Question

Let a population consist of the values 10 ​cigarettes, 19 ​cigarettes, and 20 cigarettes smoked in a day. Show that when samples of size 2 are randomly selected with​ replacement, the samples have mean absolute deviations that do not center about the value of the mean absolute deviation of the population. What does this indicate about a sample mean absolute deviation being used as an estimator of the mean absolute deviation of a​ population?

Answer 1

The mean is the sum of all values divided by the number of values :

n is the number of values in the dataset:n = 3

The population mean absolute deviation is the sum of all absolute deviations of the mean, divided by the number of data values n

Here are all possible samples having two values from the list 19,10,20

The means of all sample mean absolute deviations is 2.2222 which is not the same as the population MAD of 6.67 and thus the sample mean absolute deviation is not an unbiased estimator of the population mean absolute deviation