Question 1

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

Accepted Answer

The standard deviation is easier to interpret as it is in the same unit as the original data and can be used to determine the spread of the data. The variance, on the other hand, is more difficult to interpret as it is in squared units.

Question 2

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

Accepted Answer

Chebyshev's Theorem applies to any data distribution, not just those that are normally distributed.

Question 3

Which of the following summary measures is affected most by extreme values?
A) The median.
B) The mean.
C) The range.
D) The interquartile range.

Accepted Answer

The answer of Which of the following summary measures is...

Question 4

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 5

The smaller the spread of scores around the mean:
A) the smaller the variance of the data set.
B) the smaller the standard deviation of the data set.
C) the smaller the coefficient of variation of the data set.
D) All of these choices are true.

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 6

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 size of the standard deviation is relative to the scale of the data. In a large dataset with millions of observations, variation between individual observations may be small compared to the overall range of values, so a standard deviation of 10 would be considered small. However, in a small dataset where all observations are less than 50, a standard deviation of 10 would be considered large because it represents a significant proportion of the data range.

Question 7

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

Accepted Answer

The coefficient of variation is the ratio of the standard deviation to the mean. Therefore, two data sets with the same standard deviation but different means will have different coefficients of variation.

Question 8

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:
A) mean.
B) standard deviation.
C) coefficient of variation.
D) mean dividing by the standard deviation.

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 9

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 variance are the square of the units for the original data. For example, if the original data is in feet, the units for variance will be in square feet (ft²).

Question 10

If two data sets have the same range:
A) the distances from the smallest to largest observations in both sets will be the same.
B) the smallest and largest observations are the same in both sets.
C) both sets will have the same standard deviation.
D) both sets will have the same interquartile range.

Accepted Answer

Range is defined as the difference between the maximum and minimum values in a data set. Therefore, if two data sets have the same range, it means that the distances from the smallest to largest observation in both sets will be the same. The other options (B, C, and D) are not necessarily true if only the range is the same.

Question 11

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

The range of a data set is the difference between the largest and smallest observations. If two data sets have the same range, it means that they have the same difference between their largest and smallest observations. Therefore, the distances from the smallest to largest observations in both sets will be the same.

Question 12

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 13

Chebysheff's Theorem states that the percentage of measurements in a data set that fall within three standard deviations of their mean is:
A) 75%.
B) at least 75%.
C) 89%.
D) at least 88.9%.

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 14

Which of the following is a measure of variability?
A) The interquartile range.
B) The variance.
C) The coefficient of variation.
D) All of these choices are true.

Accepted Answer

All three options are measures of variability. The interquartile range is a measure of the spread of the middle 50% of data in a set, the variance measures how spread out the data is from the mean, and the coefficient of variation measures the relative variability of the data by expressing the standard deviation as a percentage of the mean. Therefore, all of these choices are true.

Question 15

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

Accepted Answer

The coefficient of variation (CV) is a statistical measure that expresses the standard deviation as a percentage of the mean, making it possible to compare variability between data sets with different units of measurement. Therefore, option A is correct.

Question 16

The range is considered the weakest measure of variability.

Accepted Answer

The range only takes into account the difference between the largest and smallest values in a dataset, which can be heavily influenced by outliers and does not provide information about the distribution of values between those two points. Therefore, it is considered a weak measure of variability compared to other measures such as the standard deviation or interquartile range.

Question 17

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

Accepted Answer

This is true, as the standard deviation is defined as the square root of the variance.

Question 18

The coefficient of variation is a measure of variability.

Accepted Answer

The coefficient of variation is a statistical measure used to measure the relative variability of a dataset. It is calculated by dividing the standard deviation by the mean and multiplying by 100. It is expressed as a percentage and is useful for comparing the variability of datasets with different units of measurement or different means.

Question 19

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:
A) 68%.
B) 75%.
C) 95%.
D) 99%.

Accepted Answer

The Empirical Rule, also known as the 68-95-99.7 rule, states that approximately 68% of measurements fall within one standard deviation of the mean, approximately 95% fall within two standard deviations, and approximately 99.7% fall within three standard deviations. Therefore, choice C (95%) is the best choice as it refers to the percentage of measurements that fall within two standard deviations, which is what the Empirical Rule addresses.

Question 20

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

Accepted Answer

The standard deviation is a measure of spread or variability in the data, and it is expressed in the same units as the original data. This is because it is calculated by taking the square root of the variance, which is calculated by summing the squared deviations from the mean, and dividing by the number of data points. Since the deviations and the data itself are in the same units, so is the variance and the standard deviation.

Quiz 4: A: Numerical Descriptive Techniques

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

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

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

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

The smaller the spread of scores around the mean:

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.

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

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:

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

If two data sets have the same range: