Chi- square test

x 2

Chi Square test

• Symbolized by Greek x 2

• pronounced “Ki square”

• A Test of STATISTICAL SIGNIFICANCE for TABLE data

What do tests of statistical

significance tell us?• Are OBSERVED RESULTS

•SIGNIFICANTLY DIFFERENT than would be expected

•BY CHANCE

•Criteria ά < .05

Testing HypothesisChi- square test

• Evaluates whether observed frequencies for a qualitative variable (or variables) are adequately described by hypothesized or expected frequencies.– Qualitative (or categorical) data is a set of

observations where any single observation is a word or code that represents a class or category.

Testing HypothesisChi- square test

•X2 - test for

– Test of deviation from expected frequencies: Test whether the observed frequencies deviate from expected frequencies (e.g. using a dice, there is an a priori chance of 16.67% for each number)

– Test of association: Finding relationship between two or more independent variables (e.g. test relation between gender and the use of high or low accents?)

Chi- square distribution

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Testing HypothesisChi- square test

• In the test of significance of mean we are comparing the mean of one sample with the hypothesized population mean. In the test of significance of difference between two means, we are comparing the means of two samples.

• In chi-square test, we can check the equality of more than two population parameters (like proportions, means).

• If we classify a population into several categories with respect to two attributes (such age & job performance) we can then use a chi-square test to determine whether the two attributes are independent of each other.

Testing HypothesisChi- square test

•Using the Chi- square test:

• Chi square test enables to test for the equality of several proportions.

• Chi square is test of statistics used to test a hypothesis that provides a set of theoretical frequencies with which observed frequencies are computed.

• Its really just a comparison between expected frequencies and observed frequencies among the cells in a crosstabulation table.

Testing HypothesisChi- square test

Conditions for a the application of Chi- square test (x2):

• All raw data for X2 must be frequencies / actual numbers (not percentages & proportions)

• The expected frequency of cell should be more than 5.

• The chi square test work only when the sample size is large enough (n > 50).

• The observation drawn need to be random and independent.

• A constraint must be linear.

Chi- square test•Properties of Chi- square test: • As t-distribution there is different chi- square

distribution for each different number of degree of freedom.

• For very small number of degree of freedom, the x2 distribution is severely skewed to the right. As the number of degree of freedom increases, the curve rapidly becomes more symmetrical until the number reaches large values, at which point the distribution can approximated by the normal.

• The chi-square distribution is a probability distribution.

• The chi-square distribution involves squared observations and hence it is always positive .

Chi-Square• Cannot be negative because all discrepancies are squared.

• Will be zero only in the unusual event that each observed frequency exactly equals the corresponding expected frequency.

• Other things being equal, the larger the discrepancy between the expected frequencies and their corresponding observed frequencies, the larger the observed value of chi-square.

• It is not the size of the discrepancy alone that accounts for a contribution to the value of chi-square, but the size of the discrepancy relative to the magnitude of the expected frequency.

• The value of chi-square depends on the number of discrepancies involved in its calculation.

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Chi- square as a test of Goodness of fit

•Chi- square test developed by Karl Pearson in 1990.

• Chi- square as a test of Goodness of fit, which is used to test whether or not the observed frequency results support a particular hypothesis.

• The test can be used to identify whether the deviation between the observed and estimated values can be because of a chance.

• In some situations researchers would like to see how well the observed frequency pattern will fit in to the expected frequency pattern. In such cases the chi square test is used to test whether the fit between the observed distribution and the expected distribution is good.

Testing Hypothesis Steps in Chi- square

test

An Example:1. Chi- square test

• Ex: Suppose that 60 children were asked as to which ice -cream flavour they liked out of the three flavours of vanilla, strawberry & chocolate. The answer are recorded as follows. (ά = 0.05)

• Flavours Numbers

• Vanilla 17

• Strawberry 24

• Chocolate 19

• x2 = ∑ (Fo – Fe) Fe

An Example:2. Chi- square test

• The following table depicts the expected sales (Fe) and actual sales (Fo) of television sates for company. Test whether there is a substantial difference between the observed and expected values. Using Chi- square test. (ά = 0.05)

(Fo) 57 69 51 83 44 48 35 37

(Fe) 59 76 55 75 39 53 30 48