Maths › Further Statistics 2 › Testing variances: chi-squared and the F-distribution
Testing variances: chi-squared and the F-distribution
Spread can be the thing in question rather than the nuisance. One distribution tests a single variance against a claimed value; another tests two variances against each other.
Builds on Goodness-of-fit tests and Estimators, standard error and confidence intervals.
IN THIS TOPIC
- Test a claimed variance using the chi-squared statistic and the right degrees of freedom.
- Build a confidence interval for a variance from the two chi-squared tails.
- Test two variances for equality with an F ratio.
WHAT YOU PROBABLY THINK
A chi-squared test for a variance uses a symmetric critical region, half the level in each tail.
One variance against a claim
For a sample from a normal population, the sample variance scaled by the claimed population variance has a known distribution:
The chi-squared distribution is skewed to the right and lives on the positive axis, so its two tails are not mirror images. A two-tailed test uses two different critical values, read from opposite ends of the table, and the opening claim quietly assumes a symmetry that is not there.
The same statistic inverted gives a confidence interval for σ²: divide (n − 1)s² by the upper critical value for the lower limit, and by the lower critical value for the upper limit. The larger divisor produces the smaller limit, which is why the two ends look swapped.
WORKED EXAMPLE
Testing a claimed variance
A sample of 20 gives s² = 12.5. Test at 5% whether the population variance exceeds 8.
H0: σ² = 8; H1: σ² > 8. One-tailed at 5% with 19 degrees of freedom, so the critical value is 30.144.
Statistic = 19 × 12.5/8 = 29.69.
Since 29.69 < 30.144 the result is outside the critical region: do not reject H0. There is insufficient evidence at the 5% level that the variance exceeds 8, though it is a close call.
Two variances against each other
To compare the spreads of two normal populations, take the ratio of the sample variances:
Put the larger sample variance on top, so the ratio is at least 1 and only the upper tail of the F table is needed. The degrees of freedom go in the same order as the variances, numerator first, and swapping them changes the critical value. A test at the 5% level with the larger variance on top is one-tailed by construction, so for a two-tailed question use the 2.5% column instead.
YOUR TURN
Do the spreads differ?
A sample of 10 gives s² = 18.4; an independent sample of 8 gives s² = 6.5. Test at 5% in one tail whether the first population has the larger variance, given a critical value of 3.677.
Show the working
H0: σ1² = σ2²; H1: σ1² > σ2².
F = 18.4/6.5 = 2.83, on 9 and 7 degrees of freedom.
Since 2.83 < 3.677 the result is not in the critical region.
Do not reject H0: the samples give insufficient evidence at the 5% level that the first population is more variable, despite the sample variances differing by a factor of nearly three.
THE EXAM BIT
- Use n − 1 degrees of freedom for a single variance, and both n − 1 values for an F test.
- For a two-tailed chi-squared test, read two different critical values; the distribution is not symmetric.
- For an F test, put the larger sample variance on top and keep the degrees of freedom in that order.
- State that both populations are assumed normal; these tests are sensitive to that assumption.
CHECK YOURSELF
A sample of 16 from a normal population gives s² = 20. State the test statistic for H₀: σ² = 15 and its degrees of freedom.
Show a hint
Scale the sample variance by the claimed one.
Show the answer
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(n − 1)S²/σ² is chi-squared on n − 1 degrees of freedom, and the distribution is skewed, so a two-tailed test needs two different critical values.
The ratio of two sample variances is F on the two degrees of freedom, larger variance on top, numerator degrees of freedom first.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Test a claimed variance using the chi-squared statistic and the right degrees of freedom.
- Build a confidence interval for a variance from the two chi-squared tails.
- Test two variances for equality with an F ratio.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.