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Quantitative Methods. Part 3 Chi - Squared Statistic. Recap on T-Statistic. It used the mean and standard error of a population sample The data is on an “interval” or scale Mean and standard error are the parameters This approach is known as parametric

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Quantitative methods

Quantitative Methods

Part 3

Chi - Squared Statistic

Recap on t statistic
Recap on T-Statistic

  • It used the mean and standard error of a population sample

  • The data is on an “interval” or scale

  • Mean and standard error are the parameters

  • This approach is known as parametric

  • Another approach is non-parametric testing

Introduction to chi squared
Introduction to Chi-Squared

  • It does not use the mean and standard error of a population sample

  • Each respondent can only choose one category (unlike scale in T-Statistic)

  • The expected frequency must be greater than 5 for the test to succeed.

  • If any of the categories have less than 5 for the expected frequency, then you need to increase your sample size

Example using chi squared
Example using Chi-Squared

  • “Is there a preference amongst the UW student population for a particular web browser? “ (Dr C Price’s Data)

    • They could only indicate one choice

    • These are the observed frequencies responses from the sample

Was it just chance
Was it just chance?

  • How confident am I?

    • Was the sample representative of all UW students?

    • Was it just chance?

  • Chi-Squared test for significance

    • Some variations on test

    • Simplest is Null Hypothesis

  • :The students show “no preference” for a particular browser

Chi squared goodness of fit no preference
Chi-Squared: “Goodness of fit” (No preference)

: The students show no preference for a particular browser

  • This leads to Hypothetical or Expected distribution of frequency

    • We would expect an equal number of respondents per category

    • We had 50 respondents and 5 categories

Expected frequency table

Stage1 formulation of hypothesis
Stage1: Formulation of Hypothesis

  • : There is no preference in the underlying population for the factor suggested.

  • : There is a preference in the underlying population for the factors suggested.

  • The basis of the chi-squared test is to compare the observed frequencies against the expected frequencies

Stage 2 expected distribution
Stage 2: Expected Distribution

  • As our “null- hypothesis” is no preference, we need to work out the expected frequency:

    • You would expect each category to have the same amount of respondents

    • Show this in “Expected frequency” table

    • Has to have more than 5 to be valid

Stage 3a level of confidence
Stage 3a: Level of confidence

  • Choose the level of confidence (often 0.05)

    • 0.05 means that there is 5% chance that conclusion is chance

    • 95% chance that our conclusions are certain

Stage 3b: Degree of freedom

  • We need to find the degree of freedom

  • This is calculated with the number of categories

    • We had 5 categories, df = 5-1 (4)

Stage 3 critical value of chi squared
Stage 3: Critical value of Chi-Squared

  • In order to compare our calculated chi-square value with the “critical value” in the chi-squared table we need:

    • Level of confidence (0.05)

    • Degree of freedom (4)

  • Our critical value from the table = 9.49

Stage 4 calculate statistics
Stage 4: Calculate statistics

  • We compare the observed against the expected for each category

  • We square each one

  • We add all of them up

= 52

Stage 5 decision
Stage 5: Decision

  • Can we reject the That students show no preference for a particular browser?

    • Our value of 52 is way beyond 9.49. We are 95% confident the value did not occur by chance

  • So yes we can safely reject the null hypothesis

  • Which browser do they prefer?

    • Firefox as it is way above expected frequency of 10

Chi squared no difference from a comparison population
Chi-Squared: “No Difference from a Comparison Population”.

  • RQ: Are drivers of high performance cars more likely to be involved in accidents?

    • Sample n = 50 and Market Research data of proportion of people driving these categories

    • Once null hypothesis of expected frequency has been done, the analysis is the same as no preference calculation

Chi squared test for independence
Chi-Squared test for “Independence”.

  • What makes computer games fun?

  • Review found the following

    • Factors (Mastery, Challenge and Fantasy)

    • Different opinion depending on gender

  • Research sample of 50 males and 50 females

Observed frequency table

What is the research question
What is the research question?

  • A single sample with individuals measured on 2 variables

    • RQ: ”Is there a relationship between fun factor and gender?”

    • HO : “There is no such relationship”

  • Two separate samples representing 2 populations (male and female)

    • RQ: ““Do male and female players have different preferences for fun factors?”

    • HO : “Male and female players do not have different preferences”

Chi squared analysis for independence
Chi-Squared analysis for “Independence”.

  • Establish the null hypothesis (previous slide)

  • Determine the critical value of chi-squared dependent on the confidence limit (0.05) and the degrees of freedom.

    • df = (R – 1)*(C – 1) = 1 * 2 = 2 (R=2, C=3)

  • Look up in chi-squared table

    • Chi-squared value = 5.99

Chi squared analysis for independence1
Chi-Squared analysis for “Independence”.

  • Calculate the expected frequencies

    • Add each column and divide by types (in this case 2)

    • Easier if you have equal number for each gender (if not come and see me)

Chi squared analysis for independence2
Chi-Squared analysis for “Independence”.

  • Calculate the statistics using the chi-squared formula

    • Ensure you include both male and female data

Stage 5 decision1
Stage 5: Decision

  • Can we reject the null hypothesis?

    • Our value of 24.01 is way beyond 5.99. We are 95% confident the value did not occur by chance

  • Conclusion: We are 95% confident that there is a relationship between gender and fun factor

  • But else can we get from this?

    • Significant fun factor for males = Challenge

    • Significant fun factor for females = Mastery and Fantasy


  • Work on Workshop 7 activities

  • Your journal (Homework)

  • Your Literature Review (Complete/update)


  • Dr C. Price’s notes 2010

  • Gravetter, F. and Wallnau, L. (2003) Statistics for the Behavioral Sciences, New York: West Publishing Company