Chapter 3 - Descriptive Statistics: Numerical Measures p.102

Question 1

Boxplot

Answer 1

A graphical summary of data based on a five-number summary.

Question 2

Chebyshev’s theorem p.127

Answer 2

A theorem that can be used to make statements about the proportion of data values that must be within a specified number of standard deviations of the mean.

Question 3

Coefficient of variation p.121

Answer 3

A measure of relative variability computed by dividing the standard deviation by the mean and multiplying by 100.

Question 4

Correlation coefficient p.141

Answer 4

A measure of linear association between two variables that take on values between -1 and +1. Values near +1 indicate a strong positive linear relationship; values near -1 indicate a strong negative linear relationship; and values near zero indicate the lack of a linear relationship.

Question 5

Covariance p.138

Answer 5

A measure of linear association between two variables. Positive values indicate a positive relationship; negative values indicate a negative relationship.

Question 6

Empirical rule p.128

Answer 6

A rule that can be used to compute the percentage of data values that must be within one, two, and three deviations of the mean for data that exhibits a bell-shaped distribution.

Question 7

Five-number summary p.133

Answer 7

A technique that uses five numbers to summarize the data: smallest value, first quartile, median, third quartile, and largest value.

Question 8

Geometric mean p.109

Answer 8

A measure of location that is calculated by finding the nth root of the product of n values.

Question 9

Interquartile range (IQR) p.119

Answer 9

A measure of variability, defined to be the difference between the third and first quartiles.

Question 10

Mean p.104

Answer 10

A measure of central location computed by summing the data values and dividing by the number of observations.

Question 11

Median p.107

Answer 11

A measure of central location provided by the value in the middle when the data are arranged in ascending order.

Question 12

Mode p.110

Answer 12

A measure of location, defined as the value that occurs with greatest frequency.

Question 13

Outlier p.130

Answer 13

An unusually small or unusually large data value

Question 14

Percentile p.111

Answer 14

A value such that at least p percent of the observations are greater than or equal to this value and at least (100 - p) percent of the observations are less than or equal to this value. The 50th percentile is the median.

Question 15

Point estimator p.104

Answer 15

A sample statistic used to estimate the corresponding population parameter.

Question 16

Population parameter p.104

Answer 16

A numerical value used as a summary measure for a population (e.g., the population mean, the population variance, and the population standard deviation).

Question 17

Quartiles p.112

Answer 17

The 25th, 50th, and 75th percentiles, referred to as the first quartile, the second quartile (median), and third quartile, respectively. The quartiles can be used to divide a data set into four parts, with each part containing approximately 25% of the data.

Question 18

Range p.118

Answer 18

A measure of variability, defined to be the largest value minus the smallest value.

Question 19

Sample statistics p.104

Answer 19

A numerical value used as a summary measure for a sample )e.g. the sample mean, the sample variance, and the sample standard deviation)

Question 20

Skewness p.125

Answer 20

A measure of the shape of a data distribution. Data skewed to the left result in negative skewness; a symmetric data distribution results in zero skewness; and data skewed to the right results in positive skewness.

Question 21

Standard deviation p.120

Answer 21

A measure of variability computed by taking the positive square root of the variance.

Question 22

Variance p.119

Answer 22

A measure of variability based on the squared deviations of the data values about the mean.

Question 23

Weighted mean

Answer 23

The mean obtained by assigning each observation a weight that reflects its importance.

Question 24

z-score p.126

Answer 24

A value computed by dividing the deviation about the mean (x - Mean(x)) by the standard deviation. A z-score is referred to as a standardized value and denotes the number of standard deviations x is from the mean.

The process of converting a value for a variable to a z-score is often referred to as a z-transformation.

Question 25

3.1 Sample Mean

Answer 25

Mean(x) = Sum(x)/n

Question 26

3.2 Population Mean

Answer 26

Mean(x) = Sum(x)/N

Question 27

3.3 Weighted Mean

Answer 27

Mean(x) = Sum(w * x)/Sum(w)

Where
W = weight for x

Question 28

.4 Geometric Mean

Answer 28

Mean(x) = nthRoot(Product(x)) = [Product(x)]^(1/n)

Question 29

3.5 Location of the pth Percentile

Answer 29

Location(p) = (p/100)*(n + 1)

Question 30

3.6 Interquartile Range

Answer 30

IQR = Quartile(3) - Quartile(1)

The measure of variability is the difference between the third quartile, Quartile(3), and the first quartile, Quartile(1).

Question 31

3.7 Population Variance

Answer 31

Variance(x) = Sum((x - mean)^2)/N

Question 32

3.8 Sample Variance

Answer 32

Variance(x) = Sum((X - mean)^2)/(n - 1)

Question 33

3.9 Sample Standard Deviation

Answer 33

StDev(x) = s = SquareRoot(Variance(x)) = SquareRoot(Square(s))

Question 34

3.10 Population standard deviation

Answer 34

StDev(x) = s = SquareRoot(Variance(x)) = SquareRoot(Square(s))

Question 35

3.11 Coefficient of Variation

Answer 35

A descriptive statistic that indicates ow large the standard deviation is relative to the mean.

(StDev(x)/Mean(x))%

Question 36

3.12 z-Score

Answer 36

z = (x - Mean(x))/StDev(x)

Where z is the z-score for value x

Question 37

3.13 Sample Covariance

Answer 37

Covariance(x,y) = Sum((x - Mean(x)) * (y - Mean(y)) / (n - 1)

Question 38

3.14 Population Covariance

Answer 38

Covariance(x,y) = Sum((x - Mean(x)) * (y - Mean(y)) / N

Question 39

3.15 Pearson Product Moment Correlation Coefficient: Sample Data

Answer 39

CorrelationCoefficient(x,y) = Covariance(x,y)/(StandardDeviation(x) * StandardDeviation(y))

Question 40

3.16 Pearson Product Moment Correlation Coefficient: Population data

Answer 40

CorrelationCoefficient(x,y) = Covariance(x,y)/(StandardDeviation(x) * StandardDeviation(y))

Question 41

Median

Answer 41

Arrange the data in ascending order (smallest value to largest value).

    a. For an odd number of observations, the median is the middle value.
    b. For an even number of observations, the median is the average of the two middle values.

Question 42

Mode

Answer 42

The mode is the value that occurs with greatest frequency.

Question 43

Range

Answer 43

Range = Largest value - Smallest value

Question 44

Chebyshev’s Theorem

Answer 44

At least (1 - 1/z^2) of the data values must be within z standard deviations of the mean, where z is any value greater than 1.

   - Some implications of this theorem, with z = 2,3, and 4 standard deviations, follow:
           - At least .75 or 75% of the data values must be within z = 2 standard deviations of the mean
           - At least .89 or 89% of the data values must be within z = 3 standard deviations of the mean
           - At least .94 or 94% of the data values must be within z = 4 standard deviations of the mean

Chebyshev’s theorem requires z > 1; but z need not be an integer.

Question 45

Outliers

Answer 45

Standardized values (z-score) can be used to identify outliers.
               - Any data value with a z-score less than -3 or greater than +3 is identified as an outlier.

IQR method to identify outliers

    - First compute the following lower and upper limits:
           - Lower Limit = Quartile(1) - 1.5(IQR)
           - Upper Limit = Quartile(3) + 1.5(IQR)
   - A Observation is classified as an outlier if its value is less than the lower limit or greater than the upper limit.

The approach that uses the first and third quartiles and the IQR to identify outliers does not necessarily provide the same results as the approach based upon a z-score less than -3 or greater than +3. Either or both procedures may be used.

Question 46

Five-number summary

Answer 46

1. Smallest value
2. First quartile, Quartile(1)
3. Median, Quartile(2)
4. Third quartile, Quartile(3)
5. Largest value