Dispersion- It is a statistical term that describes the size of the distribution of values expected for a particular variable and can be measured by several different statistics, such as Range, Variance and standard deviation.
Method of Dispersion-A measure of dispersion indicates the scattering of data. It explains the disparity of data from one another, delivering a precise view of their distribution.
Methods of Dispersion.
1.Relative Dispersion
a. Coefficient of Mean Deviation
b. Coefficient of Quartile Deviation
c. Coefficient of Range
d. Coefficient of Variation
2. Absolute Dispersion
a. Range
b. Quartile range
c. Standard deviation
d. Mean Deviation
Range- It is the difference between smallest & largest values in the dataset. Also the relative measure of range is known as Coefficient of Range.
Advantages and disadvantages of Range.
Calculation of Range by different Methods.
b. Quartile Range- The interquartile range of a group of observations is the interval between the values of upper quartile and the lower quartile for that group.
Advantages and Disadvantages of Quartile Range.
Calculation of Quartile Range by different Methods.
c. Standard Deviation- It measures the absolute dispersion (or) variability of a distribution. A small standard deviation means a high degree of uniformity of the observations as well as homogeneity in the series.
Advantages and Disadvantages of Quartile Range.
Calculation of Standard Deviation using.
i) Direct Method
ii) Short-cut Method
iii) Step Deviation Method.
1. BIOSTATISTICS AND RESEARCH METHODOLOGY
Unit-1: methods of dispersion
PRESENTED BY
Himanshu Rasyara
B. Pharmacy IV Year
UNDER THE GUIDANCE OF
Gangu Sreelatha M.Pharm., (Ph.D)
Assistant Professor
CMR College of Pharmacy, Hyderabad.
email: sreelatha1801@gmail.com
2. Methods Of Dispersion:
A measure of dispersion indicates the scattering of data. It explains the disparity of data from one
another, delivering a precise view of their distribution.
The measure of dispersion displays and gives us an idea about the validation and the central value of an
individual item.
DISPERSION:
It is a statistical term that describes the size of the distribution of values expected for a particular variable
and can be measured by several different statistics, such as Range, Variance and standard deviation.
Method of
Dispersion
Relative
Coefficient of
Mean
Deviation
Coefficient
of Quartile
Deviation
Coefficient of
Range
Coefficient
of Variation
Absolute
Range
Quartile
range
Standard
deviation
Mean
Deviation
3. 1) RANGE:
Denoted by ‘R’.
It is the difference between smallest & largest values in the dataset. Also the relative measure of range is
known as Coefficient of Range.
Mathematically if ‘h’ is the highest value or largest value and ‘l’ is the lowest value or smallest value then
R= H-L (or) L=S
Coefficient of range =
H−L
H+L
(OR)
L−S
L+S
Advantages:
I. It is simple to understand.
II. Easy to compute (or) calculate.
III. Helpful in statistical QC [In construction of control charts for variables] and weather forecasting.
IV. It’s units are the same as the unit of the variable being measured.
Disadvantages:
I. Not suitable for thorough analysis.
II. Do not take account of the number of observations in a sample.
III. It makes no direct use of many observations in the sample.
IV. Range also suffers from dependence upon extreme observations.
V. Range cannot be computed in case of opened distribution.
4. Examples:
1. Calculation of the Range for Individual observations.
Example: Calculate the range & COR for following data regarding Hb% of 10 patients.
8.3, 9.6, 12.3, 11.3, 9.6, 13.2, 10.1, 9.7
Solution: Range= H-L
= 13.2-8.3= 4.9
COR=
H−L
H+L
=
13.2−8.3
13.2+8.3
=
4.9
21.5
= 0.22
2. Calculation of Range in Discrete Series. Data(method 1) Frequency
0-10 1
10-20 3
20-30 10
30-40 13
40-50 9
Method 1:
Range= 50-0= 50
COR=
50−0
50=0
= 1
Method 2: (by taking mid
points)
H= 45, L= 5
Range 45-5= 40
COR=
45−5
45+5
= 0.8
Data(method 2) Frequenc
y
Mid Point
0-10 1 5
10-20 5 15
20-30 10 25
30-40 13 35
40-50 9 45
5. 2) Inter Quartile/ Quartile Ranges:
The interquartile range of a group of observations is the interval between the values of upper quartile
and the lower quartile for that group .
Upper quartile of a group is the value above which 25% of observations fall.
Lower quartile is the value below which 25% of observations fall.
This measure gives us the range which covers the middle 50% of observations in the group. If lower
quartile Q1& upper quartile is Q3 then interquartile rang
Q3-Q1
Quartile deviation/Semi-inter quartile range:
Q=
𝟏
𝟐
(Q3-Q1)
Coefficient of Semi-inter quartile range is:
𝐐𝟑 − 𝐐𝟏
𝐐𝟑 + 𝐐𝟏
Advantages:
I. It is unaffected by extreme values.
II. It is easy to understand and calculate.
III. It is quite satisfactory when only the middle half of the group is dealt with.
Disadvantages:
I. It ignores 50% of the extreme values.
II. It is not suitable for algebraic treatment.
6. Calculation of Inter Quartile Range for Individual Observation.
Example: Calculate the inter Quartile range, Quartile Deviation/ Coefficient of Quartile deviation from
the following table given.
Solution: Q1= series of [N+1/4]th item
= [199+1/4]= (50)th item
Size of 50th item= 60
Q3= Series of 3 [N+1/4]th item
= 3[199+1/4]= 3[50]= 150
Size of 150th item= 64
Range= 64-60= 4
Quartile deviation=
1
2
(4)= 2
Coefficient of Quartile range=
64−60
64+60
= 0.03
Height Frequency Cumulative Frequency
58 21 21
59 25 21+25= 46
60 28 46+28= 74
61 18 74+18= 92
62 20 92+20= 112
63 22 112+22= 134
64 24 134+24= 158
65 23 158+23= 181
66 18 181+18= 199
7. 3) Standard Deviation:
It measures the absolute dispersion (or) variability of a distribution. A small standard deviation means a high degree
of uniformity of the observations as well as homogeneity in the series.
(OR)
Standard deviation is a positive square root of the average of squared deviation taken from Arithmetic mean. It is
denoted by Greek alphabet (σ) or by S.D.
Let ‘x’ be the random variate which takes on ‘n’ values i.e. x1, x2,x3…….xn then the S.D of these ‘n’ observations is
given by
S.D =
𝚺 𝐱−𝐱′ 𝟐
𝐧−𝟏
(OR) S.D =
𝚺𝐱𝟐− 𝚺𝐱 𝟐
𝐧
𝐧 𝐨𝐫 (𝐧−𝟏)
Merits of Standard Deviation:
I. It is rigidly defined.
II. It is based on all observations.
III. It is less effected by sampling fluctuations.
IV. It represents the true measurement of dispersion of a series.
V. It is extremely useful in correlation.
Demerits of Standard Deviation:
I. It is difficult to calculate as compared to other measure of dispersion.
II. It is not simple to understand.
III. It gives more weightage to extreme failures.
IV. It consumes much time and labour while computing it.
8. Calculation of Standard Deviation of Individual Observations.
Example: Data recorded on the respiration rate per minute in 7 individuals are given below. Calculate S.D
of respiration rate.
S.D =
𝚺 𝐱−𝐱′ 𝟐
𝐧−𝟏
S.D=
35
7−1
=
35
6
= 5.83
= 2.41
Calculation of S.D in a discrete series.
a) Actual or Direct Method
b) Assumed or Short- cut Method
c) Step Deviation Method
Variable (x-x’) (x-x’)2
16 16-20=-4 16
17 17-20=-3 9
18 18-20=-2 4
19 19-20=-1 1
20 20-20=0 0
21 21-20=1 1
22 22-20=2 4
9. a) Actual or Direct Method:
The S.D for the discrete series is given by the formula.
(σ)=
𝚺𝐟 𝐱−𝐱′ 𝟐
𝐧
Where x’= Arithmetic Mean
x= Size of Item
f= Corresponding frequency
n= Σf
In practice this method is rarely used.
b) Assumed or Short- cut Method:
The formula for this method is.
(σ)=
Σ𝑓𝑑2
𝑛
−
Σ𝑓𝑑
𝑛
2
d= x-a
n= Σf
Example: Find the standard deviation of incubation period smallpox in 50 patients of the following data.
Period 10 11 12 13 14 15 16
No. of
Patients
2 7 11 15 10 4 1
11. c) Step Deviation Method:
The formula for this method is
(σ)=
Σfd2
n
−
Σfd
n
2 x i
Where, i= Common Class Interval
d=
𝑥−𝐴
𝑖
A= Assumed Mean
f= Respective Frequency
Example: Calculate the S.D of the below data blood pressure of a patient for 100 days.
B.P (mmHg) 102 106 110 114 118 122 126
No. of Days 3 9 25 35 17 10 1
12. Solution:
Let us take the assumed mean A= 114.
Let d= (x-A)/i= (x-114)/4, i=4
(σ)=
Σfd2
n
−
Σfd
n
2 x i
=
154
100
−
−12
100
2 x4
154 − 0.0144 x4= 1.235x4= 4.94 mmHg.
B.P (mmHg) No. of Days (f) d= x-114/4 fd fd2
102 3 -3 -9 27
106 9 -2 -18 36
110 25 -1 -25 25
114 35 0 0 0
118 17 1 17 17
122 10 2 20 40
126 1 3 3 9
Total N= 100 Σfd= -12 Σfd2= 154
