Measure of Dispersion, Range, Mean and Standard Deviation, Correlation and Regression Analysis

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 =
�????????????
????????????
????????????
= 5
⇒2 to 5 has deviation of 3
⇒4 to 5 has deviation of 1
⇒9to5hasdeviationof4
Differenceofthedeviationofoverallseriesfromthecentralvalueis
Dispersion.
Alsoknownasscattervalue,spreadvalueorvariation.

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=LargestObservation
S=SmallestObservation
InIndividualSeries,

5. InDiscreteSeries,
InContinuousSeries ⇒Inclusiveseries,

6. InContinuousSeries ⇒Exclusiveseries,
⇒

7. ⇒M E A N D E V I A T I O N
MeanDeviation(MD)
Mean (by default)
Mode
Median
InIndividualseries,
Step 1. Find Mean, �????????????=
�????????????
????????????
????????????
Step 2. Find absolute deviation, ( ????????????
????????????−̅????????????)
Step 3. ∑????????????
????????????−̅????????????
Step 4. MD =
�????????????
????????????−̅????????????
????????????

8. InDiscreteseries,
Step 1. Find Mean, �????????????=
�????????????
????????????????????????
????????????
�????????????
????????????
Step 2. Find absolute deviation, ( ????????????
????????????−̅????????????)
Step 3. Find, ????????????
????????????????????????
????????????−̅????????????
Step 4. MD =
�????????????
????????????????????????
????????????−̅????????????
�????????????
????????????

9. InContinuousseries,
Step 1. Find mid point.
Step 2. Find Mean, �????????????=
�????????????
????????????????????????
????????????
�????????????
????????????
Step 3. Find absolute deviation, ( ????????????
????????????−̅????????????)
Step 4. Find, ????????????
????????????????????????
????????????−̅????????????
Step 5. MD =
�????????????
????????????????????????
????????????−̅????????????
�????????????
????????????

10. ⇒S T A N D A R D D E V I A T I O N
????????????=
�????????????
????????????????????????
????????????−̅????????????
2
????????????
̅????????????=????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????
????????????=????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????
????????????=
∑????????????
????????????????????????
????????????−̅????????????
2
????????????−1
̅????????????= Sample mean
N = Total no of set in a sample
InIndividualseries,
Step 1. Calculate Mean, �????????????=
�????????????
????????????
????????????
Step 2. Calculate (????????????
????????????−̅????????????)
Step 3. Calculate ????????????
????????????−̅????????????
2
Step 4. ????????????=
�????????????
????????????−̅????????????
2
????????????

11. InDiscreteseries,
Step 1. Calculate Mean, �????????????=
∑????????????????????????
∑????????????
Step 2. Calculate (????????????
????????????−̅????????????)
Step 3. Calculate ????????????
????????????−̅????????????
2
Step 4. Calculate ????????????
????????????????????????
????????????−̅????????????
2
Step 5. ????????????=
�????????????
????????????????????????
????????????−̅????????????
2
????????????

12. InContinuousseries,
Step 1. Calculate mid point (x)
Step 2. Calculate Mean, �????????????=
∑????????????????????????
∑????????????
Step 3. Calculate (????????????
????????????−̅????????????)
Step 4. Calculate ????????????
????????????−̅????????????
2
Step 5. Calculate ????????????
????????????????????????
????????????−̅????????????
2
Step 6. ????????????=
�????????????
????????????????????????
????????????−̅????????????
2
????????????

13. WHAT IS CORRELATION ANALYSIS ?
Correlationanalysisdealswithassociation
betweentwoormorevariables.
Thedegreeofrelationshipbetweenthe
variablesunderconsiderationismeasured
throughthecorrelationanalysis.
Themeasureofcorrelationcalledthe
“CorrelationCoefficient”or“Correlation
index”summarizesinonefigurethe
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
CorrelationCoefficient(r)rangesbetween-1to1
•If r = 1 i.e. perfect positive correlation
•If r = -1 i.e. perfect negative correlation
•If r = 0 i.e. No correlation

15. ⇒????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????
′
????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????
Cov (x,y) =
∑????????????−̅????????????????????????−�????????????
????????????
⇒
⇒
????????????
????????????=
∑????????????−̅????????????
2
????????????
????????????
????????????=
∑????????????−�????????????
2
????????????
⇒
⇒
r =
To find the value of r , we require ∑????????????, ∑????????????, �????????????
2
,�????????????
2
& ∑????????????????????????from the dataset.
Where, n is the no of pair of observations
∑????????????????????????−∑????????????????????????????????????
∑????????????
2
−∑????????????
2
⋅∑????????????
2
−∑????????????
2

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

17. WHAT IS REGRESSION ANALYSIS ?
RegressionAnalysisisthetechniqueformeasuringor
estimatingtherelationshipamongvariables.
RegressionAnalysisprovidesestimatesofvaluesofthe
dependentvariablefromthevaluesoftheindependent
variables.
Theregressionlinesdescribestheaveragerelationship
existingbetweenxandy.

18. REGRESSION LINES
Regression line of y on x
•????????????−�????????????=????????????
????????????????????????????????????−̅????????????
•????????????
????????????????????????=
????????????????????????????????????????????????−∑????????????????????????????????????
????????????????????????????????????
2
−
∑????????????
2
•x = Independent variable
•y = dependent variable
•Used to calculate y for given x
Regression line of x on y
•????????????−̅????????????=????????????
????????????????????????????????????−�????????????
•????????????
????????????????????????=
????????????????????????????????????????????????−∑????????????????????????????????????
????????????????????????????????????
2
−
∑????????????
2
•x = dependent variable
•y = Independent variable
•Used to calculate x for given y

19. Step 1. Calculate ????????????
2
, ????????????
2
& ????????????????????????
Step 2. Calculate
�????????????
2
,�????????????
2
& ∑????????????????????????
Step 3. Put values in the regression
coefficient formula.
Step 4. Simplify and find the valueof
????????????
????????????????????????and ????????????
????????????????????????.
Step 5. put values in regression lines
equation to find out exact equations :
????????????−�????????????=????????????
????????????????????????????????????−̅????????????
????????????−̅????????????=????????????
????????????????????????????????????−�????????????

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