Question

2.17 income at the coffee shop. the first histogram below shows the distribution of the yearly incomes of 40 patrons at a college coffee shop. suppose two new people walk into the coffee shop: one making $225,000 and the other $250,000. the second histogram shows the new income distribution. summary statistics are also provided.
(1)
min. 1st qu. median mean 3rd qu. max. sd
60,680 63,620 65,240 65,090 66,160 69,890 2.122
(2)
60,680 63,710 65,350 73,300 66,540 250,000 37,321
(1) would the mean or the median best represent what we might think of as a typical income for the 42 patrons at this coffee shop? what does this say about the robustness of the two measures?
(2) would the standard deviation or the iqr best represent the amount of variability in the incomes of the 42 patrons at this coffee shop? what does this say about the robustness of the two measures?

Explanation:

Step1: Recall measure - of - central - tendency concepts

The mean is the sum of all values divided by the number of values. The median is the middle value when the data is ordered. The mean is affected by outliers, while the median is not.

Step2: Analyze the data for the first coffee - shop (n = 40)

The first coffee - shop has no extremely large outliers (from the histogram and summary statistics). The mean ($65,090$) and median ($65,240$) are close. So, the mean can be a good representation of a typical income as the data is relatively symmetric.

Step3: Analyze the data for the second coffee - shop (n = 42)

The second coffee - shop has a large outlier ($250,000$). The mean ($73,300$) is pulled up by this outlier, while the median ($65,350$) is not affected. So, the median is a better representation of a typical income for the second coffee - shop.

Step4: Recall measure - of - variability concepts

The standard deviation (SD) is the square root of the variance and is affected by outliers. The inter - quartile range (IQR) is the difference between the third and first quartiles and is not affected by outliers.

Step5: Analyze variability for the first coffee - shop

The first coffee - shop has a relatively small SD ($2,122$) and a certain IQR ($66,160 - 63,620=2,540$). Since there are no large outliers, the SD can represent the variability well.

Step6: Analyze variability for the second coffee - shop

The second coffee - shop has a large SD ($37,321$) mainly due to the outlier. The IQR ($66,540 - 63,710 = 2,830$) is a more robust measure of variability as it is not affected by the outlier.

(a) For the first coffee - shop, the mean is a good representation of a typical income as the data is symmetric. For the second coffee - shop, the median is a better representation of a typical income because of the outlier.
(b) For the first coffee - shop, the standard deviation can represent the variability well as there are no large outliers. For the second coffee - shop, the inter - quartile range is a more robust measure of variability due to the presence of the outlier.

Answer:

(a) For the first coffee - shop, the mean is a good representation of a typical income. For the second coffee - shop, the median is a better representation of a typical income.
(b) For the first coffee - shop, the standard deviation is a good measure of variability. For the second coffee - shop, the inter - quartile range is a more robust measure of variability.