Sample variance is a measure of the spread or dispersion of a set of data points around the sample mean. It represents the average squared deviation from the mean, providing insight into the variability within a sample. This metric is crucial in understanding the characteristics of a sample and making inferences about the corresponding population.
congrats on reading the definition of Sample Variance. now let's actually learn it.
Sample variance is used to estimate the population variance when the population mean and standard deviation are unknown.
The formula for sample variance is $\frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n-1}$, where $x_i$ are the individual data points, $\bar{x}$ is the sample mean, and $n$ is the sample size.
Sample variance is biased, meaning it tends to underestimate the true population variance, which is why the denominator is $n-1$ instead of $n$.
The sample variance is a key input for calculating the test statistic in hypothesis tests involving two population means with unknown standard deviations, such as the two-sample t-test.
In the context of testing the equality of two variances, the sample variances are used to construct the test statistic, which follows an F-distribution under the null hypothesis.
Review Questions
Explain how sample variance is used to estimate population variance when the population parameters are unknown.
When the population mean and standard deviation are unknown, the sample variance can be used to estimate the population variance. The sample variance is calculated by taking the average of the squared deviations from the sample mean, divided by the number of observations minus 1. This provides an unbiased estimate of the true population variance, which is crucial when making inferences about the population characteristics based on a sample.
Describe the role of sample variance in hypothesis testing for two population means with unknown standard deviations.
In the context of comparing two population means with unknown standard deviations, such as the two-sample t-test, the sample variances play a crucial role. The test statistic is calculated using the sample means, sample sizes, and the pooled sample variance, which is a weighted average of the individual sample variances. The pooled sample variance accounts for the variability within each sample and is used to determine the appropriate t-distribution for the hypothesis test, allowing researchers to make inferences about the difference between the two population means.
Analyze how sample variance is used in the test of two variances and explain the underlying statistical distribution.
When testing the equality of two population variances, the sample variances are used to construct the test statistic. The ratio of the two sample variances follows an F-distribution under the null hypothesis that the population variances are equal. This test, known as the F-test, allows researchers to determine whether the observed difference in sample variances is statistically significant and can be attributed to a true difference in the population variances, rather than just sampling variability. The F-distribution, with its degrees of freedom parameters derived from the sample sizes, provides the appropriate reference distribution for this hypothesis test.
The number of values in the final calculation of a statistic that are free to vary, which is important in determining the appropriate statistical distribution for hypothesis testing.