode?
8, 11, 9, 14, 9, 15, 18, 6, 9, 10
Solution:
Ordering the data from least to
greatest, we get:
6, 8, 9, 9, 9, 10, 11 14, 15, 18
Answer:
The mode is 9.

19. Statistical dispersionn
In statistics, dispersion
(also called variability, scatter, or spread) denotes
how stretched or squeezed a distribution
(theoretical or that underlying a statistical sample)
is. Common examples of measures of statistical
dispersion are the variance, standard deviation and
interquartile range.
Types of Measures of Dispersions:
Measures of dispersion are of two types
(i ) Measures of Absolute Dispersion, and
(ii) Measures of Relative Dispersion.

20. (i) Measures of Absolute Dispersion: The
actual variation or dispersion determined by the
Measures of Absolute Dispersion is called ‘absolute
dispersion’.
(ii) Measures of Relative Dispersion:
The measures of absolute dispersion cannot be
used to compare the variation of two or more
series. For
To compare the variation of two or more series, we
need a measure of relative dispersion. It is defined
as: