Question

Below is a list of gas mileage ratings for selected passenger cars in miles per gallon.

16.2 20.3 31.5 30.5 21.5 31.9 37.3 27.5 27.2 34.1 35.1 29.5 31.8 22.0 17.0 21.6

Find the mean, standard deviation, five - number summary, IQR, and identify any outliers. Use the five - number summary to sketch a boxplot. What does the boxplot tell you about the distribution of the data? (20 points)

Answer 1

Mean=

Create the following table.

data	data-mean	(data - mean)2
16.2	-10.9875	120.72515625
20.3	-6.8875	47.43765625
31.5	4.3125	18.59765625
30.5	3.3125	10.97265625
21.5	-5.6875	32.34765625
31.9	4.7125	22.20765625
37.3	10.1125	102.26265625
27.5	0.3125	0.09765625
27.2	0.0125	0.00015625
34.1	6.9125	47.78265625
35.1	7.9125	62.60765625
29.5	2.3125	5.34765625
31.8	4.6125	21.27515625
22.0	-5.1875	26.91015625
17.0	-10.1875	103.78515625
21.6	-5.5875	31.22015625

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

So

For five number

The minimum is the smallest value in a data set.

Ordering the data from least to greatest, we get:

16.2   17.0   20.3   21.5   21.6   22.0   27.2   27.5   29.5   30.5   31.5   31.8   31.9   34.1   35.1   37.3   

So, the minimum is 16.2.

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.

16.2   17.0   20.3   21.5   21.6   22.0   27.2   27.5   29.5   30.5   31.5   31.8   31.9   34.1   35.1   37.3   

So, the bottom half is

16.2   17.0   20.3   21.5   21.6   22.0   27.2   27.5   

The median of these numbers is 21.55.

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:

16.2   17.0   20.3   21.5   21.6   22.0   27.2   27.5   29.5   30.5   31.5   31.8   31.9   34.1   35.1   37.3   

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=

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.

16.2   17.0   20.3   21.5   21.6   22.0   27.2   27.5   29.5   30.5   31.5   31.8   31.9   34.1   35.1   37.3   

So, the upper half is

29.5   30.5   31.5   31.8   31.9   34.1   35.1   37.3   

The median of these numbers is 31.85.

The maximum is the greatest value in a data set.

Ordering the data from least to greatest, we get:

16.2   17.0   20.3   21.5   21.6   22.0   27.2   27.5   29.5   30.5   31.5   31.8   31.9   34.1   35.1   37.3   

So, the maximum is 37.3.

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

The third quartile is 31.85.

The first quartile is 21.55.

The interquartile range = 31.85 - 21.55 = 10.3.

Now 1.5*IQR=1.5*10.3=15.45

So Q1-1.5*IQR=6.1

1.5*IQR+Q3=47.3

Hence no outliers

As Mean is less than median it is left skewed