σ²

Definition

σ² (sigma squared) is the statistical term for the variance, which is a measure of the spread or dispersion of a dataset around its mean. It represents the average squared deviation from the mean and is a fundamental concept in probability and statistics.

5 Must Know Facts For Your Next Test

1. The variance, σ², is a key parameter in statistical analysis as it provides information about the spread of a dataset.
2. Variance is used in the calculation of other important statistical measures, such as standard deviation and z-scores.
3. When testing a single variance, the null hypothesis is that the population variance is equal to a specified value.
4. The test statistic for a single variance test is the sample variance divided by the hypothesized population variance, which follows a chi-square distribution.
5. When testing two variances, the null hypothesis is that the population variances are equal, and the test statistic follows an F-distribution.

Review Questions

• Explain the role of σ² in the context of a test of a single variance.
  • In the test of a single variance (Section 11.2), the null hypothesis is that the population variance, σ², is equal to a specified value. The test statistic is calculated as the sample variance divided by the hypothesized population variance, and this test statistic follows a chi-square distribution. The purpose of this test is to determine if the sample variance provides sufficient evidence to reject the null hypothesis and conclude that the population variance is different from the hypothesized value.
• Describe how σ² is used in the test of two variances (Section 12.1).
  • When testing two variances, the null hypothesis is that the two population variances, σ²₁ and σ²₂, are equal. The test statistic is the ratio of the two sample variances, which follows an F-distribution. The purpose of this test is to determine if there is sufficient evidence to conclude that the two population variances are different. The variance, σ², is a key parameter in this analysis, as the test statistic is directly related to the ratio of the two population variances.
• Analyze the relationship between σ² and the other measures of spread, such as standard deviation, in the context of statistical hypothesis testing.
  • The variance, σ², is directly related to the standard deviation, σ, as the standard deviation is the square root of the variance. Both measures of spread are important in statistical hypothesis testing, as they provide information about the dispersion of the data. The variance, σ², is the primary parameter used in tests of single and two variances, as the test statistics are directly based on the ratio of the sample variance to the hypothesized or compared population variance. Understanding the relationship between σ² and other measures of spread, such as standard deviation, is crucial for interpreting the results of these statistical tests.

Related terms

Variance: Variance is a measure of how spread out a set of numbers is from the mean. It is calculated as the average of the squared differences from the mean.

Standard Deviation: Standard deviation is the square root of the variance and represents the average distance of data points from the mean.

Chi-Square Distribution: The chi-square distribution is a probability distribution used in hypothesis testing, particularly in the context of testing variances and standard deviations.