In the world of data science, understanding variability is just as important as understanding central tendencies like the mean. Often, we want to know not only where our data is centered but also how spread out it is. This is where variance comes into play.
One statistical method that helps assess variability is the Test for One Variance. This test is particularly useful when we want to determine whether the variance of a population matches a hypothesized value. In practical terms, it helps answer questions such as: Is the quality control process keeping product variation within acceptable limits? or Does the variance in exam scores differ from what was expected?
What is Variance?
Variance measures the average squared deviation from the mean. In other words, it tells us how much the data points in a dataset differ from the average value.
- A small variance indicates that data points are closely clustered around the mean.
- A large variance indicates that the data points are spread out over a wider range.
The Hypothesis for One Variance Test
This test involves two hypotheses:
- Null hypothesis (H₀): The population variance is equal to a specific value (σ₀²).
- Alternative hypothesis (H₁): The population variance is not equal to (or is greater/less than) that value, depending on whether you are doing a two-tailed or one-tailed test.
Examples:
- Two-tailed: H₀: σ² = σ₀² vs H₁: σ² ≠ σ₀²
- Left-tailed: H₀: σ² = σ₀² vs H₁: σ² < σ₀²
- Right-tailed: H₀: σ² = σ₀² vs H₁: σ² > σ₀²
Test Statistic
The test statistic for a single population variance is based on the chi-square (χ²) distribution and is calculated using the formula:
χ2 = ((n - 1) * s²) / σ₀²Where:
- n = sample size
- s² = sample variance
- σ₀² = hypothesized population variance
Assumptions
- The data should come from a normally distributed population.
- The sample should be randomly selected.
Interpreting the Result
After calculating the test statistic, compare it with the critical value(s) from the chi-square distribution table based on the chosen significance level (α) and degrees of freedom (n – 1). If the test statistic falls in the rejection region, you reject the null hypothesis.
Alternatively, you can compute the p-value and compare it
Leave a Reply