Question 1

The median is a measure of center that divides an ordered array of data into two halves; if the data are arranged in ascending order from smallest to largest, all the observations below the median are smaller than or equal to it, while all the observations above the median are equal to it or larger.

Accepted Answer

This is the correct definition of the median.

Question 2

The range is considered the weakest measure of variability.

Accepted Answer

This is because the range only takes into account the difference between the largest and smallest values in a dataset and does not consider the values in between. It can be greatly affected by outliers and may not accurately represent the spread of the data. Other measures of variability, such as the variance or standard deviation, are more robust and informative.

Question 3

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

Accepted Answer

The variance is always positive or zero because it is the square of the standard deviation, which is also always positive or zero.

Question 4

In all cases, the sum of the deviations of the measurements from their mean is 0.

Accepted Answer

This is true by definition of the mean. The mean is calculated by summing all the measurements and dividing by the number of measurements, so the deviations from the mean will balance out and add up to zero.

Question 5

If the variability of a set of data is very small, then the sample variance may be negative.

Accepted Answer

The sample variance cannot be negative, as it is a measure of variation and is always non-negative. If the variability of the data is very small, then the sample variance will be close to zero, but it cannot be negative.

Question 6

In a mound-shaped distribution, there is no difference in the values of the mean, and median.

Accepted Answer

In a symmetrical distribution, the mean and median are the same value, which is the center of the distribution. Therefore, in a mound-shaped distribution (which is a symmetrical distribution), the mean and median are the same value.

Question 7

The sample variance calculated with a divisor of n gives a better estimate of the population variance,  , than does the sample variance,  , with a divisor of n - 1.

Accepted Answer

The sample variance calculated with a divisor of n-1 gives a better estimate of the population variance than does the sample variance with a divisor of n, due to the unbiasedness property of the former.

Question 8

The larger the values of the sample variance  and sample standard deviation s, the greater the variability in the data.

Accepted Answer

The sample variance and sample standard deviation measure the spread or variability of the data. The larger these values, the greater the variability in the data. Therefore, the statement is true.

Question 9

The mean is one of the most frequently used measures of variability.

Accepted Answer

The mean is a measure of central tendency, not variability. Measures of variability include the range, variance, and standard deviation.

Question 10

When all the numbers in the data set are the same, the standard deviations must be zero.

Accepted Answer

When there is no variability in the data because all the numbers in the data set are the same, the standard deviation will be zero. The standard deviation measures how spread out the data is from the mean, and if all the data points are the same, there is no variability and thus no standard deviation.

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 value in a data set, and does not consider the spread of values in between. Therefore, it is considered a weak measure of variability compared to other measures such as the standard deviation or interquartile range.

Question 12

The value of the standard deviation will always exceed that of the variance.

Accepted Answer

The standard deviation is the square root of the variance, so it is typically smaller than the variance for any given set of data, except when the variance is less than or equal to 1.

Question 13

Measures of center are values around which observations tend to cluster and that describe the location of what in some sense might be called the "center" of a data set.

Accepted Answer

Measures of center, such as mean, median, and mode, are used to describe the central tendency of a data set. They represent values around which the data tends to cluster. Therefore, the statement is true.

Question 14

Measures of variability describe typical values in the data.

Accepted Answer

Measures of variability describe how spread out or dispersed the data is, not typical values. Measures of central tendency, such as mean, median and mode, describe typical values.

Question 15

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

Accepted Answer

This statement is true. The standard deviation is the square root of the variance.

Question 16

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, thus it is expressed in the original units of measurement, whereas the variance is expressed in squared units.

Question 17

A student scores 89, 75, 94, and 88 on four exams during the semester and 97 on the final exam. If the final is weighted double and the four others weighted equally, the student's final average would be 90.

Accepted Answer

The weighted average can be calculated as:
(89 + 75 + 94 + 88)/4 * 0.5 + 97 * 0.5 = 90

So, the student's final average would be 90.

Question 18

The mode is the sum of a data set's minimum plus maximum values, divided by 2.

Accepted Answer

The statement is incorrect. The mode is the value in a data set that appears most frequently. The formula given in the statement is for finding the mean.

Question 19

In order to measure the variability in the same units as the original observations, we compute the sample variance.

Accepted Answer

To measure variability in the same units as the original observations, we compute the sample standard deviation, not the variance. The variance is measured in the units squared, while the standard deviation is in the original units.

Question 20

The sample variance is approximately the average of the squared deviations of the measurements from their mean.

Accepted Answer

This is a correct definition of the sample variance.

Quiz 2: Describing Data With Numerical Measures

The range is considered the weakest measure of variability.

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

In all cases, the sum of the deviations of the measurements from their mean is 0.

If the variability of a set of data is very small, then the sample variance may be negative.

In a mound-shaped distribution, there is no difference in the values of the mean, and median.

The sample variance calculated with a divisor of n gives a better estimate of the population variance, , than does the sample variance, , with a divisor of n - 1.

The larger the values of the sample variance and sample standard deviation s, the greater the variability in the data.

The mean is one of the most frequently used measures of variability.

When all the numbers in the data set are the same, the standard deviations must be zero.

The range is considered the weakest measure of variability.

The value of the standard deviation will always exceed that of the variance.

Measures of center are values around which observations tend to cluster and that describe the location of what in some sense might be called the "center" of a data set.

Measures of variability describe typical values in the data.

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

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

A student scores 89, 75, 94, and 88 on four exams during the semester and 97 on the final exam. If the final is weighted double and the four others weighted equally, the student's final average would be 90.

The mode is the sum of a data set's minimum plus maximum values, divided by 2.

In order to measure the variability in the same units as the original observations, we compute the sample variance.

The sample variance is approximately the average of the squared deviations of the measurements from their mean.