Chi-Square characteristics, test, computational procedure, formula, ANOVA, and their difference.

1. CHI-SQUARE AND
ANALYSIS OF VARIANCE
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2. Content Layout
› Chi-Square
› Characteristics of Chi-Square
› Chi-square Test
› Computational Procedure – Chi Square Test
› Chi Square – Test Of Independence Formulae To Be Used
› Contingency Table
› Analysis Of Variance (ANOVA)
› Assumptions
› Steps in Analysis Of Variance (ANOVA)
› Computational Procedure In ANOVA (One Way)
› Difference Between Chi Square & ANOVA
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3. Chi-square (𝜒2)
› The Chi Square statistic is commonly used for testing
relationships between categorical variables. The null
hypothesis of the Chi-Square test is that no relationship
exists on the categorical variables in the population; they
are independent.
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4. CHARACTERISTICS OF CHI SQUARE
› Every Chi square distribution extends indefinitely to right
from zero.
› It is skewed to right
› As degree of freedom increases, Chi square curve
become more bell shaped and approaches normal
distribution.
› Its mean is degree of freedom
› Its variance is twice degree of freedom
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5. CHI SQUARE (Χ2 TEST)
› Chi Square Test deals with analysis of categorical data in terms of
frequencies / proportions / percentages.
› It is primarily of three types:
• Test of Homogeneity:
To determine whether different population are similar w.r.t some
characteristics.
• Test of Independence:
Tests whether the characteristics of the elements of the same
population are related or independent.
• Test of Goodness of Fit:
To determine whether there is a significant difference between an
observed frequency distribution and theoretical probability distribution.
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6. COMPUTATIONAL PROCEDURE – CHI SQUARE TEST
› Formulate Null & Alternative Hypothesis
› State type of test
› Select LOS
› Compute expected frequencies assuming H0 to be true.
› Compute χ2 calculated value using
› 𝜒2 cal= ∑
(݂‫݋‬ −݂݁)
2
݂݁
› Extract 𝜒2 crit value from table
› Compare 𝜒2 cal & 𝜒2 crit and make decision
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7. CHI SQUARE – TEST OF INDEPENDENCE
FORMULAE TO BE USED
› Computation of expected frequency
› Fe = (RT x CT) / GT
• where RT = Row Total, CT = Column Total, GT = Grand Total
› Computation of degree of freedom
› Degree of freedom= (r – 1) (c – 1)
• r=No. of rows, c=No. of column
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8. CONTIGENCY TABLE
› A table having R rows and C columns. Each row
corresponds to a level of one variable, each column to a
level of another variable. Entries in the body of the table
are the frequencies with which each variable combination
occurred.
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9. ANALYSIS OF VARIANCE (ANOVA)
› Analysis of variance (ANOVA) is a collection of statistical
models and their associated estimation procedures (such
as the "variation" among and between groups) used to
analyze the differences among group means in a
sample. ANOVA was developed by statistician and
evolutionary biologist Ronald Fisher.
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10. ANALYSIS OF VARIANCE (ANOVA)
› It enables us to test for the significance of the differences
among more than two sample means.
› Using ANOVA, we will be able to make inferences about
whether our samples are drawn from population having the
same mean.
› Examples:
• Comparing the mileage of five different brands of cars
• Testing which of the four different training methods produces the fastest learning
record
• Comparing the average salary of three different companies
› In each of these cases, we would compare the means of more
than two sample means.
› F-Distribution is used to analyze certain situations
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11. ASSUMPTIONS
› Populations are normally distributed
› Samples are random and independent
› Population Variances are equal.
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12. STEPS IN ANALYSIS OF VARIANCE
› Determine one estimate of the population variance from
the variance among the sample means.
› Determine second estimate of the population variance
from the variance within the sample means.
› Compare these two estimates. If they are approximately
equal in value, accept the null hypotheses.
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13. COMPUTATIONAL PROCEDURE IN ANOVA (ONE WAY)
› Define Null & Alternative Hypothesis
› Select Significance Level
› Calculate Sum of all observations: T = Ʃx𝑖
› Calculate correction factor: CF = T2 / nT where nT = sample size
› Calculate Sum of squares total, SST = Σ(Σ𝑥𝑖2) − CF
› Calculate Sum of squares between columns, SSB = Σ((Σ𝑥𝑖)2/𝑛𝑖) − CF
› Calculate Sum of squares within columns, SSW = SST -SSB
› Calculate 𝑓𝑐𝑎𝑙= 𝑠2𝑏 ∕ 𝑠2w OR 𝑓𝑐𝑎𝑙= 𝜎2𝑏 ∕ 𝜎2w
where s=variance
› Calculate Mean of squares between groups, MSB = SSB / (k – 1) where k = no.
of samples
› Calculate Mean of squares within groups, MSW = SSW / (nT – k)
› Calculate Fcal = MSB / MSW
› Calculate Fcrit = F(dfnum, dfden, α) where dfnum = k – 1, dfden = nT – k
› Compare Fcal & Fcrit and make your statistical & managerial decisions
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14. DIFFERENCE BETWEEN CHI SQUARE & ANOVA
CHI SQUARE (Χ2 TEST)
› It enables us to test whether more than
two population proportions can be
considered equal
ANOVA (F TEST)
› Analysis of Variance (ANOVA) enables us
to test whether more than two population
means can be considered equal.
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15. THANK YOU!!!
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