Chapter 3, Part B Descriptive Statistics: Numerical Measures n Measures of Distribution Shape, Relative Location, and Detecting Outliers n Exploratory Data Analysis n Measures of Association Between Two Variables n The Weighted Mean and Working with Grouped Data Working with Grouped Data

Measures of Distribution Shape, Relative Location, and Detecting Outliers n Distribution Shape n z-Scores n Chebyshev’s Theorem n Empirical Rule n Detecting Outliers

Distribution Shape: Skewness n An important measure of the shape of a distribution is called skewness. n The formula for the skewness of sample data is n Skewness can be easily computed using statistical software.

Distribution Shape: Skewness n Symmetric (not skewed) Relative Frequency Skewness = 0 Skewness = 0 Skewness is zero. Skewness is zero. Mean and median are equal. Mean and median are equal.

Relative Frequency Distribution Shape: Skewness n Moderately Skewed Left Skewness = .31 Skewness = .31 Skewness is negative. Skewness is negative. Mean will usually be less than the median. Mean will usually be less than the median.

Distribution Shape: Skewness n Moderately Skewed Right Relative Frequency Skewness =.31 Skewness =.31 Skewness is positive. Skewness is positive. Mean will usually be more than the median. Mean will usually be more than the median.

Distribution Shape: Skewness n Highly Skewed Right Relative Frequency Skewness = 1.25 Skewness = 1.25 Skewness is positive (often above 1.0). Skewness is positive (often above 1.0). Mean will usually be more than the median. Mean will usually be more than the median.

Seventy apartments were randomly Seventy apartments were randomly sampled in Suva. The monthly rent prices for the apartments are listed below in ascending order. Distribution Shape: Skewness n Example: Apartment Rents

Relative Frequency Skewness =.92 Skewness =.92 Distribution Shape: Skewness n Example: Apartment Rents

The z-score is often called the standardized value. The z-score is often called the standardized value. It denotes the number of standard deviations a data It denotes the number of standard deviations a data value x i is from the mean. value x i is from the mean. It denotes the number of standard deviations a data It denotes the number of standard deviations a data value x i is from the mean. value x i is from the mean. z-Scores Excel’s STANDARDIZE function can be used to Excel’s STANDARDIZE function can be used to compute the z-score. compute the z-score. Excel’s STANDARDIZE function can be used to Excel’s STANDARDIZE function can be used to compute the z-score. compute the z-score.

z-Scores A data value less than the sample mean will have a A data value less than the sample mean will have a z-score less than zero. z-score less than zero. A data value greater than the sample mean will have A data value greater than the sample mean will have a z-score greater than zero. a z-score greater than zero. A data value equal to the sample mean will have a A data value equal to the sample mean will have a z-score of zero. z-score of zero. An observation’s z-score is a measure of the relative An observation’s z-score is a measure of the relative location of the observation in a data set. location of the observation in a data set.

z-Score of Smallest Value (425) z-Score of Smallest Value (425) z-Scores Standardized Values for Apartment Rents Example: Apartment Rents Example: Apartment Rents

Chebyshev’s Theorem At least (1 - 1/ z 2 ) of the items in any data set will be At least (1 - 1/ z 2 ) of the items in any data set will be within z standard deviations of the mean, where z is within z standard deviations of the mean, where z is any value greater than 1. any value greater than 1. At least (1 - 1/ z 2 ) of the items in any data set will be At least (1 - 1/ z 2 ) of the items in any data set will be within z standard deviations of the mean, where z is within z standard deviations of the mean, where z is any value greater than 1. any value greater than 1. Chebyshev’s theorem requires z > 1, but z need not Chebyshev’s theorem requires z > 1, but z need not be an integer. be an integer. Chebyshev’s theorem requires z > 1, but z need not Chebyshev’s theorem requires z > 1, but z need not be an integer. be an integer.

At least of the data values must be At least of the data values must be within of the mean. within of the mean. At least of the data values must be At least of the data values must be within of the mean. within of the mean. 75%75% z = 2 standard deviations z = 2 standard deviations Chebyshev’s Theorem At least of the data values must be At least of the data values must be within of the mean. within of the mean. At least of the data values must be At least of the data values must be within of the mean. within of the mean.89%89% z = 3 standard deviations z = 3 standard deviations At least of the data values must be At least of the data values must be within of the mean. within of the mean. At least of the data values must be At least of the data values must be within of the mean. within of the mean. 94%94% z = 4 standard deviations z = 4 standard deviations

Chebyshev’s Theorem Let z = 1.5 with = and s = At least (1  1/(1.5) 2 ) = 1  0.44 = 0.56 or 56% of the rent values must be between - z ( s ) =  1.5(54.74) = z ( s ) =  1.5(54.74) = 409and + z ( s ) = (54.74) = z ( s ) = (54.74) = 573 (Actually, 86% of the rent values are between 409 and 573.) are between 409 and 573.) Example: Apartment Rents Example: Apartment Rents

Empirical Rule When the data are believed to approximate a When the data are believed to approximate a bell-shaped distribution … bell-shaped distribution … The empirical rule is based on the normal The empirical rule is based on the normal distribution, which is covered in Chapter 6. distribution, which is covered in Chapter 6. The empirical rule is based on the normal The empirical rule is based on the normal distribution, which is covered in Chapter 6. distribution, which is covered in Chapter 6. The empirical rule can be used to determine the The empirical rule can be used to determine the percentage of data values that must be within a percentage of data values that must be within a specified number of standard deviations of the specified number of standard deviations of the mean. mean. The empirical rule can be used to determine the The empirical rule can be used to determine the percentage of data values that must be within a percentage of data values that must be within a specified number of standard deviations of the specified number of standard deviations of the mean. mean.

Empirical Rule For data having a bell-shaped distribution: For data having a bell-shaped distribution: of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean %68.26% +/- 1 standard deviation of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean.95.44%95.44% +/- 2 standard deviations of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean. of the values of a normal random variable of the values of a normal random variable are within of its mean. are within of its mean %99.72% +/- 3 standard deviations

Empirical Rule x  – 3   – 1   – 2   + 1   + 2   + 3  68.26% 95.44% 99.72%

Detecting Outliers An outlier is an unusually small or unusually large An outlier is an unusually small or unusually large value in a data set. value in a data set. A data value with a z-score less than -3 or greater A data value with a z-score less than -3 or greater than +3 might be considered an outlier. than +3 might be considered an outlier. It might be: It might be: an incorrectly recorded data value an incorrectly recorded data value a data value that was incorrectly included in the a data value that was incorrectly included in the data set data set a correctly recorded data value that belongs in a correctly recorded data value that belongs in the data set the data set

Detecting Outliers The most extreme z-scores are and 2.27 The most extreme z-scores are and 2.27 Using | z | > 3 as the criterion for an outlier, there Using | z | > 3 as the criterion for an outlier, there are no outliers in this data set. are no outliers in this data set. Standardized Values for Apartment Rents Example: Apartment Rents Example: Apartment Rents

Exploratory Data Analysis Exploratory data analysis procedures enable us to use Exploratory data analysis procedures enable us to use simple arithmetic and easy-to-draw pictures to simple arithmetic and easy-to-draw pictures to summarize data. summarize data. Exploratory data analysis procedures enable us to use Exploratory data analysis procedures enable us to use simple arithmetic and easy-to-draw pictures to simple arithmetic and easy-to-draw pictures to summarize data. summarize data. We simply sort the data values into ascending order We simply sort the data values into ascending order and identify the five-number summary and then and identify the five-number summary and then construct a box plot. construct a box plot. We simply sort the data values into ascending order We simply sort the data values into ascending order and identify the five-number summary and then and identify the five-number summary and then construct a box plot. construct a box plot.

Five-Number Summary 11 Smallest Value Smallest Value First Quartile First Quartile Median Median Third Quartile Third Quartile Largest Value Largest Value

Five-Number Summary Lowest Value = 425 First Quartile = 445 Median = 475 Third Quartile = 525 Largest Value = 615 Example: Apartment Rents Example: Apartment Rents

Box Plot A box plot is a graphical summary of data that is A box plot is a graphical summary of data that is based on a five-number summary. based on a five-number summary. A box plot is a graphical summary of data that is A box plot is a graphical summary of data that is based on a five-number summary. based on a five-number summary. A key to the development of a box plot is the A key to the development of a box plot is the computation of the median and the quartiles Q 1 and computation of the median and the quartiles Q 1 and Q 3. Q 3. A key to the development of a box plot is the A key to the development of a box plot is the computation of the median and the quartiles Q 1 and computation of the median and the quartiles Q 1 and Q 3. Q 3. Box plots provide another way to identify outliers. Box plots provide another way to identify outliers.

A box is drawn with its ends located at the first and A box is drawn with its ends located at the first and third quartiles. third quartiles. Box Plot A vertical line is drawn in the box at the location of A vertical line is drawn in the box at the location of the median (second quartile). the median (second quartile). Q1 = 445 Q3 = 525 Q2 = 475 Example: Apartment Rents Example: Apartment Rents

Box Plot Limits are located (not drawn) using the interquartile range (IQR). Limits are located (not drawn) using the interquartile range (IQR). Data outside these limits are considered outliers. Data outside these limits are considered outliers. The locations of each outlier is shown with the symbol *. The locations of each outlier is shown with the symbol *.continued

Box Plot Lower Limit: Q (IQR) = (80) = 325 Upper Limit: Q (IQR) = (80) = 645 The lower limit is located 1.5(IQR) below Q 1. The lower limit is located 1.5(IQR) below Q 1. The upper limit is located 1.5(IQR) above Q 3. The upper limit is located 1.5(IQR) above Q 3. There are no outliers (values less than 325 or There are no outliers (values less than 325 or greater than 645) in the apartment rent data. greater than 645) in the apartment rent data. Example: Apartment Rents Example: Apartment Rents

Box Plot Whiskers (dashed lines) are drawn from the ends Whiskers (dashed lines) are drawn from the ends of the box to the smallest and largest data values of the box to the smallest and largest data values inside the limits. inside the limits Smallest value inside limits = 425 Largest value inside limits = 615 Example: Apartment Rents Example: Apartment Rents

Box Plot An excellent graphical technique for making comparisons among two or more groups. comparisons among two or more groups.

Measures of Association Between Two Variables Thus far we have examined numerical methods used Thus far we have examined numerical methods used to summarize the data for one variable at a time. to summarize the data for one variable at a time. Thus far we have examined numerical methods used Thus far we have examined numerical methods used to summarize the data for one variable at a time. to summarize the data for one variable at a time. Often a manager or decision maker is interested in Often a manager or decision maker is interested in the relationship between two variables. the relationship between two variables. Often a manager or decision maker is interested in Often a manager or decision maker is interested in the relationship between two variables. the relationship between two variables. Two descriptive measures of the relationship Two descriptive measures of the relationship between two variables are covariance and correlation between two variables are covariance and correlation coefficient. coefficient. Two descriptive measures of the relationship Two descriptive measures of the relationship between two variables are covariance and correlation between two variables are covariance and correlation coefficient. coefficient.

Covariance Positive values indicate a positive relationship. Positive values indicate a positive relationship. Negative values indicate a negative relationship. Negative values indicate a negative relationship. The covariance is a measure of the linear association The covariance is a measure of the linear association between two variables. between two variables. The covariance is a measure of the linear association The covariance is a measure of the linear association between two variables. between two variables.

Covariance The covariance is computed as follows: The covariance is computed as follows: forsamples forpopulations

Correlation Coefficient Just because two variables are highly correlated, it Just because two variables are highly correlated, it does not mean that one variable is the cause of the does not mean that one variable is the cause of the other. other. Just because two variables are highly correlated, it Just because two variables are highly correlated, it does not mean that one variable is the cause of the does not mean that one variable is the cause of the other. other. Correlation is a measure of linear association and not Correlation is a measure of linear association and not necessarily causation. necessarily causation. Correlation is a measure of linear association and not Correlation is a measure of linear association and not necessarily causation. necessarily causation.

The correlation coefficient is computed as follows: The correlation coefficient is computed as follows: forsamplesforpopulations Correlation Coefficient

Values near +1 indicate a strong positive linear Values near +1 indicate a strong positive linear relationship. relationship. Values near +1 indicate a strong positive linear Values near +1 indicate a strong positive linear relationship. relationship. Values near -1 indicate a strong negative linear Values near -1 indicate a strong negative linear relationship. relationship. Values near -1 indicate a strong negative linear Values near -1 indicate a strong negative linear relationship. relationship. The coefficient can take on values between -1 and +1. The coefficient can take on values between -1 and +1. The closer the correlation is to zero, the weaker the The closer the correlation is to zero, the weaker the relationship. relationship. The closer the correlation is to zero, the weaker the The closer the correlation is to zero, the weaker the relationship. relationship.

A golfer is interested in investigating the A golfer is interested in investigating the relationship, if any, between driving distance and 18-hole score Average Driving Distance (yds.) Average 18-Hole Score Covariance and Correlation Coefficient Example: Golfing Study Example: Golfing Study

Covariance and Correlation Coefficient xy Average Std. Dev Total Example: Golfing Study Example: Golfing Study

Sample Covariance Sample Covariance Sample Correlation Coefficient Sample Correlation Coefficient Covariance and Correlation Coefficient Example: Golfing Study Example: Golfing Study

The Weighted Mean and Working with Grouped Data Weighted Mean Weighted Mean Mean for Grouped Data Mean for Grouped Data Variance for Grouped Data Variance for Grouped Data Standard Deviation for Grouped Data Standard Deviation for Grouped Data

Weighted Mean When the mean is computed by giving each data When the mean is computed by giving each data value a weight that reflects its importance, it is value a weight that reflects its importance, it is referred to as a weighted mean. referred to as a weighted mean. In the computation of a grade point average (GPA), In the computation of a grade point average (GPA), the weights are the number of credit hours earned for the weights are the number of credit hours earned for each grade. each grade. When data values vary in importance, the analyst When data values vary in importance, the analyst must choose the weight that best reflects the must choose the weight that best reflects the importance of each value. importance of each value.

Weighted Mean where: x i = value of observation i x i = value of observation i w i = weight for observation i w i = weight for observation i

Grouped Data The weighted mean computation can be used to The weighted mean computation can be used to obtain approximations of the mean, variance, and obtain approximations of the mean, variance, and standard deviation for the grouped data. standard deviation for the grouped data. To compute the weighted mean, we treat the To compute the weighted mean, we treat the midpoint of each class as though it were the mean midpoint of each class as though it were the mean of all items in the class. of all items in the class. We compute a weighted mean of the class midpoints We compute a weighted mean of the class midpoints using the class frequencies as weights. using the class frequencies as weights. Similarly, in computing the variance and standard Similarly, in computing the variance and standard deviation, the class frequencies are used as weights. deviation, the class frequencies are used as weights.

Mean for Grouped Data where: f i = frequency of class i f i = frequency of class i M i = midpoint of class i M i = midpoint of class i Sample Data Sample Data Population Data Population Data

The previously presented sample of apartment rents is shown here as grouped data in the form of a frequency distribution. Sample Mean for Grouped Data Sample Mean for Grouped Data Example: Apartment Rents Example: Apartment Rents

Sample Mean for Grouped Data Sample Mean for Grouped Data This approximation differs by $2.41 from the actual sample mean of $ Example: Apartment Rents Example: Apartment Rents

Variance for Grouped Data For sample data For sample data For population data For population data

Sample Variance for Grouped Data Sample Variance for Grouped Datacontinued Example: Apartment Rents Example: Apartment Rents

s 2 = 208,234.29/(70 – 1) = 3, This approximation differs by only $.20 from the actual standard deviation of $ Sample Variance Sample Variance Sample Standard Deviation Sample Standard Deviation Example: Apartment Rents Example: Apartment Rents Sample Variance for Grouped Data Sample Variance for Grouped Data

End of Chapter 3, Part B