It is simply the difference between the maximum value and the minimum value given in a data set. Example: 1, 3,5, 6, 7 => Range = 7 -1= 6 Standard Deviation: The square root of the variance is known as the standard deviation i.e. S.D. = √σ Mean and Mean Deviation: The average of numbers is known as the mean and the arithmetic mean of the absolute deviations of the observations from a measure of central tendency is known as the mean deviation (also called mean absolute deviation). There are two main types of dispersion methods in statistics which are: Absolute Measure of Dispersion Relative Measure of Dispersion The most commonly used techniques for investigating the relationship between two quantitative variables are correlation and linear regression. Correlation quantifies the strength of the linear relationship between a pair of variables, whereas regression expresses the relationship in the form of an equation.

1. Measure of dispersion,
Range, Mean and Standard
Deviation, Correlation and
Regression Analysis
Presented by:
Parth Chauhan
M.Sc. Forensic Science
LNJN NICFS, MHA
FS-105

2. WHAT IS DISPERSION ?
Dispersion measures the extent to which the items vary from some central
value.
central value = Mean / Mode / Median
Example : 2 , 4 , 9 Arithmetic Mean =
�𝑥𝑥𝑖𝑖
𝑛𝑛
= 5
⇒ 2 to 5 has deviation of 3
⇒ 4 to 5 has deviation of 1
⇒ 9 to 5 has deviation of 4
Difference of the deviation of overall series from the central value is
Dispersion.
Also known as scatter value, spread value or variation.

3. MEASUREMENTS OF DISPERSION
• Range
• Quartile Deviation
• Mean Deviation
• Standard Deviation
• Variance
• Coeff. of Range
• Coeff. of Quartile Deviation
• Coeff. of Mean Deviation
• Coeff. of Standard Deviation
• Coeff. of Variance
ABSOLUTE MEASURE RELATIVE MEASURE
(UNIT SAME) (UNIT FREE)

4. ⇒ R A N G E
⇒ RANGE = L – S Where,
L = Largest Observation
S = Smallest Observation
In Individual Series,

5. In Discrete Series,
In Continuous Series ⇒ Inclusive series ,

6. In Continuous Series ⇒ Exclusive series ,
⇒

7. ⇒ M E A N D E V I A T I O N
Mean Deviation (MD)
Mean (by default)
Mode
Median
In Individual series,
Step 1. Find Mean, �
𝑥𝑥 =
�𝑥𝑥𝑖𝑖
𝑛𝑛
Step 2. Find absolute deviation, ( 𝑥𝑥𝑖𝑖 − ̅
𝑥𝑥 )
Step 3. ∑ 𝑥𝑥𝑖𝑖 − ̅
𝑥𝑥
Step 4. MD =
� 𝑥𝑥𝑖𝑖− ̅
𝑥𝑥
𝑛𝑛

8. In Discrete series,
Step 1. Find Mean, �
𝑥𝑥 =
� 𝑓𝑓𝑖𝑖𝑋𝑋𝑖𝑖
� 𝑓𝑓𝑖𝑖
Step 2. Find absolute deviation, ( 𝑥𝑥𝑖𝑖 − ̅
𝑥𝑥 )
Step 3. Find, 𝑓𝑓𝑖𝑖 𝑥𝑥𝑖𝑖 − ̅
𝑥𝑥
Step 4. MD =
� 𝑓𝑓𝑖𝑖 𝑥𝑥𝑖𝑖− ̅
𝑥𝑥
� 𝑓𝑓𝑖𝑖

9. In Continuous series,
Step 1. Find mid point.
Step 2. Find Mean, �
𝑥𝑥 =
� 𝑓𝑓𝑖𝑖𝑋𝑋𝑖𝑖
� 𝑓𝑓𝑖𝑖
Step 3. Find absolute deviation, ( 𝑥𝑥𝑖𝑖 − ̅
𝑥𝑥 )
Step 4. Find, 𝑓𝑓𝑖𝑖 𝑥𝑥𝑖𝑖 − ̅
𝑥𝑥
Step 5. MD =
� 𝑓𝑓𝑖𝑖 𝑥𝑥𝑖𝑖− ̅
𝑥𝑥
� 𝑓𝑓𝑖𝑖

10. ⇒ S T A N D A R D D E V I A T I O N
𝜎𝜎 =
� 𝑓𝑓𝑖𝑖 𝑥𝑥𝑖𝑖− ̅
𝑥𝑥 2
𝑁𝑁
̅
𝑥𝑥 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚
𝑁𝑁 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑖𝑖𝑖𝑖 𝑎𝑎 𝑠𝑠𝑠𝑠𝑠𝑠
𝜎𝜎 =
∑ 𝑓𝑓𝑖𝑖 𝑥𝑥𝑖𝑖 − ̅
𝑥𝑥 2
𝑁𝑁 − 1
̅
𝑥𝑥 = Sample mean
N = Total no of set in a sample
In Individual series, Step 1. Calculate Mean, �
𝑥𝑥 =
� 𝑋𝑋𝑖𝑖
𝑛𝑛
Step 2. Calculate (𝑥𝑥𝑖𝑖 − ̅
𝑥𝑥)
Step 3. Calculate 𝑥𝑥𝑖𝑖 − ̅
𝑥𝑥 2
Step 4. 𝜎𝜎 =
� 𝑥𝑥𝑖𝑖− ̅
𝑥𝑥 2
𝑁𝑁

11. In Discrete series,
Step 1. Calculate Mean, �
𝑥𝑥 =
∑ 𝑓𝑓𝑓𝑓
∑ 𝑓𝑓
Step 2. Calculate (𝑥𝑥𝑖𝑖 − ̅
𝑥𝑥)
Step 3. Calculate 𝑥𝑥𝑖𝑖 − ̅
𝑥𝑥 2
Step 4. Calculate 𝑓𝑓𝑖𝑖 𝑥𝑥𝑖𝑖 − ̅
𝑥𝑥 2
Step 5. 𝜎𝜎 =
� 𝑓𝑓𝑖𝑖 𝑥𝑥𝑖𝑖− ̅
𝑥𝑥 2
𝑁𝑁

12. In Continuous series,
Step 1. Calculate mid point (x)
Step 2. Calculate Mean, �
𝑥𝑥 =
∑ 𝑓𝑓𝑓𝑓
∑ 𝑓𝑓
Step 3. Calculate (𝑥𝑥𝑖𝑖 − ̅
𝑥𝑥)
Step 4. Calculate 𝑥𝑥𝑖𝑖 − ̅
𝑥𝑥 2
Step 5. Calculate 𝑓𝑓𝑖𝑖 𝑥𝑥𝑖𝑖 − ̅
𝑥𝑥 2
Step 6. 𝜎𝜎 =
� 𝑓𝑓𝑖𝑖 𝑥𝑥𝑖𝑖− ̅
𝑥𝑥 2
𝑁𝑁

13. WHAT IS CORRELATION ANALYSIS ?
Correlation analysis deals with association
between two or more variables.
The degree of relationship between the
variables under consideration is measured
through the correlation analysis.
The measure of correlation called the
“Correlation Coefficient” or “Correlation
index” summarizes in one figure the
direction & correlation.

14. • Direct correlation
• If both the variables varies
in same direction.
• X and Y both increase or
decrease
• value lies between 0 to 1
Positive Correlation Negative Correlation
• Inverse correlation
• If varies in opposite direction.
• X increase and Y decrease or
vice versa
• Value lies between -1 to 0
Correlation Coefficient (r) ranges between -1 to 1
• If r = 1 i.e. perfect positive correlation
• If r = -1 i.e. perfect negative correlation
• If r = 0 i.e. No correlation

15. ⇒ 𝐊𝐊𝐊𝐊𝐊𝐊𝐊𝐊 𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐩𝐧𝐧′𝐬𝐬 𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏𝐏 𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌 𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌𝐌
Cov (x,y) =
∑ 𝑥𝑥− ̅
𝑥𝑥 𝑦𝑦− �
𝑦𝑦
𝑛𝑛
⇒
⇒ 𝜎𝜎𝑥𝑥 =
∑ 𝑥𝑥 − ̅
𝑥𝑥 2
𝑛𝑛
𝜎𝜎𝑦𝑦 =
∑ 𝑦𝑦 − �
𝑦𝑦 2
𝑛𝑛
⇒
⇒ r =
To find the value of r , we require ∑ 𝑥𝑥 , ∑ 𝑦𝑦 , � 𝑥𝑥2
,� 𝑦𝑦2
& ∑ 𝑥𝑥𝑥𝑥 from the dataset.
Where, n is the no of pair of observations
∑ 𝑋𝑋𝑋𝑋 − ∑ 𝑥𝑥𝑥𝑥𝑥𝑥
∑ 𝑥𝑥2 − ∑ 𝑥𝑥 2 ⋅ ∑ 𝑦𝑦2 − ∑ 𝑦𝑦 2

16. Step 1. Calculate 𝑥𝑥2
, 𝑦𝑦2
& 𝑥𝑥𝑥𝑥
Step 2. Calculate � 𝑥𝑥2
,� 𝑦𝑦2
& ∑ 𝑥𝑥𝑥𝑥
Step 3. Put values in the correlation
formula.
Step 4. Simplify and find the value.

17. WHAT IS REGRESSION ANALYSIS ?
Regression Analysis is the technique for measuring or
estimating the relationship among variables.
Regression Analysis provides estimates of values of the
dependent variable from the values of the independent
variables.
The regression lines describes the average relationship
existing between x and y.

18. REGRESSION LINES
Regression line of y on x
• 𝑦𝑦 − �
𝑦𝑦 = 𝑏𝑏𝑦𝑦𝑦𝑦 𝑥𝑥 − ̅
𝑥𝑥
• 𝑏𝑏𝑦𝑦𝑦𝑦 =
𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛−∑ 𝑥𝑥𝑥𝑥𝑥𝑥
𝑛𝑛𝑛𝑛𝑥𝑥2− ∑ 𝑥𝑥 2
• x = Independent variable
• y = dependent variable
• Used to calculate y for given x
Regression line of x on y
• 𝑥𝑥 − ̅
𝑥𝑥 = 𝑏𝑏𝑥𝑥𝑥𝑥 𝑦𝑦 − �
𝑦𝑦
• 𝑏𝑏𝑥𝑥𝑥𝑥 =
𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛−∑ 𝑥𝑥𝑥𝑥𝑥𝑥
𝑛𝑛𝑛𝑛𝑦𝑦2− ∑ 𝑦𝑦 2
• x = dependent variable
• y = Independent variable
• Used to calculate x for given y

19. Step 1. Calculate 𝑥𝑥2, 𝑦𝑦2 & 𝑥𝑥𝑥𝑥
Step 2. Calculate � 𝑥𝑥2
,� 𝑦𝑦2
& ∑ 𝑥𝑥𝑥𝑥
Step 3. Put values in the regression
coefficient formula.
Step 4. Simplify and find the value of
𝑏𝑏𝑦𝑦𝑦𝑦 and 𝑏𝑏𝑥𝑥𝑥𝑥.
Step 5. put values in regression lines
equation to find out exact equations :
𝑦𝑦 − �
𝑦𝑦 = 𝑏𝑏𝑦𝑦𝑦𝑦 𝑥𝑥 − ̅
𝑥𝑥
𝑥𝑥 − ̅
𝑥𝑥 = 𝑏𝑏𝑥𝑥𝑥𝑥 𝑦𝑦 − �
𝑦𝑦

20. R E F E R E N C E S
• Craig Adam; “Essential Mathematics and Statistics for Forensic Science”, Wiley
Blackwell, 2010
• N.M Shah; “Statistics and Economics”, APC Publications, 2019

21. T H A N K Y O U