Statistics 202
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Transcript of Statistics 202
Correlation & Correlation & RegressionRegression
CorrelationCorrelation
Finding the relationship between two quantitative variables without being able to infer causal relationships
Correlation is a statistical technique used to determine the degree to which two variables are related
• Rectangular coordinate• Two quantitative variables• One variable is called independent (X) and
the second is called dependent (Y)• Points are not joined • No frequency table
Scatter diagram
Example
Scatter diagram of weight and systolic blood Scatter diagram of weight and systolic blood pressurepressure
8 0
1 0 0
1 2 0
1 4 0
1 6 0
1 8 0
2 0 0
2 2 0
6 0 7 0 8 0 9 0 1 0 0 1 1 0 1 2 0w t (k g )
8 0
1 0 0
1 2 0
1 4 0
1 6 0
1 8 0
2 0 0
2 2 0
6 0 7 0 8 0 9 0 1 0 0 1 1 0 1 2 0W t ( k g )
Scatter diagram of weight and systolic blood pressure
Scatter plots
The pattern of data is indicative of the type of relationship between your two variables:
positive relationship negative relationship no relationship
Positive relationshipPositive relationship
0
2
4
6
8
10
12
14
16
18
0 10 20 30 40 50 60 70 80 90
Age in Weeks
Heig
ht in
CM
Negative relationshipNegative relationship
Reliability
Age of Car
No relationNo relation
Correlation CoefficientCorrelation Coefficient
Statistic showing the degree of relation between two variables
Simple Correlation coefficient Simple Correlation coefficient (r)(r)
It is also called Pearson's correlation It is also called Pearson's correlation or product moment correlation or product moment correlationcoefficient. coefficient.
It measures the It measures the naturenature and and strengthstrength between two variables ofbetween two variables ofthe the quantitativequantitative type. type.
The The signsign of of rr denotes the nature of denotes the nature of association association
while the while the valuevalue of of rr denotes the denotes the strength of association.strength of association.
If the sign is If the sign is +ve+ve this means the relation this means the relation is is direct direct (an increase in one variable is (an increase in one variable is associated with an increase in theassociated with an increase in theother variable and a decrease in one other variable and a decrease in one variable is associated with avariable is associated with adecrease in the other variable).decrease in the other variable).
While if the sign is While if the sign is -ve-ve this means an this means an inverse or indirectinverse or indirect relationship (which relationship (which means an increase in one variable is means an increase in one variable is associated with a decrease in the other).associated with a decrease in the other).
The value of r ranges between ( -1) and ( +1)The value of r ranges between ( -1) and ( +1) The value of r denotes the strength of the The value of r denotes the strength of the
association as illustratedassociation as illustratedby the following diagram.by the following diagram.
-1 10-0.25-0.75 0.750.25
strong strongintermediate intermediateweak weak
no relation
perfect correlation
perfect correlation
Directindirect
If If rr = Zero = Zero this means no association or this means no association or correlation between the two variables.correlation between the two variables.
If If 0 < 0 < rr < 0.25 < 0.25 = weak correlation. = weak correlation.
If If 0.25 ≤ 0.25 ≤ rr < 0.75 < 0.75 = intermediate correlation. = intermediate correlation.
If If 0.75 ≤ 0.75 ≤ rr < 1 < 1 = strong correlation. = strong correlation.
If If r r = l= l = perfect correlation. = perfect correlation.
ny)(
y.nx)(
x
nyx
xyr
22
22
How to compute the simple correlation coefficient (r)
ExampleExample:: A sample of 6 children was selected, data about their A sample of 6 children was selected, data about their
age in years and weight in kilograms was recorded as age in years and weight in kilograms was recorded as shown in the following table . It is required to find the shown in the following table . It is required to find the correlation between age and weight.correlation between age and weight.
serial No
Age (years)
Weight (Kg)
17122683812451056116913
These 2 variables are of the quantitative type, one These 2 variables are of the quantitative type, one variable (Age) is called the independent and variable (Age) is called the independent and denoted as (X) variable and the other (weight)denoted as (X) variable and the other (weight)is called the dependent and denoted as (Y) is called the dependent and denoted as (Y) variables to find the relation between age and variables to find the relation between age and weight compute the simple correlation coefficient weight compute the simple correlation coefficient using the following formula:using the following formula:
ny)(
y.nx)(
x
nyx
xyr
22
22
Serial n.
Age (years)
(x)
Weight (Kg)
(y)xyX2Y2
17128449144268483664381296641444510502510056116636121691311781169
Total∑x=41
∑y=66
∑xy= 461
∑x2=291
∑y2=742
r = 0.759r = 0.759strong direct correlation strong direct correlation
6(66)742.
6(41)291
66641461
r22
EXAMPLE: Relationship between Anxiety and EXAMPLE: Relationship between Anxiety and Test ScoresTest Scores
AnxietyAnxiety ))XX((
Test Test score (Y)score (Y)
XX22YY22XYXY
101022100100442020883364649924242299448181181811771149497755662525363630306655363625253030
∑∑X = 32X = 32∑∑Y = 32Y = 32∑∑XX22 = 230 = 230∑∑YY22 = 204 = 204∑∑XY=129XY=129
Calculating Correlation CoefficientCalculating Correlation Coefficient
94.)200)(356(
102477432)204(632)230(6
)32)(32()129)(6(22
r
r = - 0.94
Indirect strong correlation
exerciseexercise
Regression AnalysesRegression Analyses
Regression: technique concerned with predicting some variables by knowing others
The process of predicting variable Y using variable X
RegressionRegression Uses a variable (x) to predict some outcome Uses a variable (x) to predict some outcome
variable (y)variable (y) Tells you how values in y change as a function Tells you how values in y change as a function
of changes in values of xof changes in values of x
Correlation and RegressionCorrelation and Regression
Correlation describes the strength of a Correlation describes the strength of a linear relationship between two variables
Linear means “straight line”
Regression tells us how to draw the straight line described by the correlation
Regression Calculates the “best-fit” line for a certain set of dataCalculates the “best-fit” line for a certain set of dataThe regression line makes the sum of the squares of The regression line makes the sum of the squares of
the residuals smaller than for any other linethe residuals smaller than for any other lineRegression minimizes residuals
8 0
1 0 0
1 2 0
1 4 0
1 6 0
1 8 0
2 0 0
2 2 0
6 0 7 0 8 0 9 0 1 0 0 1 1 0 1 2 0W t ( k g )
By using the least squares method (a procedure By using the least squares method (a procedure that minimizes the vertical deviations of plotted that minimizes the vertical deviations of plotted points surrounding a straight line) we arepoints surrounding a straight line) we areable to construct a best fitting straight line to the able to construct a best fitting straight line to the scatter diagram points and then formulate a scatter diagram points and then formulate a regression equation in the form of:regression equation in the form of:
nx)(
x
nyx
xyb 2
21)xb(xyy b
bXay
Regression Equation
Regression equation describes the regression line mathematically Intercept Slope 8 0
1 0 0
1 2 0
1 4 0
1 6 0
1 8 0
2 0 0
2 2 0
6 0 7 0 8 0 9 0 1 0 0 1 1 0 1 2 0W t ( k g )
Linear EquationsLinear EquationsY
Y = bX + a
a = Y-interceptX
Changein Y
Change in Xb = Slope
bXay
Hours studying and Hours studying and gradesgrades
Regressing grades on hours grades on hours
Linear Regression
2.00 4.00 6.00 8.00 10.00
Number of hours spent studying
70.00
80.00
90.00
Final grade in course = 59.95 + 3.17 * studyR-Square = 0.88
Predicted final grade in class =
59.95 + 3.17*(number of hours you study per week)
Predict the final grade ofPredict the final grade of……
Someone who studies for 12 hours Final grade = 59.95 + (3.17*12) Final grade = 97.99
Someone who studies for 1 hour: Final grade = 59.95 + (3.17*1) Final grade = 63.12
Predicted final grade in class = 59.95 + 3.17*(hours of study)
ExerciseExercise
A sample of 6 persons was selected the A sample of 6 persons was selected the value of their age ( x variable) and their value of their age ( x variable) and their weight is demonstrated in the following weight is demonstrated in the following table. Find the regression equation and table. Find the regression equation and what is the predicted weight when age is what is the predicted weight when age is 8.5 years8.5 years..
Serial no.Age (x)Weight (y)123456
768569
128
12101113
AnswerAnswer
Serial no.Age (x)Weight (y)xyX2Y2
123456
768569
128
12101113
8448965066
117
493664253681
14464
144100121169
Total4166461291742
6.83641x 11
666
y
92.0
6)41(
291
666414612
b
Regression equation
6.83)0.9(x11y (x)
0.92x4.675y (x)
12.50Kg8.5*0.924.675y (8.5)
Kg58.117.5*0.924.675y (7.5)
11.411.611.8
1212.212.412.6
7 7.5 8 8.5 9
Age (in years)
Wei
ght (
in K
g)
we create a regression line by plotting two estimated values for y against their X component,
then extending the line right and left.
Exercise 2Exercise 2
The following are the The following are the age (in years) and age (in years) and systolic blood systolic blood pressure of 20 pressure of 20 apparently healthy apparently healthy adults.adults.
Age (x)
B.P (y)
Age (x)
B.P (y)
20436326533158465870
120128141126134128136132140144
46536020634326193123
128136146124143130124121126123
Find the correlation between age Find the correlation between age and blood pressure using simple and blood pressure using simple and Spearman's correlation and Spearman's correlation coefficients, and comment.coefficients, and comment.Find the regression equation?Find the regression equation?What is the predicted blood What is the predicted blood pressure for a man aging 25 years?pressure for a man aging 25 years?
Serialxyxyx21201202400400243128550418493631418883396942612632766765531347102280963112839689617581367888336484613260722116958140812033641070144100804900
Serialxyxyx21146128588821161253136720828091360146876036001420124248040015631439009396916431305590184917261243224676181912122993611931126390696120231232829529
Total852263011448
641678
nx)(
x
nyx
xyb 2
21 4547.0
2085241678
2026308521144862
=
=112.13 + 0.4547 x
for age 25
B.P = 112.13 + 0.4547 * 25=123.49 = 123.5 mm hg
y