Fifteen randomly selected college students were asked to...

Fifteen randomly selected college students were asked to state the number of hours they slept the previous night. The resulting data are 5,6,6,8,7,7,9,5,4,8,11 $6,7,8,7.$ Find the following: a. Variance $\left.s^{2}, \text { using formula ( } 2.5\right)$ b. $\left. \text { Variance } s^{2}, \text { using formula ( } 2.9\right)$ c. Standard deviation, $s$

Fifteen randomly selected college students were asked to state the number of hours they slept the previous night. The resulting data are 5,6,6,8,7,7,9,5,4,8,11
$6,7,8,7.$ Find the following:
a. Variance $\left.s^{2}, \text { using formula ( } 2.5\right)$
b. $\left. \text { Variance } s^{2}, \text { using formula ( } 2.9\right)$
c. Standard deviation, $s$

Key Concepts

Variance

Variance is a statistical measure that quantifies the average of the squared differences between each data value and the mean. It provides an indication of how much the values in a data set are spread out and is fundamental in assessing variability within a dataset.

Degrees of Freedom

Degrees of freedom refer to the number of independent values in a calculation that are free to vary. In the context of sample variance, the division is often done by (n - 1) instead of n to correct for bias, providing a better, unbiased estimate of the population variance.

Standard Deviation

Standard deviation is the square root of variance and measures the average distance of each data point from the mean. It is used to quantify the amount of variation or dispersion in a set of data and provides a more interpretable metric since it is in the same units as the original data.