Sample Variance

Definition

Sample variance is a statistical measure that quantifies the degree of dispersion or variability of a set of sample data points around their mean. It is calculated by averaging the squared differences between each data point and the sample mean, giving insight into how spread out the values are in relation to the mean. This measure is critical in point estimation, helping to assess the reliability and precision of estimators used to infer population parameters.

5 Must Know Facts For Your Next Test

1. Sample variance is denoted as 's^2' and is calculated using the formula $$s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2$$, where 'n' is the sample size, 'x_i' represents each data point, and '\bar{x}' is the sample mean.
2. Using 'n-1' in the denominator instead of 'n' is known as Bessel's correction, which corrects the bias in estimating the population variance from a sample.
3. A low sample variance indicates that the data points tend to be close to the mean, while a high sample variance suggests that the data points are more spread out.
4. Sample variance is essential for constructing confidence intervals and conducting hypothesis tests, as it provides information about variability in sample data.
5. In point estimation, understanding sample variance helps determine the reliability of estimators and assess how well they approximate population parameters.

Review Questions

• How does sample variance relate to the accuracy of point estimators?
  • Sample variance directly influences the accuracy of point estimators by providing a measure of how much variability exists within a sample. High sample variance indicates greater dispersion among data points, which can lead to less reliable estimates of population parameters. Conversely, low sample variance suggests that data points are more clustered around the mean, enhancing the precision of estimators used for inference.
• Compare and contrast sample variance with population variance in terms of their calculations and implications for statistical analysis.
  • Sample variance differs from population variance primarily in its calculation method; sample variance uses 'n-1' in its denominator to account for potential bias from sampling. While population variance considers all members of a population, making it a definitive measure, sample variance estimates the population's variability based on a subset. This distinction impacts statistical analysis because conclusions drawn from sample variance must consider potential inaccuracies due to sampling error.
• Evaluate how an increase in sample size affects the reliability of sample variance as an estimator for population variance.
  • As sample size increases, the reliability of sample variance as an estimator for population variance generally improves. Larger samples tend to provide a more accurate representation of the population, reducing sampling error and leading to more stable estimates of variability. This enhanced precision allows researchers to draw more confident conclusions about population characteristics, reinforcing the importance of appropriate sample sizes in statistical studies.

Related terms

Population Variance: The variance calculated from all data points in a population, providing a complete measure of dispersion without sampling error.

Standard Deviation:

The square root of the variance, representing average deviation from the mean and providing a more interpretable measure of variability.

Estimator: A rule or formula used to calculate an estimate of a population parameter based on sample data.