Question 1

Chebyshev's theorem states that the percentage of observations in a data set that should fall within 3 standard deviations of their mean is at least 88.9%.

Accepted Answer

This statement is true according to Chebyshev's theorem, which states that at least 1 - (1/k^2) of the data falls within k standard deviations of the mean, where k is any positive number greater than 1. When k=3, 1 - (1/k^2) = 1 - (1/9) = 8/9 = 0.8889, or 88.9%.

Question 2

While Chebyshev's theorem applies to any distribution, regardless of shape, the empirical rule applies only to distributions that are bell-shaped and symmetrical.

Accepted Answer

Chebyshev's theorem can be applied to any distribution, whereas the empirical rule applies only to bell-shaped and symmetrical distributions.

Question 3

Chebyshev's theorem relates to only to symmetric histograms.

Accepted Answer

Chebyshev's theorem can be applied to any histogram or data set, regardless of whether it is symmetric or not.

Question 4

The width of the box in a boxplot is the mean of the data set.

Accepted Answer

The width of the box in a boxplot represents the interquartile range (IQR), which is the difference between the third quartile (Q3) and the first quartile (Q1). The mean is not represented in the boxplot.

Question 5

A courier company is reviewing their delivery times. The following descriptive statistics relates to this courier company. The mean time for package delivery is 5 hours, the first quartile is 4 hours, the third quartile is 17 hours. This means that the median must lie between 4 hours and 17 hours.

Accepted Answer

This is true because the median, also known as the second quartile, is located between the first and third quartiles. Therefore, the median must lie between 4 hours and 17 hours in this case.

Question 6

The boxplot may be used for ordinal data or numerical data.

Accepted Answer

The boxplot is commonly used to display ordinal and numerical data. It shows the distribution of the data by indicating the median, quartiles, and outliers.

Question 7

When the distribution is skewed, the interquartile range is a better measure of variability than the standard deviation.

Accepted Answer

When a distribution is skewed, the standard deviation can be affected by outliers and extreme values. The interquartile range, which measures the range of the middle 50% of the data, is less affected by these extreme values and provides a more robust measure of variability.

Question 8

The empirical rule states that the percentage of observations in a data set (provided that the data set has a bell-shaped and symmetric distribution) that fall within one standard deviation of their mean is
approximately 75%.

Accepted Answer

The empirical rule actually states that approximately 68% of observations fall within one standard deviation of the mean in a bell-shaped and symmetric distribution.

Question 9

The interquartile range is found by taking the difference between the 1st and 3rd quartiles and dividing that value by 2.

Accepted Answer

The interquartile range is found by simply taking the difference between the 3rd quartile (Q3) and the 1st quartile (Q1), without dividing by 2.

Question 10

A courier company is reviewing their delivery times. The following descriptive statistics relates to this courier company. The mean time for package delivery is 5 hours, the first quartile is 4 hours, the third quartile is 17 hours. This means that 50% of deliveries take more than 5 hours?

Accepted Answer

The statement is incorrect. The first quartile represents the 25th percentile, and the third quartile represents the 75th percentile. This means that 50% of deliveries fall between 4 and 17 hours (between the first and third quartiles), and the remaining 50% fall outside of this range. Therefore, we cannot conclude that 50% of deliveries take more than 5 hours.

Question 11

The left side of the box in a boxplot is the first quartile and the right side of the box in a boxplot is the third quartile, so the width of the box is the interquartile range.

Accepted Answer

This statement is true. The interquartile range (IQR) is the range of the middle 50% of the data, which is represented by the box in a boxplot. The left side of the box represents the first quartile (Q1), which is the value separating the lowest 25% of the data from the rest, and the right side represents the third quartile (Q3), which is the value separating the highest 25% of the data from the rest. Therefore, the width of the box is equal to the IQR, which is the difference between Q3 and Q1.

Question 12

The coefficient of variation is the standard deviation divided by the mean.

Accepted Answer

The formula for coefficient of variation is standard deviation divided by the mean, which means the statement is true.

Question 13

A courier company is reviewing their delivery times. The following descriptive statistics relates to this courier company. The mean time for package delivery is 5 hours, the first quartile is 4 hours, the third quartile is 17 hours. This means that 75% of deliveries take more than 4 hours?

Accepted Answer

The first quartile (Q1) represents the 25th percentile, meaning 25% of deliveries take less than 4 hours, so 75% take more than 4 hours.

Question 14

Chebyshev's theorem applies only to data sets that have a mound-shaped distribution.

Accepted Answer

Chebyshev's theorem applies to all data sets regardless of their distribution shape, but it provides a less precise bound for non-mound-shaped distributions, such as skewed or bimodal ones.

Question 15

There are no measures of variability for nominal data.

Accepted Answer

Variability measures such as standard deviation and range require numerical values and cannot be calculated for nominal data, which only includes categories or labels.

Question 16

The coefficient of variation gives an indication of the magnitude of the standard deviation relative to the mean.

Accepted Answer

The coefficient of variation is defined as the ratio of the standard deviation to the mean, so it provides information on the variability of the data relative to its average. A larger coefficient of variation indicates greater relative variability or dispersion, while a smaller coefficient of variation indicates greater relative consistency or homogeneity of the data. Therefore, the statement is true.

Question 17

A courier company is reviewing their delivery times. The following descriptive statistics relates to this courier company. The mean time for package delivery is 5 hours, the first quartile is 4 hours, the third quartile is 17 hours. This means that the 25% of the deliveries take more than 17 hours.

Accepted Answer

The third quartile is 17 hours, which means that 25% of the deliveries take more than 17 hours.

Question 18

The standard deviation is expressed in terms of the original units of measurement, but the variance is not.

Accepted Answer

The standard deviation is the square root of the variance and is expressed in the original units of measurement, whereas the variance is expressed in the units squared.

Question 19

The mean of fifty sales receipts is $65.75 and the standard deviation is $10.55. Using Chebyshev's theorem, at least 75% of the sales receipts were between $44.65 and $86.85.

Accepted Answer

According to Chebyshev's theorem, at least 75% of the data lies within 2 standard deviations from the mean. In this case, 2 standard deviations from the mean are $65.75 + (2 x $10.55) = $86.85 and $65.75 - (2 x $10.55) = $44.65. Therefore, at least 75% of the sales receipts must have been between $44.65 and $86.85.

Question 20

The value of the standard deviation may be either positive or negative, while the value of the variance will always be positive.

Accepted Answer

The variance is always non-negative (i.e. it can be zero but not negative), while the standard deviation can be positive, zero or undefined (if the variance is undefined).

Quiz 5: Numerical Descriptive Measures