AP Statistics 1.7: Identifying Outliers

Identify potential outliers in quantitative data using quartiles and the interquartile range, then describe their context.

A company sells a certain type of whistle. The price of the whistle varies from store to store. Julio, a statistician at the company, wants to estimate the mean price, in dollars ($), of this type of whistle at all stores that sell the whistle.

Julio wants to examine some characteristics of the distribution of the sample of whistle prices.

Julio called the managers of 20 randomly selected stores that sell the whistle and recorded the price of the whistle at each store. Following is a dotplot of Julio's data.

The summary statistics for Julio's data are shown in the following table.

Sample Size	Mean	Standard Deviation	Minimum	$\mathrm{Q}_{1}$	Median	$\mathrm{Q}_{3}$	Maximum
20	5.12	0.743	4.25	4.51	4.885	5.475	6.58

Summary Statistics for Julio's Data

Using the $1.5 \times$ IQR rule, determine whether there are any outliers in the sample of whistle prices. Justify your response.

Based on the $1.5 \times$ IQR rule, there are not whistle prices in this sample that would be considered outliers. A whistle price is an outlier using this method if it is more than 1.5 × IQR below the first quartile $\left(\mathrm{Q}_{1}\right)$ or more than $1.5 \times$ IQR above the third quartile $\left(\mathrm{Q}_{3}\right)$ . Because

\begin{aligned} & \mathrm{Q}_{1}-1.5 \times \mathrm{IQR} \\ & =4.51-1.5(5.475-4.51) \\ & =3.0625 \end{aligned}

and the minimum value (4.25) is greater than 3.0625, there are no outliers to the left. Because

\begin{aligned} & \mathrm{Q}_{3}+1.5 \times \mathrm{IQR} \\ & =5.475+1.5(5.475-4.51) \\ & =6.9225 \end{aligned}

and the maximum value (6.58) is less than 6.9225, there are no outliers to the right.

Scoring
Essentially correct (E) if the response satisfies the following four components:

In part (b-i) the response indicates the distribution is skewed to the right or approximately symmetric

In part (b-i) the response justifies the response to component 1 with appropriate reasoning based on the summary statistics

In part (b-ii) the response provides a justification for stating there are no outliers by correctly calculating the lower and upper outlier criteria with work shown

In part (b-ii) the response states whether there are outliers based on the calculated lower and upper outlier criteria

Partially correct (P) if the response satisfies three of the four components required for E.

OR
satisfies components 1 and 2 OR components 3 and 4 of components 1-4 required for E.

Incorrect (I) if the response does not meet the criteria for E or P.

Additional Notes:

- If the response states skewed right, to satisfy component 2 the response should refer to the median being less than the mean. If the response states approximately symmetric, the response should refer to the median and mean being close together.

- A response that indicates skewed left does not satisfy components 1 or 2.

- If the response says the distribution is skewed to the right, examples of responses that satisfy component 2 include:

the mean is higher than the median,

the ratio between the mean and median is greater than 1,

the difference between the maximum and median is greater than the difference between the median and minimum, and

the difference between the maximum and third quartile is greater than the difference between the first quartile and minimum.

- For responses that score P, the number of components satisfied should be considered if holistic scoring is required.

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1.7.D—Calculate outliers for quantitative data question 1