Question

1) The mean of 10 numbers is 8. If an 11th number is now included in the results, the mean becomes 9. What is the value of the 11th number? Produce an algebraic solution. Guess and check is unacceptable.

2) The ages of video renters at Kartye’s Quick Video last Saturday were as follows:

21 15 19 29 16 19 23 23 12 39
22 15 19 37 49 46 19 22 15 10

a) Compute the actual value of the mean
b) Identify the type of skewness of the data.

3) get an example that demonstrates why measures of central tendency can be misleading and thus measures of the variability of data are important

Answer 1

Q.1

Mean of k numbers is sum of those k numbers divided by k.

The sum of the first 10 numbers is = 10 * 8 = 80

The sum of the first 11 numbers is = 11 * 9 = 99

Hence, the 11th number is = 99 - 80 = 19

Q.2

(a) Based on the provided data, mean = 470 / 20 = 23.5

(b) Here, the 3rd order central moment is = 27690 / 20 = 1384.5 > 0

Hence, the distribution is leptokurtic.

Q.3

Suppose the data is of such a form that majority of the observations are in either of the tail parts. Then the common measures of central tendency, like mean or median will still be in the midle part of the dataset, which is not a representative value of the data.

Consider the datapoints: 0, 5, 10, 50, 90, 95, 100.

Here, both mean and median will be 50, but it is not a good measure to reflect the U shaped distribution. Here, sample variance is 12250, which is huge, and indicates that the sample data points are widely apart from the sample mean.