Question

identifying an outlier. the data set below represents the number of televisions repaired in a service shop over an 11 - week period: 1, 48, 50, 25, 21, 19, 26, 30, 18, 17, 3. the outlier of the data set is ___.

Explanation:

Step1: Sort the data set

$3, 17, 18, 19, 21, 25, 26, 30, 48, 50$

Step2: Calculate the first quartile ($Q_1$)

The lower half of the data is $3, 17, 18, 19, 21$. The median of this lower - half (i.e., $Q_1$) is $18$.

Step3: Calculate the third quartile ($Q_3$)

The upper half of the data is $26, 30, 48, 50$. The median of this upper - half (i.e., $Q_3$) is $39$.

Step4: Calculate the inter - quartile range (IQR)

$IQR = Q_3 - Q_1=39 - 18 = 21$

Step5: Determine the lower and upper bounds for non - outliers

Lower bound $=Q_1-1.5\times IQR=18 - 1.5\times21=18 - 31.5=- 13.5$
Upper bound $=Q_3 + 1.5\times IQR=39+1.5\times21=39 + 31.5 = 70.5$

Step6: Identify the outlier

All the values $3, 17, 18, 19, 21, 25, 26, 30, 48, 50$ are within the bounds $-13.5$ and $70.5$. There is no outlier in the traditional sense using the $1.5\times IQR$ rule. But if we consider values that are far from the general cluster, $48$ and $50$ are relatively large compared to most of the other values in the set. Among them, $50$ is the most extreme value.

Answer:

$50$