Slides by JOHN LOUCKS St. Edward’s University

Chapter 3, Part A Descriptive Statistics: Numerical Measures Measures of Location Measures of Variability

Measures of Location Mean If the measures are computed for data from a sample, they are called sample statistics. Median Mode Percentiles If the measures are computed for data from a population, they are called population parameters. Quartiles A sample statistic is referred to as the point estimator of the corresponding population parameter.

Mean The mean of a data set is the average of all the data values. The sample mean is the point estimator of the population mean m.

Sample Mean Sum of the values of the n observations Number of in the sample

Population Mean m Sum of the values of the N observations Number of observations in the population

Sample Mean Example: Apartment Rents Seventy efficiency apartments were randomly sampled in a small college town. The monthly rent prices for these apartments are listed below.

Sample Mean Example: Apartment Rents

Median The median of a data set is the value in the middle when the data items are arranged in ascending order. Whenever a data set has extreme values, the median is the preferred measure of central location. The median is the measure of location most often reported for annual income and property value data. A few extremely large incomes or property values can inflate the mean.

Median For an odd number of observations: 26 18 27 12 14 27 19 in ascending order the median is the middle value. Median = 19

Median For an even number of observations: 26 18 27 12 14 27 30 19 in ascending order the median is the average of the middle two values. Median = (19 + 26)/2 = 22.5

Averaging the 35th and 36th data values: Median Example: Apartment Rents Averaging the 35th and 36th data values: Median = (475 + 475)/2 = 475 Note: Data is in ascending order.

Mode The mode of a data set is the value that occurs with greatest frequency. The greatest frequency can occur at two or more different values. If the data have exactly two modes, the data are bimodal. If the data have more than two modes, the data are multimodal.

450 occurred most frequently (7 times) Mode Example: Apartment Rents 450 occurred most frequently (7 times) Mode = 450 Note: Data is in ascending order.

Using Excel to Compute the Mean, Median, and Mode Excel Formula Worksheet Note: Rows 7-71 are not shown.

Using Excel to Compute the Mean, Median, and Mode Value Worksheet Note: Rows 7-71 are not shown.

Percentiles A percentile provides information about how the data are spread over the interval from the smallest value to the largest value. Admission test scores for colleges and universities are frequently reported in terms of percentiles. The pth percentile of a data set is a value such that at least p percent of the items take on this value or less and at least (100 - p) percent of the items take on this value or more.

Percentiles Arrange the data in ascending order. Compute index i, the position of the pth percentile. i = (p/100)n If i is not an integer, round up. The p th percentile is the value in the i th position. If i is an integer, the p th percentile is the average of the values in positions i and i +1.

Averaging the 56th and 57th data values: 80th Percentile Example: Apartment Rents i = (p/100)n = (80/100)70 = 56 Averaging the 56th and 57th data values: 80th Percentile = (535 + 549)/2 = 542 Note: Data is in ascending order.

80th Percentile Example: Apartment Rents “At least 80% of the items take on a value of 542 or less.” “At least 20% of the items take on a value of 542 or more.” 56/70 = .8 or 80% 14/70 = .2 or 20%

Using Excel’s Rank and Percentile Tool to Compute Percentiles and Quartiles Using Excel’s Percentile Function The formula Excel uses to compute the location (Lp) of the pth percentile is Lp = (p/100)n + (1 – p/100) Excel would compute the location of the 80th percentile for the apartment rent data as follows: L80 = (80/100)70 + (1 – 80/100) = 56 + .2 = 56.2 The 80th percentile would be 535 + .2(549 - 535) = 535 + 2.8 = 537.8

Using Excel’s Rank and Percentile Tool to Compute Percentiles and Quartiles Excel Formula Worksheet 80th percentile It is not necessary to put the data in ascending order. Note: Rows 7-71 are not shown.

Using Excel’s Rank and Percentile Tool to Compute Percentiles and Quartiles Excel Value Worksheet Note: Rows 7-71 are not shown.

Quartiles Quartiles are specific percentiles. First Quartile = 25th Percentile Second Quartile = 50th Percentile = Median Third Quartile = 75th Percentile

Third quartile = 75th percentile Example: Apartment Rents Third quartile = 75th percentile i = (p/100)n = (75/100)70 = 52.5 = 53 Third quartile = 525 Note: Data is in ascending order.

Third Quartile Using Excel’s Quartile Function Excel computes the locations of the 1st, 2nd, and 3rd quartiles by first converting the quartiles to percentiles and then using the following formula to compute the location (Lp) of the pth percentile: Lp = (p/100)n + (1 – p/100) Excel would compute the location of the 3rd quartile (75th percentile) for the rent data as follows: L75 = (75/100)70 + (1 – 75/100) = 52.5 + .25 = 52.75 The 3rd quartile would be 515 + .75(525 - 515) = 515 + 7.5 = 522.5

Third Quartile Excel Formula Worksheet 3rd quartile It is not necessary to put the data in ascending order. Note: Rows 7-71 are not shown.

Third Quartile Excel Value Worksheet Note: Rows 7-71 are not shown.

Excel’s Rank and Percentile Tool Step 1 Click the Data tab on the Ribbon Step 2 In the Analysis group, click Data Analysis Step 3 Choose Rank and Percentile from the list of Analysis Tools Step 4 When the Rank and Percentile dialog box appears (see details on next slide)

Excel’s Rank and Percentile Tool Step 4 Complete the Rank and Percentile dialog box as follows:

Excel’s Rank and Percentile Tool Excel Value Worksheet Note: Rows 11-71 are not shown.

Averages and Standard Deviation Ch 3 Geometric Mean (GM) The Geometric Mean is useful in finding the averages of increases in: Percents Ratios Indexes Growth Rates The Geometric Mean will always be less than or equal to (never more than) the arithmetic mean The GM gives a more conservative figure that is not drawn up by large values in the set Statistics Is Fun! & Statistics Are Fun!

Averages and Standard Deviation Ch 3 Geometric Mean The GM of a set of n positive numbers is defined as the nth root of the product of n values. The formula is: Statistics Is Fun! & Statistics Are Fun!

Geometric Mean Example 1: Percentage Increase Averages and Standard Deviation Ch 3 Geometric Mean Example 1: Percentage Increase Statistics Is Fun! & Statistics Are Fun!

Verify Geometric Mean Example Averages and Standard Deviation Ch 3 Verify Geometric Mean Example Statistics Is Fun! & Statistics Are Fun!

Another Use Of GM: Ave. % Increase Over Time Averages and Standard Deviation Ch 3 Another Use Of GM: Ave. % Increase Over Time Another use of the geometric mean is to determine the percent increase in sales, production or other business or economic series from one time period to another Where n = number of periods Statistics Is Fun! & Statistics Are Fun!

Example for GM: Ave. % Increase Over Time Averages and Standard Deviation Ch 3 Example for GM: Ave. % Increase Over Time The total number of females enrolled in American colleges increased from 755,000 in 1992 to 835,000 in 2000. That is, the geometric mean rate of increase is 1.27%. The annual rate of increase is 1.27% For the years 1992 through 2000, the rate of female enrollment growth at American colleges was 1.27% per year Statistics Is Fun! & Statistics Are Fun!

Measures of Variability It is often desirable to consider measures of variability (dispersion), as well as measures of location. For example, in choosing supplier A or supplier B we might consider not only the average delivery time for each, but also the variability in delivery time for each.

Measures of Variability Range Interquartile Range Variance Standard Deviation Coefficient of Variation

Range The range of a data set is the difference between the largest and smallest data values. It is the simplest measure of variability. It is very sensitive to the smallest and largest data values.

Range = largest value - smallest value Example: Apartment Rents Range = largest value - smallest value Range = 615 - 425 = 190 Note: Data is in ascending order.

Interquartile Range The interquartile range of a data set is the difference between the third quartile and the first quartile. It is the range for the middle 50% of the data. It overcomes the sensitivity to extreme data values.

Interquartile Range = Q3 - Q1 = 525 - 445 = 80 Example: Apartment Rents 3rd Quartile (Q3) = 525 1st Quartile (Q1) = 445 Interquartile Range = Q3 - Q1 = 525 - 445 = 80 Note: Data is in ascending order.

Variance The variance is a measure of variability that utilizes all the data. It is based on the difference between the value of each observation (xi) and the mean ( for a sample, m for a population).

Variance The variance is the average of the squared differences between each data value and the mean. The variance is computed as follows: for a sample for a population

Standard Deviation The standard deviation of a data set is the positive square root of the variance. It is measured in the same units as the data, making it more easily interpreted than the variance.

Standard Deviation The standard deviation is computed as follows: for a sample for a population

Coefficient of Variation The coefficient of variation indicates how large the standard deviation is in relation to the mean. The coefficient of variation is computed as follows: for a sample for a population

Sample Variance, Standard Deviation, And Coefficient of Variation Example: Apartment Rents Variance Standard Deviation the standard deviation is about 11% of the mean Coefficient of Variation

Using Excel to Compute the Sample Variance, Standard Deviation, and Coefficient of Variation Formula Worksheet Note: Rows 8-71 are not shown.

Using Excel to Compute the Sample Variance, Standard Deviation, and Coefficient of Variation Value Worksheet Note: Rows 8-71 are not shown.

Using Excel’s Descriptive Statistics Tool Step 1 Click the Data tab on the Ribbon Step 2 In the Analysis group, click Data Analysis Step 3 Choose Descriptive Statistics from the list of Analysis Tools Step 4 When the Descriptive Statistics dialog box appears: (see details on next slide)

Using Excel’s Descriptive Statistics Tool Excel’s Descriptive Statistics Dialog Box

Using Excel’s Descriptive Statistics Tool Excel Value Worksheet (Partial) Note: Rows 9-71 are not shown.

Using Excel’s Descriptive Statistics Tool Excel Value Worksheet (Partial) Note: Rows 1-8 and 17-71 are not shown.

End of Chapter 3, Part A

Chapter 3, Part B Descriptive Statistics: Numerical Measures Measures of Distribution Shape, Relative Location, and Detecting Outliers Exploratory Data Analysis

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

Distribution Shape: Skewness An important measure of the shape of a distribution is called skewness. The formula for computing skewness for a data set is somewhat complex. Skewness can be easily computed using statistical software. Excel’s SKEW function can be used to compute the skewness of a data set.

Distribution Shape: Skewness Symmetric (not skewed) Skewness is zero. Mean and median are equal. Skewness = 0 .05 .10 .15 .20 .25 .30 .35 Relative Frequency

Distribution Shape: Skewness Moderately Skewed Left Skewness is negative. Mean will usually be less than the median. Skewness = - .31 Relative Frequency .05 .10 .15 .20 .25 .30 .35

Distribution Shape: Skewness Moderately Skewed Right Skewness is positive. Mean will usually be more than the median. Skewness = .31 Relative Frequency .05 .10 .15 .20 .25 .30 .35

Distribution Shape: Skewness Highly Skewed Right Skewness is positive (often above 1.0). Mean will usually be more than the median. Skewness = 1.25 Relative Frequency .05 .10 .15 .20 .25 .30 .35

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

Distribution Shape: Skewness Example: Apartment Rents .05 .10 .15 .20 .25 .30 .35 Skewness = .92 Relative Frequency

z-Scores The z-score is often called the standardized value. It denotes the number of standard deviations a data value xi is from the mean.

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

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

Chebyshev’s Theorem At least (1 - 1/z2) of the items in any data set will be within z standard deviations of the mean, where z is any value greater than 1.

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

Chebyshev’s Theorem Example: Apartment Rents Let z = 1.5 with = 490.80 and s = 54.74 At least (1 - 1/(1.5)2) = 1 - 0.44 = 0.56 or 56% of the rent values must be between - z(s) = 490.80 - 1.5(54.74) = 409 and + z(s) = 490.80 + 1.5(54.74) = 573 (Actually, 86% of the rent values are between 409 and 573.)

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

Empirical Rule x 99.72% 95.44% 68.26% m m – 3s m – 1s m + 1s m + 3s 99.72% 95.44% 68.26% x m m – 3s m – 1s m + 1s m + 3s m – 2s m + 2s

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

Standardized Values for Apartment Rents Detecting Outliers Example: Apartment Rents The most extreme z-scores are -1.20 and 2.27 Using |z| > 3 as the criterion for an outlier, there are no outliers in this data set. Standardized Values for Apartment Rents

Exploratory Data Analysis Five-Number Summary Box Plot

Five-Number Summary 1 Smallest Value 2 First Quartile 3 Median 4 Third Quartile 5 Largest Value

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

Box Plot Example: Apartment Rents A box is drawn with its ends located at the first and third quartiles. A vertical line is drawn in the box at the location of the median (second quartile). 400 425 450 475 500 525 550 575 600 625 Q1 = 445 Q3 = 525 Q2 = 475

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

Box Plot Example: Apartment Rents The lower limit is located 1.5(IQR) below Q1. Lower Limit: Q1 - 1.5(IQR) = 445 - 1.5(80) = 325 The upper limit is located 1.5(IQR) above Q3. Upper Limit: Q3 + 1.5(IQR) = 525 + 1.5(80) = 645 There are no outliers (values less than 325 or greater than 645) in the apartment rent data.

Box Plot Example: Apartment Rents Whiskers (dashed lines) are drawn from the ends of the box to the smallest and largest data values inside the limits. 400 425 450 475 500 525 550 575 600 625 Smallest value inside limits = 425 Largest value inside limits = 615

End of Chapter 3, Part B