Descriptive statistics - Unit quiz

When a set of data has suspect outliers, which of the following are preferred measures of central tendency and of variability?

• mean and standard deviation

• mean and variance

• mean and range

• median and range

• median and interquartile range

Answer explanation

The mean, standard deviation, variance, and range are all affected by outliers; the median and interquartile range are not.

If the standard deviation of a set of observations is 0, you can conclude

• that there is no relationship between the observations.

• that the average value is 0.

• that all observations are the same value.

• that a mistake in arithmetic has been made.

• none of the above.

Answer explanation

Because of the squaring operation in the definition, the standard deviation (and also the variance) can be zero only if all the values in the set are equal.

Data on the number of text messages sent one weekend by junior and senior high school students in a large high school is summarized as follows:

An AP Statistics student checking the calculations finds that the message counts for all the students were underreported by 5. When the numbers are corrected, what are the corrected IQR and standard deviation?

• IQR = 71 and standard deviation = 41.2

• IQR = 71 and standard deviation = 46.2

• IQR = 76 and standard deviation = 41.2

• IQR = 76 and standard deviation = 46.2

• IQR = 146 and standard deviation = 41.2

Answer explanation

Adding the same constant to every value will increase the Min, Q1, Median, Q3, Max, and Mean by that constant. However, measures of variability, including the IQR and the standard deviation, will remain unchanged.

Suppose the starting salaries of a graduating class are as follows:

What is the mean starting salary?

• $30,000

• $30,533

• $32,500

• $32,533

• $35,000

Answer explanation

There are a total of 10 + 17 + 25 + 38 + 27 + 21 + 12 = 150 students. Their total salary is 10(15,000) + 17(20,000) + 25(25,000) + 38(30,000) + 27(35,000) + 21(40,000) + 12(45,000) = $4,580,000. The mean is 4,580,000/150 = $30,533

For which of the following variables would it be most appropriate to construct a histogram?

• Television brand

• Insect species

• Gender

• Eye color

• Phone call length

Answer explanation

Histograms refer to numerical (quantitative) data, not categorical (qualitative) data.

Given that the median of a set is 350 and the interquartile range is 50, which of the following statements is true?

• Fifty percent of the data are less than or equal to 350.

• Fifty percent of the data are between 325 and 375.

• Seventy-five percent of the data are less than or equal to 375.

• The mean is between 325 and 375.

• All of the above statements are true.

Answer explanation

Fifty percent of the data are on either side of the median. The interquartile range gives the distance between Q1 and Q3 but doesn’t say where the median falls between Q1 and Q3. For example, it is possible that Q1 = 345 and Q3 = 395. Depending on the shape of the distribution, the mean could be anything.

• 0

• 1

• 2

• At least 1

• At least 2

Answer explanation

The interquartile range is IQR = Q3 – Q1 = 139 – 126 = 13. Outliers are any values less than Q1 – 1.5(IQR) = 106.5 or greater than Q3 + 1.5(IQR) = 158.5. Look at the minimum and maximum values. We see that there are no outliers on the lower end and at least one outlier on the upper end.

On a large campus, 500 students were surveyed about their belief in an afterlife. The following table shows their responses categorized by class.

Of the following graphical displays, which is the most appropriate to compare the proportions of students in each class who believe in an afterlife?

• Back-to-back stem plot

• Parallel boxplots

• Scatterplot

• Segmented bar chart

• Cumulative frequency plot

Answer explanation

With this categorical data, the only reasonable display is a segmented bar chart.

Counts of the numbers of spam messages received by a company’s executives during one week are summarized below.

How many outliers are there?

• 0

• 1

• 2

• 3

• 4

Answer explanation

Outliers are values less than Q1 – 1.5(IQR) and greater than Q3 + 1.5(IQR). First calculate the IQR = 21.5 – 15.5 = 6. The outliers are values below 15.5 – 1.5(6) = 15.5 – 9 = 6.5 and greater than 21.5 + 1.5(6) = 21.5 + 9 = 30.5.

In which of the following histograms is the mean greater than the median?

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• 
• 
• 
• 

Answer explanation

In general, when histograms are skewed to the right (to the higher values), the mean is greater than the median. However, be careful as this is not a hard and fast rule.