When to use a Chi-Square test:

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1 When to use a Chi-Square test: Usually in psychological research, we aim to obtain one or more scores from each participant. However, sometimes data consist merely of the frequencies with which certain categories of event occurred. Instead of each participant providing a score, participants contribute to a head count : they fall into one category or another. In these circumstances, where one has categorical data, one of the family of Chi-Square tests is the appropriate test to use. The Chi-Square test allows us to compare observed frequencies with what we might expect to find by chance (expected frequencies). Chi-Square Goodness of Fit test: This test is used when you want to compare an observed frequency distribution to a theoretical frequency distribution. The most common situation is where your data consist of the observed frequencies for a number of mutually exclusive categories, and you want to know if they all occur equally frequently (so that our theoretical frequency-distribution contains categories that all have the same number of individuals in them). Worked Example: We select a random sample of 100 A Level students, and ask them their favourite A- Level. We find that they are distributed across five subjects (fictional data!). Favourite A Level Subject Total Observed Frequency Are all A-Levels equally popular, as far as students are concerned? Or do some A- Levels attract more students than others? Looking at the observed frequencies, it appears that Psychology and Maths are much more popular than the other A-Levels. We can perform a Chi-Square test on these data to find out if this is true, i.e. to see the pattern of preference deviates significantly from what we might expect to obtain if all A-Levels were equally popular with students. Patricia George / Coursework / Statistics / Chi-Square Test / Page 1 of 5

2 Formula for Chi-Square: χ 2 ( 0 ) = E 2 E Step 1: Calculate the expected frequencies (the frequencies which we would expect to obtain if there were no preferences): total number of instances Expected Frequency = number of categories We have 100 applications and 5 categories, so the Expected Frequency for each category is 20 (i.e. 100/5) Step 2: Subtract, from each observed frequency, its associated expected frequency (e.g. work out (O-E): O - E Step 3: Square each value of (O-E): (O E) Step 4: Divide each of the values obtained in step 3, by its associated expected frequency: (O E) 2 E Patricia George / Coursework / Statistics / Chi-Square Test / Page 2 of 5

3 Step 5: Add together all the values obtained in step 4, to get your value of Chi- Square: Χ 2 = = This is our obtained value of Chi-Square: It is a single number summary of the discrepancy between our obtained frequencies, and the frequencies, which we would expect if all the categories had equal frequencies. The bigger our Chi-Square, the greater the difference between the observed and the expected frequencies. How do we assess whether this value represents a real departure of our obtained frequencies from the expected frequencies? To do this we need to know how likely it is to obtain various values of Chi-Square by chance. Patricia George / Coursework / Statistics / Chi-Square Test / Page 3 of 5

4 Assessing the size of our obtained Chi-Square value: In a nutshell, you a) Find a table of critical values (at the back of most statistics books) b) Work out how many degrees of freedom (d.f.) you have. For this test it is simply the number of categories minus one. c) Decide on the probability level. (p<0.05) d) Find the critical value in the table (at the intersection of the appropriate d.f. row and probability column). If your obtained Chi-Square value is bigger than the one in the table, then you conclude that your obtained Chid-Square value is too large to have arisen by chance: it is more likely to seem from the fact that there were real differences between the observed and expected frequencies. If on the other hand, your obtained Chi-Square value is smaller than the one in the table, you conclude that there is no reason to think that the observed pattern of frequencies is not due simply to chance (i.e. we retain our initial assumption that the discrepancies between the observed and expected frequencies are due merely to random sampling variations, and hence we have no reason to believe that the categories did not occur with equal frequency. For our worked example (a) We have an obtained Chi-Square value of 52.5 (b) We have five categories, so there are 5-1 = 4 degrees of freedom. (c) Consult a table of critical values of Chi-Square. Here is a typical example: Probability level: d.f etc etc (d) The values in each column are the critical values of Chi-Square. These values would be expected to occur by chance with the probability shown at the top of the column. We are interested in the 4.d.f. row, since our obtained Chi-Square has 4 d.f. Patricia George / Coursework / Statistics / Chi-Square Test / Page 4 of 5

5 With 4 d.f. obtained values of Chi-Square as large as 9.49 or more will occur by chance with a probability of 0.05: in other words, you would expect to get a Chi-Square value of 9.49 or more, only 5 times in a hundred Chi-Squared tests. Thus, values of Chi-Square that are this large do occur by chance, but not very often. Looking at the same row, we find that Chi-Square values of are even more uncommon, they occur by chance once in a hundred times (0.01). (e) We compare our obtained Chi-Square to these tabulated valued. If our obtained Chi-Square is larger than a value in the table, it implies that our obtained value is even less likely to occur by chance than is the value in the table. Our obtained value of 52.5 is much bigger than the critical value having occurred by chance are less than one in a thousand (written formally, p < We can therefore be relatively confident to conclude that our observed frequencies are significantly different from the frequencies that we would expect to obtain if all categories were equally popular. In other words, not all A-Levels are equally popular with A-Level students. Patricia George / Coursework / Statistics / Chi-Square Test / Page 5 of 5

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