Question

1. The following set of data is from a sample of n = 8

           171   169   158   164   158   135   150 141

a) Find the mean, median, and mode.

b) Calculate the variance, standard deviation, range, inter-quartile range, and coefficient of variation.

c) Describe the shape of the data.

d) List the five-number summary and build a boxplot.

Answer 1

a)

Mean = (171 + 169 + 158 + 164 + 158 + 135 + 150 + 141)/8
= 1246/8
Mean = 155.75

The median is the middle number in a sorted list of numbers. So, to find the median, we need to place the numbers in value order and find the middle number.

Ordering the data from least to greatest, we get:

135   141   150   158   158   164   169   171   

As you can see, we do not have just one middle number but we have a pair of middle numbers, so the median is the average of these two numbers:

Medain = 158 +158 /2 =158

The mode of a set of data is the value in the set that occurs most often.

Ordering the data from least to greatest, we get:

135   141   150   158   158   164   169   171   

We see that the mode is 158 .

b)

Variance = 166.835

Standard Deviation σ = √(1/8 - 1) x ((171 - 155.75)2 + (169 - 155.75)2 + (158 - 155.75)2 + (164 - 155.75)2 + (158 - 155.75)2 + (135 - 155.75)2 + (150 - 155.75)2 + (141 - 155.75)2)
= √(1/7) x ((15.25)2 + (13.25)2 + (2.25)2 + (8.25)2 + (2.25)2 + (-20.75)2 + (-5.75)2 + (-14.75)2)
= √(0.1429) x ((232.5625) + (175.5625) + (5.0625) + (68.0625) + (5.0625) + (430.5625) + (33.0625) + (217.5625))
= √(0.1429) x (1167.5)
= √(166.83575)
= 12.9146

The range is the difference between the highest and lowest values in the data set.

Ordering the data from least to greatest, we get:

135   141   150   158   158   164   169   171   

The lowest value is 135.

The highest value is 171.

The range = 171 - 135 = 36.

The interquartile range is the difference between the third and first quartiles.

The third quartile is 166.5.

The first quartile is 145.5.

The interquartile range = 166.5 - 145.5 = 21.

Cofficient of Varaiance =σ/μ
=12.9146/155.75
Coefficient of Variance = 0.0829

c)

The shape of the distribution is Bimodal

d)

The 5 number summary of the data values:

Min: 135
1st quartile: 145.5
Median: 158
3rd quartile: 166.5
Max: 171
Interquartile range: 21