Question

The following data set is the ages of 15 random professors.

46 51 60 58 37 65 43 55 30 68 25 62 56 42 59

a) Find the mean

b) Find the mode

c) Give the five number summary

d) Are they any outliers? If so, what are they?

Answer 1

a) Since we know that

Where n is the number of data points
Now

and n = 15
This implies that


b) Mode is the number which appears most often in a set of numbers.
Mode = 65.0

c) Since we know that
The lower quartile(Q1) is the median of the lower half of the data set while upper quartile(Q3) is the median of the upper half of the data set.
Also, median for a list of even number of data point is the mean of 2 middle most values if we sort the list in increasing order while for a list of odd number it is the middle most value if the list is sorted in increasing order.
Since our list have odd number of data points, this implies that
Median = 55.0
Lower half of our list is [25.0, 30.0, 37.0, 42.0, 43.0, 46.0, 51.0]
Since our lower half list have even number of data points, this implies that
Q1 = 42.0
Upper half of our list is [56.0, 58.0, 59.0, 60.0, 62.0, 65.0, 68.0]
Since our upper half list have even number of data points, this implies that
Q3 = 60.0

The maximum and the minimum values are as follows
Maximum value = 68.0
Minimum value = 25.0

Interquartile range is equal to the distance between quartile 1(Q1) and quartile 3(Q3).
IQR = Q3 - Q2
IQR = 60.0-42.0
IQR = 18.0

FIVE POINT SUMMARY
Min = 25
Q1 = 42
Median = 55
Q3 = 60
Max = 68

d)
Since we know that the outliers are those points in the data set which are either less than 1.5 times the IQR from Quartile 1 or more than 1.5 times the IQR from Quartile 3
There are NO outliers