Sample variance

Definition

Sample variance is a measure of how much individual data points in a sample differ from the sample mean. It provides an understanding of the dispersion or spread of data, which is crucial when assessing the reliability and variability of statistical estimates derived from the sample.

5 Must Know Facts For Your Next Test

1. Sample variance is calculated using the formula $$s^2 = \frac{\sum_{i=1}^{n}(x_i - \bar{x})^2}{n - 1}$$, where $$s^2$$ is the sample variance, $$x_i$$ are individual sample points, $$\bar{x}$$ is the sample mean, and $$n$$ is the number of observations in the sample.
2. The use of $$n - 1$$ instead of $$n$$ in the formula corrects for bias in the estimation of population variance from a sample, known as Bessel's correction.
3. A larger sample variance indicates a wider spread of data points around the mean, while a smaller variance suggests that the data points are closely clustered around the mean.
4. Sample variance is sensitive to outliers; even one extreme value can significantly affect the computed variance.
5. In statistical inference, sample variance plays a vital role in hypothesis testing and constructing confidence intervals, helping assess how well a sample represents its population.

Review Questions

• How does sample variance relate to population variance, and why is it important to differentiate between them?
  • Sample variance and population variance both measure dispersion within data sets but apply to different contexts. Sample variance is calculated from a subset (sample) of data to estimate how much variation exists compared to the entire population. Understanding this difference is important because estimates derived from samples can be biased; thus, using sample variance helps gauge that bias when inferring properties about a larger population.
• What are the consequences of using $n$ instead of $n - 1$ when calculating sample variance?
  • Using $n$ instead of $n - 1$ when calculating sample variance leads to an underestimation of the actual population variance. This occurs because dividing by $n$ does not account for the degrees of freedom lost by estimating the mean from the same data set. As a result, relying on this incorrect method can yield misleading conclusions in statistical analysis and hypothesis testing.
• Evaluate how sample variance impacts statistical inference processes like confidence intervals and hypothesis testing.
  • Sample variance significantly influences statistical inference by providing essential information about variability in data. In constructing confidence intervals, a larger sample variance results in wider intervals, indicating less certainty about where the true population parameter lies. Similarly, in hypothesis testing, sample variance affects test statistics that determine whether to reject or fail to reject a null hypothesis. Hence, accurate calculation of sample variance is crucial for reliable conclusions drawn from statistical analyses.

Related terms

Population variance: Population variance measures the dispersion of a full population’s data points from the population mean, serving as a benchmark for comparing sample variance.

Standard deviation: Standard deviation is the square root of variance, offering a measure of spread in the same units as the original data, making it easier to interpret.

Mean: The mean is the average value of a set of numbers, which serves as a central point to calculate both variance and standard deviation.