Statistics - 3.2 Measures of Dispersion

Chebyshev's Inequality

For any data set or distribution, at least (1 - 1/k^2)=100% of the obserationslie within k standard deviations of the mean, where k is any number greater than 1.

In a statistics​ class, the standard deviation of the heights of all students was 3.9 inches. The standard deviation of the heights of males was 3.4 inches and the standard deviation of females was 3.3 inches. Why is the standard deviation of the entire class more than the standard deviation of the males and females considered​ separately?

The standard deviation of the entire class is more than the standard deviation of the males and females considered separately because the distribution of the entire class has more dispersion.

True or​ False: When comparing two​ populations, the larger the standard​ deviation, the more dispersion the distribution​ has, provided that the variable of interest from the two populations has the same unit of measure.

​True, because the standard deviation describes how​ far, on​ average, each observation is from the typical value. A larger standard deviation means that observations are more distant from the typical​ value, and​ therefore, more dispersed.

True or​ False: Chebyshev's inequality applies to all distributions regardless of​ shape, but the empirical rule holds only for distributions that are bell shaped.

True, Chebyshev's inequality is less precise than the empirical rule but will work for any​ distribution, while the empirical rule only works for​ bell-shaped distributions.

The sum of the deviations about the mean always equals

Zero

Empirical Rule

If data have a distribution that is bell shaped, the Emperical Rule can be used to determine the percentage of data that will lie within k standard deviations of the mean. (pg 139)

The standard deviation is used in conjunction with the​ ______ to numerically describe distributions that are bell shaped. The​ ______ measures the center of the​ distribution, while the standard deviation measures the​ ______ of the distribution.

mean, mean, spread