Question 1

The smaller the spread of scores around the mean:

Accepted Answer

All of these statements are true because the spread of scores around the mean directly affects the variance, standard deviation, and coefficient of variation of a data set. A smaller spread indicates that the data points are closer to the mean, leading to a smaller variance and standard deviation. Since the coefficient of variation is a standardized measure of dispersion that takes into account the mean of the dataset, a smaller spread around the mean also results in a smaller coefficient of variation.

Question 2

A population with 200 elements has a variance of 20.From this information,it can be shown that the population standard deviation is 10.

Accepted Answer

The standard deviation is the square root of the variance, so the correct standard deviation for a variance of 20 is the square root of 20, which is approximately 4.47, not 10.

Question 3

If two data sets have the same standard deviation,they must have the same coefficient of variation.

Accepted Answer

The coefficient of variation is calculated as the ratio of the standard deviation to the mean. Therefore, two data sets can have the same standard deviation but different means, resulting in different coefficients of variation.

Question 4

The Empirical Rule states that the percentage of observations in a data set (providing that the data set is bell shaped)that fall within one standard deviation of their mean is approximately 75%.

Accepted Answer

The Empirical Rule actually states that approximately 68% of observations in a bell-shaped distribution fall within one standard deviation of the mean.

Question 5

Which of the following is a measure of variability?

Accepted Answer

All of these choices are measures of variability. The interquartile range and the variance both measure how spread out the data is, while the coefficient of variation represents the ratio of the standard deviation to the mean, and provides a measure of the relative variability of the data.

Question 6

If two data sets have the same range,the distances from the smallest to largest observations in both sets will be the same.

Accepted Answer

Yes, the range is the difference between the smallest and largest observations. If two data sets have the same range, it means that the difference between the smallest and largest observations in both sets is the same.

Question 7

The coefficient of variation allows us to compare two sets of data based on different measurement units.

Accepted Answer

The coefficient of variation is a standard statistical measure that allows us to compare two sets of data based on different measurement units by normalizing the standard deviation relative to the mean of the data. Therefore, the coefficient of variation is particularly useful when comparing the variability of data sets that have different ranges or units of measurement.

Question 8

The units for the variance are the same as the units for the original data (for example,feet,inches,etc.).

Accepted Answer

The units for the variance are squared units of the original data. For example, if the original data was measured in feet, the variance will be measured in square feet (ft^2).

Question 9

Chebysheff's Theorem states that the percentage of measurements in a data set that fall within three standard deviations of their mean is:

Accepted Answer

Chebyshev's Theorem guarantees that for any distribution, at least $1 - \frac{1}{k^2}$ of the data within $k$ standard deviations from the mean. For $k=3$, this gives at least $1 - \frac{1}{3^2} = 1 - \frac{1}{9} = \frac{8}{9} = 88.9\%$.

Question 10

The variance is more meaningful and easier to interpret compared to the standard deviation.

Accepted Answer

The standard deviation is more commonly used and easier to understand compared to the variance. The standard deviation is the square root of the variance, making it more convenient to work with in statistical analyses.

Question 11

The range is considered the weakest measure of variability.

Accepted Answer

The range only takes into account the difference between the highest and lowest values in a dataset and may not accurately represent the spread of values in between. Therefore, it is considered the weakest measure of variability. More robust measures of variability include the standard deviation and interquartile range.

Question 12

The coefficient of variation is a measure of variability.

Accepted Answer

The coefficient of variation (CV) is a statistical measure of the dispersion of data points in a data series around the mean. It is calculated as the ratio of the standard deviation to the mean, expressed as a percentage. Therefore, it is a measure of variability.

Question 13

If the observations are in the millions,a standard deviation of 10 would be considered small.If the observations are all less than 50,a standard deviation of 10 would be considered large.

Accepted Answer

The standard deviation is affected by the scale or range of the observations. When the observations are in millions, a standard deviation of 10 is very small compared to the scale of the data. However, if the observations are all less than 50, a standard deviation of 10 is large in proportion to the range of the data.

Question 14

The units for the standard deviation are the same as the units for the original data (for example,feet,inches,etc.).

Accepted Answer

This is true. Since the standard deviation is calculated based on the original data, it will have the same units as the original data.

Question 15

The Empirical Rule states that the approximate percentage of measurements in a data set (providing that the data set has a bell shaped distribution)that fall within two standard deviations of their mean is approximately:

Accepted Answer

The Empirical Rule states that approximately 95% of measurements in a data set that follows a bell-shaped distribution fall within two standard deviations of the mean.

Question 16

Is a standard deviation of 10 a large number indicating great variability,or is it small number indicating little variability? To answer this question correctly,one should look carefully at the value of the:

Accepted Answer

The coefficient of variation (C) is the correct answer because it is a measure that indicates the extent of variability in relation to the mean of the population. It is calculated as the ratio of the standard deviation to the mean, and it provides a standardized measure of dispersion of a probability distribution or frequency distribution. This is useful for comparing the degree of variation from one data series to another, even if the means are drastically different from each other.

Question 17

If two data sets have the same range:

Accepted Answer

Range is the difference between the largest and smallest observation in a data set. If two data sets have the same range, it means the distance between the smallest and largest observations is the same for both data sets. However, it does not imply that the smallest and largest observations are the same, or that the two sets have the same standard deviation or interquartile range.

Question 18

The standard deviation is the positive square root of the variance.

Accepted Answer

This is a commonly accepted definition of standard deviation.

Question 19

Which of the following summary measures is affected most by extreme values?

Accepted Answer

The range is affected most by extreme values because it is the difference between the maximum and minimum values. Extreme values can significantly increase the range, while they have less of an effect on the median, mean, and interquartile range.

Question 20

Chebysheff's Theorem applies only to data sets that have a bell shaped distribution.

Accepted Answer

Chebyshev's Theorem applies to any data set, regardless of its distribution. It gives bounds on how much data can differ from the mean.

Quiz 4: Numerical Descriptive Techniques