Question

The COVID-19 pandemic has forced people to work from home. The following are age data from a sample of people who work from home: 18 54 20 46 25 48 53 27 26 37 40 36 42 25 27 33 28 40 45 25
a. Calculate the average, median, quartile 1 and quartile 3
b. Calculate range, interquartile range, variance, standard deviation, and coefficient of variation.
c. Draw a box plot, give a conclusion

Note: no handwritten answer would be very appreciated. thanks.

Answer 1

a.Average is

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:

18   20   25   25   25   26   27   27   28   33   36   37   40   40   42   45   46   48   53   54   

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:

Median is

The first quartile (or lower quartile or 25th percentile) is the median of the bottom half of the numbers. So, to find the first quartile, we need to place the numbers in value order and find the bottom half.

18   20   25   25   25   26   27   27   28   33   36   37   40   40   42   45   46   48   53   54   

So, the bottom half is

18   20   25   25   25   26   27   27   28   33   

The median of these numbers is 25.5.

The third quartile (or upper quartile or 75th percentile) is the median of the upper half of the numbers. So, to find the third quartile, we need to place the numbers in value order and find the upper half.

18   20   25   25   25   26   27   27   28   33   36   37   40   40   42   45   46   48   53   54   

So, the upper half is

36   37   40   40   42   45   46   48   53   54   

The median of these numbers is 43.5.

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

Ordering the data from least to greatest, we get:

18   20   25   25   25   26   27   27   28   33   36   37   40   40   42   45   46   48   53   54   

The lowest value is 18.

The highest value is 54.

The range = 54 - 18 = 36.

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

The third quartile is 43.5.

The first quartile is 25.5.

The interquartile range = 43.5 - 25.5 = 18.

Create the following table.

data	data-mean	(data - mean)2
18	-16.75	280.5625
54	19.25	370.5625
20	-14.75	217.5625
46	11.25	126.5625
25	-9.75	95.0625
48	13.25	175.5625
53	18.25	333.0625
27	-7.75	60.0625
26	-8.75	76.5625
37	2.25	5.0625
40	5.25	27.5625
36	1.25	1.5625
42	7.25	52.5625
25	-9.75	95.0625
27	-7.75	60.0625
33	-1.75	3.0625
28	-6.75	45.5625
40	5.25	27.5625
45	10.25	105.0625
25	-9.75	95.0625

Find the sum of numbers in the last column to get.

So variance is

So standard deviation is

Coefficient of variation is

c. Box plot is

Here

The minimum is the smallest value in a data set.

Ordering the data from least to greatest, we get:

18   20   25   25   25   26   27   27   28   33   36   37   40   40   42   45   46   48   53   54   

So, the minimum is 18.

The maximum is the greatest value in a data set.

Ordering the data from least to greatest, we get:

18   20   25   25   25   26   27   27   28   33   36   37   40   40   42   45   46   48   53   54   

So, the maximum is 54.

And so from box plot we see that is no outliers.