Making Sense of ChiSquared Test – Finding Differences in Proportions
 03
 Apr
 2018
 Posted ByUK
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Every blood donor of a large blood bank has to go through five process steps. These steps are Registration, Screening, HB Test, Donation and Refreshment. At the end of the process, that often takes around an hour, feedback forms are available for the donors. In one week, 210 donors have returned these forms with their satisfaction score for each process step. This satisfaction score is measured using a sixpoint scale. The desired rating is either 5 (satisfied) or 6 (very satisfied). The results are summarised in Figure 1.
From the results obtained, there is an obvious majority of “Not 5 or 6” ratings for Step 5. The question is, is there a significant difference to the other steps’ satisfaction score or are we observing just random variation. How can we test this?
One way is to use the socalled ChiSquared test (also written Χ^{2} test, another method is Fisher’s Exact test). The ChiSquared test is a statistical tool to check whether there is a significant difference between observed frequencies (discrete data) and expected frequencies for two or more groups.
To perform the ChiSquared test, the following steps are necessary:
1. Plot the Data
For any statistical application, it is essential to combine it with a graphical representation of the data. The selection of tools for this purpose is limited. They include pie chart, column chart and bar chart.
The column chart in Figure 2 reveals, that the customer satisfaction for Step 5 seems drastically lower (more “Not 5 or 6” ratings) compared to the other steps. Although, there does not seem to be any doubt about a worse rating for Step 5, it is a good practice to confirm the finding with a statistical test. Additionally, a statistical test can help to calculate the risk for this decision.
2. Formulate the Hypothesis for ANOVA
In this case, the parameter of interest is an average, i.e. the nullhypothesis is
H_{0}: P_{1} = P_{2} = P_{3} = P_{4} = P_{5}, with all P being the population proportions of the five process steps in this blood donation process.
This means, the alternative hypothesis is
H_{A}: At least one P is different to at least one other P.
3. Decide on the Acceptable Risk
Since there is no reason for changing the commonly used acceptable risk of 5%, i.e. 0.05, we use this risk as our threshold for making our decision.
4. Select the Right Tool
If there is a need for comparing two or more proportions, the popular test for this situation is the ChiSquared test.
5. Test the Assumptions
Finally, there is only one prerequisite for the ChiSquared test to work properly: All expected values need to be at least 5. After the test, we will know whether this assumption holds true.
6. Conduct the ChiSquared Test
Using the ChiSquared test, statistics software SigmaXL generates the output in Figure 3.
Since the pvalue is 0.0000, i.e. less than 0.05 (or 5 percent), we reject H_{0} and accept H_{A}. The risk for this decision is zero.
From the test result at Figure 3 we can conclude, that all expected counts are larger than five. This means, the test result is valid.
7. Make a Decision
With this, the ChiSquared test statistics means that there is at least one significant difference.
In order to interpret the ChiSquared test results, three steps are necessary:
 Check the pvalue. As mentioned earlier, the pvalue = 0. This means there is a difference.
 Lock for the highest standardised residual (Std. Residual). The highest standardised residual is 15.703 at step 5. This means, step 5 is significantly different at level “Not 5o6”.
 Check the nature of the deviation. This means, compare Observed Counts and Expected Counts and conclude what the deviation means. In our case, Step 5 is supposed to produce 57.71 “Not 5o6” satisfaction ratings if it was following the average of all steps. Instead, Step 5 delivers 177 of these ratings. Ergo, Step 5 has received statistically worse ratings from the donors.
With this result, it is quite obvious, that interventions are needed to increase the customer satisfaction with Step 5.
Interested in the stats? Read here.

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