Question

Use the accompanying data set to complete the following actions. a. Find the quartiles. b. Find the interquartile range. c. Identify any outliers. 40 52 35 43 42 37 41 48 45 38 34 56 42 35 15 52 38 49 30 29

Answer 1

GIVEN:

The 20 data points 40 52 35 43 42 37 41 48 45 38 34 56 42 35 15 52 38 49 30 29.

(a) QUARTILES:

SECOND QUARTILE (MEDIAN):

The median of a set of data is the middlemost number in the set. The median is also the number that is halfway into the set. To find the median, the data should first be arranged in order from least to greatest.

First, we write data in increasing order: 15, 29, 30, 34, 35, 35, 37, 38, 38, 40, 41, 42, 42, 43, 45, 48, 49, 52, 52, 56

Since there is an even number of items in the data set, we compute the median by taking the mean of the two middlemost numbers.

The two middlemost numbers are 40 and 41.

Thus second quartile or median $=(40+41)/2$

  $=40.5$

FIRST QUARTILE:

The first quartile, denoted by Q₁ , is the median of the lower half of the data set. This means that about 25% of the numbers in the data set lie below Q₁ and about 75% lie above Q₁ .

The lower half of the data is 15, 29, 30, 34, 35, 35, 37, 38, 38, 40.

The first quartile, Q₁, is the median of 15, 29, 30, 34, 35, 35, 37, 38, 38, 40.

Since there is an even number of values, we need the mean of the middle two values to find the first quartile:

$Q_{1}=(35+35)/2$

$=35$

THIRD QUARTILE:

The third quartile, denoted by Q₃ , is the median of the upper half of the data set. This means that about 75% of the numbers in the data set lie below Q₃ and about 25% lie above Q₃ .

The upper half of the data is: 41, 42, 42, 43, 45, 48, 49, 52, 52, 56

The third quartile, Q₃, is the median of 41, 42, 42, 43, 45, 48, 49, 52, 52, 56

Since there is an even number of values, we need the mean of the middle two values to find the third quartile:

$Q_{3}=(45+48)/2$

$=46.5$

(b) INTERQUARTILE RANGE:

The interquartile range is a value that is the difference between the upper quartile (third quartile) value and the lower quartile (first quartile) value.

The first quartile (Q₁) is 35 and the third quartile (Q₃) is 46.5.

$IQR=Q_{3}-Q_{1}$

  $=46.5-35$

  $=11.5$

The interquartile range is 11.5.

(c) FINDING OUTLIERS:

A commonly used rule says that a data point is an outlier if it is more than 1.5⋅IQR above the third quartile or below the first quartile. Said differently, low outliers are below Q1−1.5⋅IQR and high outliers are above Q3+1.5⋅IQR.

$Q_{1}-(1.5*IQR)=$$35-(1.5*11.5)$

$=17.75$

$Q_{3}+(1.5*IQR)=$$46.5+(1.5*11.5)$

  $=63.75$

Thus there is one outlier 15 in the given dataset as it is below Q1−1.5⋅IQR.