Question

The following variable (X) represents the number of coupons used over a 6 month period by a sample of 11 shoppers: 80, 63, 70, 69, 70, 54, 61, 70, 65, 76, 60. Use this data to compute: The mean, the median, the mode, the range, the variance, the standard deviation, the 80th percentile, the sum of the values of (X), and the sum of the squared deviations of each value of (X) from the mean. In addition, please explain what information is provided to us about this variable by your answers to: (1) the sample mean and (2) the sample standard deviation.

Answer 1

Mean = (sum of the observations)/(number of observations)

Mean = (80 + 63...)/11 = 67.09

Here sum of x = 80 + 63 + .... = 738

To find the variance, we need to follow the below steps

First subtract mean from each and every observation and then take the square and then finally add them

= (80-67.09)^2 + (63 - 67.09)^2....

= 554.9091 (this is sum of the squared deviations of each value of x from mean)

Now we need to divide this sum by n-1 , 11-1, 10

Variance = 554.9091/10 = 55.49091

We know that s.d is √variance

So, s.d = √55.49091 = 7.4492

To find median we need to arrange the data in ascending order then the middlemost value of the arranged data is our median

54, 60, 61, 63,65, 69, 70, 70, 70, 76, 80

We have 11 data points

So, median is the middlemost number which is 6th data point

= 69

Mode is the number which occurs most of the tjmes

= 70

80th percentile

Now 80% of 11 is 8.8 ~ 9

So, 9th data point is 80th percentile = 76

Sample mean = 67.09

S.d = 7.4492