Question

Consider the following information related to a set of quantitative sample data:

Standard deviation of the sample data is 14.92, and the mean is 22.0 0th quartile is -24, 1st quartile of the data is 14.5, 2nd quartile is 24.5, 3rd quartile is 33.5, and 4th quartile is 64.

a. Is there any outlier(s) in the data?
b. If there is an outlier(s), what is the value(s) of the outlier?

Answer 1

(a) There are outliers

(b)
`x <-14` and
`x >62`

Explanation:

Given

`\sigma = 14.92`

`\bar x = 22.0`

`q_0 = -24`

`q_1 = 14.5`

`q_2 = 24.5`

`q_3 = 33.5`

`q_4 = 64`

Solving (a): Check for outliers

This is calculated using:

`Lower = Q_1 - (1.5 * IQR)` --- lower bound of outlier

`Upper = Q_3 +(1.5 * IQR)` --- upper bound of outlier

Where

`IQR = Q_3 - Q_1`

So, we have:

`IQR = 33.5 - 14.5`

`IQR = 19`

The lower bound of outlier becomes

`Lower = Q_1 - (1.5 * IQR)`

`Lower = 14.5 - (1.5 * 19)`

`Lower = 14.5 - 28.5`

`Lower = -14`

The upper bound of outlier becomes

`Upper = Q_3 +(1.5 * IQR)`

`Upper = 33.5 + 1.5 * 19`

`Upper = 33.5 + 28.5`

`Upper = 62`

So, we have:

`-14 \le x \le 62` --- the range without outlier

Given that:

`q_0 = -24` --- This represents the lowest data

`q_4 = 64` --- This represents the highest data

-24 and 64 are out of range of
`-14 \le x \le 62`.

Hence, there are outliers

Solving (b): The outliers

The outliers are data less than the lower bound (i.e. less than -14) or greater than the upper bound (i.e. 62)

So, the outliers are:

`x <-14` and
`x >62`