Q 94. - The Practice of Statistics for AP

Question

Electoral votes To become president of the United States, a candidate does not have to receive a majority of the popular vote. The candidate does have to win a majority of the $538$ electoral votes that are cast in the Electoral College. Here is a stemplot of the number of electoral votes for each of the $50$ states and the District of Columbia. 

(a) Make a boxplot of these data by hand. Be sure to check for outliers.

(b) Which measure of center and spread would you use to summarize the distribution—the mean and standard deviation or the median and IQR? Justify your answer.

Step-by-Step Solution

Answer

Part (b) It is best to summarize the distribution by using the median and the IQR.

Part (a) A boxplot of the given data is drawn. 

Part (a) Step 1: Given information

Given 

Part (a) Step 2: Concept

A boxplot is a frequent data analysis graph.

Part (a) Step 3: Explanation

Sort the data values: $3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 9, 9, 9,10,10, 10, 10, 11, 11, 11, 11, 12,13, 15, 15, 15, 17,20,21,21,27,31, 34,55$

The median is the middle value of the sorted data collection since the number of data values is odd:

MEDIAN = $8$

The third quartile is the median of all data values above the median. The first quartile is the $13th$ data value since the data set has $25$ values below the median. $Q_1=4$

The median of all data values above the median is the third quartile. Because there are $12$ values above the median in the data set, the $39th$ data value is the third quartile. $Q_3=12$

Outliers are observations that are more than $1.5$ times the IQR above or below $Q_3$, or more than $1.5$ times the IQR below Ql. The difference between the third and first quartiles is the interquartile range.

$$\begin{gathered} Q_3 + 1.5IQR = 12 + 1.5 (12 - 4) = 24 \\ {Q}_1 + 1.5IQR = 4 - 1.5 (12 - 4) = -9 \end{gathered}$$

The minimum is three, while the maximum is fifty-five. The numbers $27, 31, 34, 55$ are then noted as outliers.

Make a boxplot. 

An $X$ is used to represent outliers. The quartiles' bounds have been drawn in the box, and a line has been drawn at the median. The whiskers reflect the minimum and maximum values of the data points (that are not outliers). As a result, a boxplot of the given data is required.

Part (b) Step 1: Explanation

We discovered four outliers in the data in exercise $94a$ Because the median and IQR are unaffected by outliers, they are the best ways to describe the distribution. The mean and standard deviation, on the other hand, are affected by outliers.