A shopkeeper noted down the number of drinks sold each day. After 10 days, the following values were obtained, arranged in ascending order: 35, r, 223, 257, 285, 292, 300, 344, 350, 374 where r is an unknown number. Based on the definition of an outlier for a boxplot, if 35 is the only outlier in this data set, what is the minimum possible value of r?
Question
A shopkeeper noted down the number of drinks sold each day. After 10 days, the following values were obtained, arranged in ascending order: 35, r, 223, 257, 285, 292, 300, 344, 350, 374 where r is an unknown number. Based on the definition of an outlier for a boxplot, if 35 is the only outlier in this data set, what is the minimum possible value of r?
Solution
To find the minimum possible value of r, we need to understand the definition of an outlier in the context of a boxplot. An outlier is a data point that falls outside of the "whiskers" of the boxplot. These whiskers are determined by the interquartile range (IQR), which is the range between the first quartile (Q1) and the third quartile (Q3).
The lower whisker extends to the smallest data point greater than Q1 - 1.5IQR, and the upper whisker extends to the largest data point less than Q3 + 1.5IQR. Any data point outside these ranges is considered an outlier.
Given that 35 is the only outlier in the data set, it must be the case that 35 < Q1 - 1.5*IQR.
To find Q1, we need to find the median of the lower half of the data. Since we have 10 data points, the lower half is the first 5 data points. But since we don't know r, we can't find the exact value of Q1. However, we know that r must be greater than 35 (since 35 is an outlier and r is not), and less than 223 (the third smallest value).
So, without knowing the exact value of Q1, we can't find the exact minimum possible value of r. But we know that it must be greater than 35 and less than 223.
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