Population Variance

Definition

Population variance is a measure of the spread or dispersion of a population around its mean. It quantifies the average squared deviation of each data point from the population mean, providing a way to understand the variability within a given population.

5 Must Know Facts For Your Next Test

1. Population variance is denoted by the symbol $\sigma^2$, where $\sigma$ represents the population standard deviation.
2. Population variance is calculated as the sum of the squared deviations of each data point from the population mean, divided by the total number of data points in the population.
3. Population variance is a key parameter in hypothesis testing, as it is used to determine the sampling distribution of the sample mean and other statistics.
4. The population variance is an important measure in the context of descriptive statistics, as it provides information about the spread and dispersion of the data within the population.
5. When testing a single population variance or comparing two population variances, the population variance is a crucial statistic used in the test statistic calculations.

Review Questions

• Explain the relationship between population variance and population standard deviation.
  • Population variance and population standard deviation are closely related concepts. Population variance, denoted as $\sigma^2$, is the average squared deviation of each data point from the population mean. Population standard deviation, denoted as $\sigma$, is the square root of the population variance. In other words, the population standard deviation is the positive square root of the population variance, and the two measures provide complementary information about the spread and dispersion of the data within the population.
• Describe how population variance is used in the context of descriptive statistics.
  • In the context of descriptive statistics, population variance is an important measure that provides information about the spread and dispersion of data within a population. Population variance quantifies the average squared deviation of each data point from the population mean, giving a sense of how much the individual data points vary from the central tendency. This information is crucial for understanding the characteristics of the population and can be used to make inferences about the distribution of the data, identify outliers, and compare the variability of different populations.
• Explain the role of population variance in hypothesis testing, specifically in the context of testing a single population variance and comparing two population variances.
  • Population variance plays a crucial role in hypothesis testing, particularly in the context of testing a single population variance and comparing two population variances. When testing a single population variance, the population variance is used to calculate the test statistic, which is then compared to a critical value to determine if the observed variance is significantly different from the hypothesized variance. Similarly, when comparing two population variances, the population variances are used to calculate the test statistic, which is used to determine if the variances of the two populations are significantly different. The population variance is a key parameter in these statistical tests, as it directly influences the sampling distribution of the test statistic and, consequently, the conclusions drawn about the population characteristics.

Related terms

Population Mean: The average or central value of a population, calculated by summing all the values and dividing by the total number of data points.

Standard Deviation: The square root of the population variance, representing the average distance of data points from the population mean.

Sample Variance: The variance calculated from a sample of a population, used to estimate the population variance when the full population data is not available.