How do you find the expected value in chi-square?

When conducting a chi-square test, it is crucial to calculate the expected values to determine whether the observed frequencies significantly deviate from what would be expected if the null hypothesis were true. The expected values represent the frequencies or counts that would be anticipated under the assumption of independence between variables. Let’s explore the process of finding the expected values in chi-square.

Calculation of Expected Values

The expected value for each cell in a chi-square table can be calculated using the following formula:

Expected value = (Sum of row total) x (Sum of column total) / (Grand total)

To better understand this, let’s consider an example. Suppose we are analyzing a dataset with categorical variables: A and B. Both variables have two categories, A1 and A2 for variable A and B1 and B2 for variable B. We have collected data and obtained the following observed frequencies:

To calculate the expected values, we need to determine the row and column totals as well as the grand total:

– Row Total for A1 = 30 + 45 = 75
– Row Total for A2 = 25 + 50 = 75
– Column Total for B1 = 30 + 25 = 55
– Column Total for B2 = 45 + 50 = 95
– Grand Total = 150

Now, using the formula mentioned above, we can calculate the expected values:

Expected value for A1 and B1 = (75 * 55) / 150 = 27.5

Similarly, we can calculate the expected values for the remaining cells:

FAQs

1. How is the expected value different from the observed value in chi-square?

The observed values represent the actual frequencies obtained from the data, while the expected values are the hypothetical frequencies assuming independence.

2. Can the expected value be zero in chi-square?

No, the expected value cannot be zero. If any expected value is zero, it poses a problem for the chi-square test and indicates a lack of independence between variables.

3. Are the expected values always integers?

No, the expected values can be decimal numbers. The calculated value depends on the number of decimal places retained in the calculations.

4. How does the chi-square test make use of expected values?

The chi-square test compares the observed frequencies to the expected frequencies to determine whether there is a significant association between variables.

5. What does it mean if the observed frequency is much higher than the expected frequency?

If an observed frequency is substantially greater than the expected frequency, it suggests a significant association between the variables being examined.

6. Is it possible to have a negative expected value?

No, the expected value cannot be negative. A negative expected value would not make logical sense in the context of frequencies.

7. Are the expected values affected by the sample size?

Yes, the expected values are influenced by the sample size as they are calculated based on the row and column totals.

8. What happens if the observed frequency and expected frequency are the same?

If the observed frequency is equal to the expected frequency, it indicates that there is no association between the variables being studied.

9. Can the expected values differ significantly from the observed values?

In some cases, the expected values may differ significantly from the observed values. This discrepancy may point to a significant association between the variables.

10. Is it possible to calculate the expected values by hand for larger datasets?

Calculating expected values by hand for larger datasets can be time-consuming and prone to errors. It is more practical to use statistical software to perform these calculations.

11. What if the sample size is small and expected values are too low?

In cases where the sample size is small and the expected values are too low, it can be difficult to draw meaningful conclusions from the chi-square analysis.

12. Are the expected values affected if the variables have more than two categories?

Yes, the number of categories in the variables influence the expected values. In such cases, the calculations become more complex and require additional steps to find the expected frequencies for each cell.

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