Calculate the expected frequencies

Oct 13, 2025 by windailys.com 104 views

Now you flipped your 2 coins again, but you did it 100 times. You collected the data below. Use the chart to calculate the Chi-square value

Introduction

In statistics, the Chi-square test is a widely used method for determining whether there is a significant association between two categorical variables. The test is based on the Chi-square statistic, which is calculated from the observed frequencies of the variables and their expected frequencies under the null hypothesis. In this article, we will use a real-life example to demonstrate how to calculate the Chi-square value from a given dataset.

The Data

The following table shows the observed frequencies of the outcomes of flipping two coins 100 times:

Outcome	Observed (Obs)	Expected (Exp)	Obs - Exp	(Obs - Exp)^2
HH	22	25	-3	9
HT	30	25	5	25
TH	26	25	1	1
TT	22	25	-3	9

Calculating the Chi-square Value

To calculate the Chi-square value, we need to follow these steps:

Calculate the expected frequencies: The expected frequencies are calculated by multiplying the total number of observations by the probability of each outcome. In this case, the probability of each outcome is 0.25 (1/4), since there are four possible outcomes (HH, HT, TH, TT) and each outcome is equally likely.
Calculate the observed minus expected frequencies: We subtract the expected frequencies from the observed frequencies to get the observed minus expected frequencies.
Calculate the squared observed minus expected frequencies: We square the observed minus expected frequencies to get the squared observed minus expected frequencies.
Calculate the Chi-square value: We sum up the squared observed minus expected frequencies to get the Chi-square value.

Calculating the Chi-square Value (continued)

Let's calculate the Chi-square value using the data from the table:

Outcome	Observed (Obs)	Expected (Exp)	Obs - Exp	(Obs - Exp)^2
HH	22	25	-3	9
HT	30	25	5	25
TH	26	25	1	1
TT	22	25	-3	9

The Chi-square value is calculated as follows:

Chi-square = (9 + 25 + 1 + 9) = 44

Interpreting the Chi-square Value

The Chi-square value is a measure of the difference between the observed frequencies and the expected frequencies. A large Chi-square value indicates a significant difference between the observed frequencies and the expected frequencies.

In this case, the Chi-square value is 44, which is a relatively large value. This suggests that there is a significant difference between the observed frequencies and the expected frequencies.

Conclusion

In this article, we used a real-life example to demonstrate how to calculate the Chi-square value from a given dataset. We calculated the expected frequencies, observed minus expected frequencies, and squared observed minus expected frequencies, and then calculated the Chi-square value. We also interpreted the Chi-square value and concluded that there is a significant difference between the observed frequencies and the expected frequencies.

References

[1] Agresti, A. (2013). An Introduction to Categorical Data Analysis. Wiley.
[2] Everitt, B. S. (2002). The Cambridge Dictionary of Statistics. Cambridge University Press.

Discussion

The Chi-square test is a widely used method for determining whether there is a significant association between two categorical variables. The test is based on the Chi-square statistic, which is calculated from the observed frequencies of the variables and their expected frequencies under the null hypothesis.

In conclusion, the Chi-square test is a useful tool for determining whether there is a significant association between two categorical variables. By calculating the Chi-square value from a given dataset, we can determine whether there is a significant difference between the observed frequencies and the expected frequencies.

Future Work

In future work, we can use the Chi-square test to determine whether there is a significant association between other categorical variables. We can also use the test to determine whether there is a significant difference between the observed frequencies and the expected frequencies in other datasets.

Limitations

The Chi-square test has some limitations. The test assumes that the data are independent and identically distributed, which may not always be the case. Additionally, the test assumes that the expected frequencies are greater than or equal to 5, which may not always be the case.

Conclusion

Final Thoughts

References

[1] Agresti, A. (2013). An Introduction to Categorical Data Analysis. Wiley.
[2] Everitt, B. S. (2002). The Cambridge Dictionary of Statistics. Cambridge University Press.
Q&A: Chi-square Test

Introduction

The Chi-square test is a widely used statistical method for determining whether there is a significant association between two categorical variables. In our previous article, we demonstrated how to calculate the Chi-square value from a given dataset. In this article, we will answer some frequently asked questions about the Chi-square test.

Q1: What is the Chi-square test?

A1: The Chi-square test is a statistical method for determining whether there is a significant association between two categorical variables. It is based on the Chi-square statistic, which is calculated from the observed frequencies of the variables and their expected frequencies under the null hypothesis.

Q2: What are the assumptions of the Chi-square test?

A2: The Chi-square test assumes that the data are independent and identically distributed, and that the expected frequencies are greater than or equal to 5.

Q3: How do I calculate the Chi-square value?

A3: To calculate the Chi-square value, you need to follow these steps:

Calculate the expected frequencies by multiplying the total number of observations by the probability of each outcome.
Calculate the observed minus expected frequencies by subtracting the expected frequencies from the observed frequencies.
Calculate the squared observed minus expected frequencies by squaring the observed minus expected frequencies.
Sum up the squared observed minus expected frequencies to get the Chi-square value.

Q4: What does the Chi-square value represent?

A4: The Chi-square value represents the difference between the observed frequencies and the expected frequencies. A large Chi-square value indicates a significant difference between the observed frequencies and the expected frequencies.

Q5: How do I interpret the Chi-square value?

A5: To interpret the Chi-square value, you need to compare it to the critical value from a Chi-square distribution table. If the calculated Chi-square value is greater than the critical value, you reject the null hypothesis and conclude that there is a significant association between the variables.

Q6: What are the limitations of the Chi-square test?

A6: The Chi-square test has some limitations. It assumes that the data are independent and identically distributed, and that the expected frequencies are greater than or equal to 5. Additionally, the test may not be suitable for small sample sizes.

Q7: Can I use the Chi-square test for continuous data?

A7: No, the Chi-square test is only suitable for categorical data. If you have continuous data, you may need to use a different statistical method, such as the t-test or the ANOVA.

Q8: How do I choose the correct Chi-square test?

A8: There are several types of Chi-square tests, including the Pearson Chi-square test, the likelihood ratio Chi-square test, and the Mantel-Haenszel Chi-square test. You need to choose the correct test based on the research question and the type of data you have.

Q9: What are the advantages of the Chi-square test?

A9: The Chi-square test has several advantages, including its simplicity, flexibility, and ability to handle large datasets. It is also a widely used and well-established statistical method.

Q10: What are the disadvantages of the Chi-square test?

A10: The Chi-square test has several disadvantages, including its assumption of independence and identical distribution, and its sensitivity to outliers. Additionally, the test may not be suitable for small sample sizes.

Conclusion

In this article, we answered some frequently asked questions about the Chi-square test. We discussed the assumptions, calculations, and interpretation of the Chi-square value, as well as the limitations and advantages of the test. We hope that this article has provided you with a better understanding of the Chi-square test and its applications.

References

[1] Agresti, A. (2013). An Introduction to Categorical Data Analysis. Wiley.
[2] Everitt, B. S. (2002). The Cambridge Dictionary of Statistics. Cambridge University Press.

Discussion

The Chi-square test is a widely used statistical method for determining whether there is a significant association between two categorical variables. In this article, we answered some frequently asked questions about the Chi-square test, including its assumptions, calculations, and interpretation.

The Chi-square test has several advantages, including its simplicity, flexibility, and ability to handle large datasets. However, it also has several disadvantages, including its assumption of independence and identical distribution, and its sensitivity to outliers.

In conclusion, the Chi-square test is a useful tool for determining whether there is a significant association between two categorical variables. By understanding the assumptions, calculations, and interpretation of the Chi-square value, you can use the test to make informed decisions in your research.

Future Work

Limitations

The Chi-square test has some limitations. It assumes that the data are independent and identically distributed, and that the expected frequencies are greater than or equal to 5. Additionally, the test may not be suitable for small sample sizes.

Conclusion

Final Thoughts

References

[1] Agresti, A. (2013). An Introduction to Categorical Data Analysis. Wiley.
[2] Everitt, B. S. (2002). The Cambridge Dictionary of Statistics. Cambridge University Press.