Wize University Statistics Textbook > Inference about a Population Variance
Inference for Population Variance & Chi-square Distribution
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Inference for Population Variance Using the Chi-square Distribution
Variance (and standard deviation) measures spread of data. The larger the variation, the bigger the spread. A large variance could entail high risk or great uncertainty, depending on context.
Examples of large variance:
- The annual salaries in this company ranges from $15,000 to $3,200,000.
- This headphone could last between 2 months to 6 years.
- Your commute to school could be anywhere from 15 minutes to 1 hour.
Because variances and standard deviations cannot be negative, they do not follow a normal distribution, which has a positive side and a negative side. Instead, they follow a Chi-Square distribution, which goes from .
Properties of a Chi-Square Distribution:
- Not symmetric (skewed to the right)
- Ranges from 0 (no negative values)
- Total area under the curve = 100%
We use the Chi-square distribution to make inferences about the population variance .
Inference of Population Variance Formula
We established that variances and standard deviations cannot be negative and they do not follow a normal distribution. Instead, they follow a Chi- Square distribution.
If the data in the population is normally distributed (or we can assume it is), then the following expression has a Chi-square distribution with degrees of freedom
where sample size, sample variance, and population variance.
Watch Out!
Students tend to get confused and ask "Why is the data normally distributed but we are using the Chi-square Distribution?"
That is not what we're doing! The data in the population is normally distributed but, to make inferences about the variance of the data in the population, we are using the Chi-square distribution. If we're making inferences about the population mean, then we use the t-distribution. If we're making inferences about the population variance, then we use the Chi-square distribution.