Question

How is a sample's variance related to its standard deviation and vice versa?

A) They are independent measures
B) Variance is the square of the standard deviation
C) Standard deviation is the square of the variance
D) They are inversely proportional

Answer 1

The variance of a sample is the square of its standard deviation, and the standard deviation is the square root of the variance, making answer B) correct. These statistics measure the spread of data points.

Step-by-step explanation:

The relationship between a sample's variance and its standard deviation is that the variance is the square of the standard deviation, and the standard deviation is the square root of the variance. This correlation means that variance and standard deviation are measures of spread or dispersion within a set of data points. The variance is calculated as the mean of the squared deviations from the mean. Specifically, for a sample, this means squaring each value's deviation from the sample mean, summing these squares up, and then dividing by the sample size minus one. The standard deviation, denoted as 's' for a sample, is the square root of this variance. It is useful because it is in the same units as the data, which makes it easier to interpret compared to the variance.

To calculate the standard deviation, one must first calculate the variance by averaging the squared differences between each data point and the mean. The formula for sample variance (s²) includes dividing the sum of the squared deviations by one less than the number of data points in the sample (n - 1), which is a method known as Bessel's correction. Once the variance is found, the standard deviation is the square root of the variance, and it is represented as 's' for a sample.

Answer 2

The variance of a sample is the square of its standard deviation, and the standard deviation is the square root of the variance, making answer B) correct. These statistics measure the spread of data points.

Step-by-step explanation:

The relationship between a sample's variance and its standard deviation is that the variance is the square of the standard deviation, and the standard deviation is the square root of the variance. This correlation means that variance and standard deviation are measures of spread or dispersion within a set of data points.

The variance is calculated as the mean of the squared deviations from the mean. Specifically, for a sample, this means squaring each value's deviation from the sample mean, summing these squares up, and then dividing by the sample size minus one. The standard deviation, denoted as 's' for a sample, is the square root of this variance. It is useful because it is in the same units as the data, which makes it easier to interpret compared to the variance.

To calculate the standard deviation, one must first calculate the variance by averaging the squared differences between each data point and the mean. The formula for sample variance (s²) includes dividing the sum of the squared deviations by one less than the number of data points in the sample (n - 1), which is a method known as Bessel's correction. Once the variance is found, the standard deviation is the square root of the variance, and it is represented as 's' for a sample.