Edexcel IAL - Statistics 3- 4.2 Degrees of Freedom- Study notes - New syllabus
Edexcel IAL – Statistics 3- 4.2 Degrees of Freedom -Study notes- New syllabus
Edexcel IAL – Statistics 3- 4.2 Degrees of Freedom -Study notes -Edexcel A level Maths- per latest Syllabus.
Key Concepts:
- 4.2 Degrees of Freedom
Degrees of Freedom in Chi-squared Tests
In a chi-squared goodness of fit test, the degrees of freedom determine which critical value is used from the chi-squared distribution. Students are expected to calculate the degrees of freedom correctly, especially when one or more parameters are estimated from the data.
Meaning of Degrees of Freedom
The degrees of freedom represent the number of independent pieces of information available after accounting for constraints imposed by the model.
In goodness of fit tests, constraints arise because:
- The total of the expected frequencies must equal the total of the observed frequencies
- Some parameters of the distribution may be estimated from the data
Formula for Degrees of Freedom
For a chi-squared goodness of fit test, the degrees of freedom are given by
\( \mathrm{df = k – 1 – m} \)
where:
\( \mathrm{k} \) = number of categories (after any grouping)
\( \mathrm{m} \) = number of parameters estimated from the data
Effect of Estimating Parameters
Each parameter estimated from the data reduces the degrees of freedom by 1.
Typical cases include:
Discrete uniform distribution: no parameters estimated, so \( \mathrm{m = 0} \)
Poisson distribution: the mean \( \mathrm{\lambda} \) is estimated, so \( \mathrm{m = 1} \)
Normal distribution: the mean and variance are estimated, so \( \mathrm{m = 2} \)
Binomial distribution (with unknown \( \mathrm{p} \)): one parameter estimated, so \( \mathrm{m = 1} \)
Combining Cells
A key condition for the validity of the chi-squared test is that expected frequencies should not be too small.
If any expected frequency satisfies
\( \mathrm{E_i < 5} \)
then adjacent categories should be combined until all expected frequencies are at least 5.
After combining cells:
The number of categories \( \mathrm{k} \) is reduced
The degrees of freedom must be recalculated using the new value of \( \mathrm{k} \)
Yates’ correction is not required in this syllabus.
Key Points to Remember
- Degrees of freedom depend on the number of categories and estimated parameters
- Estimating parameters reduces the degrees of freedom
- Cells must be combined if expected frequencies are less than 5
- Yates’ correction is not used
Example:
Observed data are grouped into 6 categories and are tested for goodness of fit to a Poisson distribution.
The mean of the Poisson distribution is estimated from the data. Two categories are combined because their expected frequencies are less than 5.
Find the degrees of freedom for the test.
▶️ Answer/Explanation
After combining, the number of categories is
\( \mathrm{k = 5} \)
Since the Poisson mean is estimated from the data,
\( \mathrm{m = 1} \)
Degrees of freedom:
\( \mathrm{df = k – 1 – m = 5 – 1 – 1 = 3} \)
Conclusion: The chi-squared test should be carried out with 3 degrees of freedom.
Example :
A goodness of fit test is carried out to check whether data follow a binomial distribution with unknown probability \( \mathrm{p} \).
The observed data are grouped into 7 categories. One category has expected frequency less than 5 and is combined with a neighbouring category.
Find the appropriate degrees of freedom.
▶️ Answer/Explanation
After combining cells, the number of categories becomes
\( \mathrm{k = 6} \)
For a binomial distribution with unknown \( \mathrm{p} \), one parameter is estimated from the data:
\( \mathrm{m = 1} \)
Degrees of freedom:
\( \mathrm{df = k – 1 – m = 6 – 1 – 1 = 4} \)
Conclusion: The chi-squared test should be carried out with 4 degrees of freedom.
Example :
Data are grouped into 8 classes and are tested for goodness of fit to a Normal distribution.
Both the mean and variance of the Normal distribution are estimated from the data. Two adjacent classes are combined because their expected frequencies are less than 5.
Determine the degrees of freedom for the test.
▶️ Answer/Explanation
After combining, the number of categories reduces to
\( \mathrm{k = 6} \)
For a Normal distribution with both parameters estimated:
\( \mathrm{m = 2} \)
Degrees of freedom:
\( \mathrm{df = k – 1 – m = 6 – 1 – 2 = 3} \)
Conclusion: The chi-squared test should be performed with 3 degrees of freedom.