4A-1

Descriptive Statistics (Part 1) Numerical Description Numerical Description Central Tendency Central Tendency Dispersion Chapter 4A4A McGraw-Hill/Irwin© 2008 The McGraw-Hill Companies, Inc. All rights reserved.

4A-3 Statistics are descriptive measures derived from a sample (n items).Statistics are descriptive measures derived from a sample (n items). Parameters are descriptive measures derived from a population (N items).Parameters are descriptive measures derived from a population (N items). Numerical Description

4A-4 Three key characteristics of numerical data:Three key characteristics of numerical data: CharacteristicInterpretation Central Tendency Where are the data values concentrated? What seem to be typical or middle data values? Numerical Description Dispersion How much variation is there in the data? How spread out are the data values? Are there unusual values? Shape Are the data values distributed symmetrically? Skewed? Sharply peaked? Flat? Bimodal?

4A-5 Numerical statistics can be used to summarize this random sample of brands.Numerical statistics can be used to summarize this random sample of brands. Defect rate = total no. defectsDefect rate = total no. defects no. inspected x 100 Must allow for sampling error since the analysis is based on sampling.Must allow for sampling error since the analysis is based on sampling. Numerical Description  Example: Vehicle Quality Consider the data set of vehicle defect rates from J. D. Power and Associates.Consider the data set of vehicle defect rates from J. D. Power and Associates.

4A-6 Numerical Description Number of defects per 100 vehicles, 1004 models.Number of defects per 100 vehicles, 1004 models.

4A-7 Sorted data provides insight into central tendency and dispersion.Sorted data provides insight into central tendency and dispersion. Numerical Description

4A-8 The dot plot offers a visual impression of the data.The dot plot offers a visual impression of the data.  Visual Displays Numerical Description

4A-9 The central tendency is the middle or typical values of a distribution.The central tendency is the middle or typical values of a distribution. Central tendency can be assessed using a dot plot, histogram or more precisely with numerical statistics.Central tendency can be assessed using a dot plot, histogram or more precisely with numerical statistics. Central Tendency

4A-10 StatisticFormula Excel Formula ProCon Mean=AVERAGE(Data) Familiar and uses all the sample information. Influenced by extreme values. Central Tendency  Six Measures of Central Tendency Median Middle value in sorted array =MEDIAN(Data) Robust when extreme data values exist. Ignores extremes and can be affected by gaps in data values.

4A-11 StatisticFormula Excel Formula ProCon Mode Most frequently occurring data value =MODE(Data) Useful for attribute data or discrete data with a small range. May not be unique, and is not helpful for continuous data. Central Tendency  Six Measures of Central Tendency Midrange=0.5*(MIN(Data)+MAX(Data)) Easy to understand and calculate. Influenced by extreme values and ignores most data values.

4A-12 StatisticFormula Excel Formula ProCon Geometric mean (G) =GEOMEAN(Data) Useful for growth rates and mitigates high extremes. Less familiar and requires positive data. Trimmed mean Same as the mean except omit highest and lowest k% of data values (e.g., 5%) =TRMEAN(Data, %) Mitigates effects of extreme values. Excludes some data values that could be relevant. Central Tendency  Six Measures of Central Tendency

4A-13 A familiar measure of central tendency.A familiar measure of central tendency. In Excel, use function =AVERAGE(Data) where Data is an array of data values.In Excel, use function =AVERAGE(Data) where Data is an array of data values. Population FormulaSample Formula Central Tendency  Mean

4A-14 For the sample of n = 37 car brands:For the sample of n = 37 car brands: Central Tendency  Mean

4A-15 Arithmetic mean is the most familiar average.Arithmetic mean is the most familiar average. Affected by every sample item.Affected by every sample item. The balancing point or fulcrum for the data.The balancing point or fulcrum for the data. Central Tendency  Characteristics of the Mean

4A-16 Regardless of the shape of the distribution, absolute distances from the mean to the data points always sum to zero.Regardless of the shape of the distribution, absolute distances from the mean to the data points always sum to zero. Central Tendency  Characteristics of the Mean Consider the following asymmetric distribution of quiz scores whose mean = 65.Consider the following asymmetric distribution of quiz scores whose mean = 65. = (42 – 65) + (60 – 65) + (70 – 65) + (75 – 65) + (78 – 65) = (-23) + (-5) + (5) + (10) + (13) = = 0

4A-17 The median (M) is the 50 th percentile or midpoint of the sorted sample data.The median (M) is the 50 th percentile or midpoint of the sorted sample data. M separates the upper and lower half of the sorted observations.M separates the upper and lower half of the sorted observations. If n is odd, the median is the middle observation in the data array.If n is odd, the median is the middle observation in the data array. If n is even, the median is the average of the middle two observations in the data array.If n is even, the median is the average of the middle two observations in the data array. Central Tendency  Median

4A-18 Central Tendency  Median For n = 8, the median is between the fourth and fifth observations in the data array.For n = 8, the median is between the fourth and fifth observations in the data array.

4A-19 Central Tendency  Median For n = 9, the median is the fifth observation in the data array.For n = 9, the median is the fifth observation in the data array.

4A-20 Consider the following n = 6 data values: Consider the following n = 6 data values: What is the median?What is the median? M = (x 3 +x 4 )/2 = (15+17)/2 = For even n, Median = n/2 = 6/2 = 3 and n/2+1 = 6/2 + 1 = 4 Central Tendency  Median

4A-21 Consider the following n = 7 data values: Consider the following n = 7 data values: What is the median?What is the median? M = x 4 = For odd n, Median = (n+1)/2 = (7+1)/2 = 8/2 = 4 Central Tendency  Median

4A-22 Use Excel’s function =MEDIAN(Data) where Data is an array of data values.Use Excel’s function =MEDIAN(Data) where Data is an array of data values. For the 37 vehicle quality ratings (odd n) the position of the median is (n+1)/2 = (37+1)/2 = 19.For the 37 vehicle quality ratings (odd n) the position of the median is (n+1)/2 = (37+1)/2 = 19. So, the median is x 19 = 121.So, the median is x 19 = 121. When there are several duplicate data values, the median does not provide a clean “50-50” split in the data.When there are several duplicate data values, the median does not provide a clean “50-50” split in the data. Central Tendency  Median

4A-23 The median is insensitive to extreme data values.The median is insensitive to extreme data values. For example, consider the following quiz scores for 3 students:For example, consider the following quiz scores for 3 students: Tom’s scores: 20, 40, 70, 75, 80 Mean =57, Median = 70, Total = 285 Jake’s scores: 60, 65, 70, 90, 95 Mean = 76, Median = 70, Total = 380 Mary’s scores: 50, 65, 70, 75, 90 Mean = 70, Median = 70, Total = 350 What does the median for each student tell you?What does the median for each student tell you? Central Tendency  Characteristics of the Median

4A-24 The most frequently occurring data value.The most frequently occurring data value. Similar to mean and median if data values occur often near the center of sorted data.Similar to mean and median if data values occur often near the center of sorted data. May have multiple modes or no mode.May have multiple modes or no mode. Central Tendency  Mode

4A-25 Lee’s scores: 60, 70, 70, 70, 80Mean =70, Median = 70, Mode = 70 Pat’s scores: 45, 45, 70, 90, 100Mean = 70, Median = 70, Mode = 45 Sam’s scores: 50, 60, 70, 80, 90Mean = 70, Median = 70, Mode = none Xiao’s scores: 50, 50, 70, 90, 90Mean = 70, Median = 70, Modes = 50,90 Central Tendency  Mode For example, consider the following quiz scores for 3 students:For example, consider the following quiz scores for 3 students: What does the mode for each student tell you?What does the mode for each student tell you?

4A-26 Easy to define, not easy to calculate in large samples.Easy to define, not easy to calculate in large samples. Use Excel’s function =MODE(Array) - will return #N/A if there is no mode. - will return first mode found if multimodal.Use Excel’s function =MODE(Array) - will return #N/A if there is no mode. - will return first mode found if multimodal. May be far from the middle of the distribution and not at all typical.May be far from the middle of the distribution and not at all typical. Central Tendency  Mode

4A-27 Generally isn’t useful for continuous data since data values rarely repeat.Generally isn’t useful for continuous data since data values rarely repeat. Best for attribute data or a discrete variable with a small range (e.g., Likert scale).Best for attribute data or a discrete variable with a small range (e.g., Likert scale). Central Tendency  Mode

4A-28 Consider the following P/E ratios for a random sample of 68 Standard & Poor’s 500 stocks.Consider the following P/E ratios for a random sample of 68 Standard & Poor’s 500 stocks. What is the mode?What is the mode? Central Tendency  Example: Price/Earnings Ratios and Mode

4A-29 Distribution’s Shape Histogram Appearance Statistics Skewed left (negative skewness) Long tail of histogram points left (a few low values but most data on right) Mean < Median Central Tendency  Symptoms of Skewness Symmetric Tails of histogram are balanced (low/high values offset) (low/high values offset) Mean  Median Skewed right (positive skewness) Long tail of histogram points right (most data on left but a few high values) Mean > Median

4A-30 The geometric mean (G) is a multiplicative average.The geometric mean (G) is a multiplicative average. For the J. D. Power quality data (n=37):For the J. D. Power quality data (n=37): In Excel use =GEOMEAN(Array)In Excel use =GEOMEAN(Array) The geometric mean tends to mitigate the effects of high outliers.The geometric mean tends to mitigate the effects of high outliers. Central Tendency  Geometric Mean

4A-31 The midrange is the point halfway between the lowest and highest values of X.The midrange is the point halfway between the lowest and highest values of X. Easy to use but sensitive to extreme data values.Easy to use but sensitive to extreme data values. Midrange = For the J. D. Power quality data (n=37):For the J. D. Power quality data (n=37): Midrange = = Here, the midrange (130) is higher than the mean (125.38) or median (121).Here, the midrange (130) is higher than the mean (125.38) or median (121). Central Tendency  Midrange

4A-32 To calculate the trimmed mean, first remove the highest and lowest k percent of the observations.To calculate the trimmed mean, first remove the highest and lowest k percent of the observations. For example, for the n = 68 P/E ratios, we want a 5 percent trimmed mean (i.e., k =.05).For example, for the n = 68 P/E ratios, we want a 5 percent trimmed mean (i.e., k =.05). To determine how many observations to trim, multiply k x n = 0.05 x 68 = 3.4 or 3 observations.To determine how many observations to trim, multiply k x n = 0.05 x 68 = 3.4 or 3 observations. So, we would remove the three smallest and three largest observations before averaging the remaining values.So, we would remove the three smallest and three largest observations before averaging the remaining values. Central Tendency  Trimmed Mean

4A-33 Here is a summary of all the measures of central tendency for the n = 68 P/E values.Here is a summary of all the measures of central tendency for the n = 68 P/E values. The trimmed mean mitigates the effects of very high values, but still exceeds the median.The trimmed mean mitigates the effects of very high values, but still exceeds the median. Mean:22.72 =AVERAGE(PERatio) =AVERAGE(PERatio) Median:19.00 =MEDIAN(PERatio) =MEDIAN(PERatio) Mode:13.00 =MODE(PERatio) =MODE(PERatio) Geometric Mean: =GEOMEAN(PERatio) =GEOMEAN(PERatio) Midrange:49.00 =(MIN(PERatio)+MAX(PERatio))/2 =(MIN(PERatio)+MAX(PERatio))/2 5% Trim Mean: =TRIMMEAN(PERatio,0.1) =TRIMMEAN(PERatio,0.1) Central Tendency  Trimmed Mean

4A-34 Variation is the “spread” of data points about the center of the distribution in a sample. Consider the following measures of dispersion:Variation is the “spread” of data points about the center of the distribution in a sample. Consider the following measures of dispersion: StatisticFormulaExcelProCon Range x max – x min =MAX(Data)- MIN(Data) Easy to calculate Sensitive to extreme data values. DispersionDispersion Variance (s 2 ) =VAR(Data) Plays a key role in mathematical statistics. Non-intuitive meaning.  Measures of Variation

4A-35 StatisticFormulaExcelProCon Standard deviation (s) =STDEV(Data) Most common measure. Uses same units as the raw data ($, £, ¥, etc.). Non-intuitive meaning. DispersionDispersion  Measures of Variation Coef- ficient. of variation (CV) None Measures relative variation in percent so can compare data sets. Requires non- negative data.

4A-36 StatisticFormulaExcelProCon Mean absolute deviation (MAD) =AVEDEV(Data) Easy to understand. Lacks “nice” theoretical properties. DispersionDispersion  Measures of Variation

4A-37 The difference between the largest and smallest observation.The difference between the largest and smallest observation. Range = x max – x min For example, for the n = 68 P/E ratios,For example, for the n = 68 P/E ratios, Range = 91 – 7 = 84 DispersionDispersion  Range

4A-38 The population variance (  2 ) is defined as the sum of squared deviations around the mean  divided by the population size.The population variance (  2 ) is defined as the sum of squared deviations around the mean  divided by the population size. For the sample variance (s 2 ), we divide by n – 1 instead of n, otherwise s 2 would tend to underestimate the unknown population variance  2.For the sample variance (s 2 ), we divide by n – 1 instead of n, otherwise s 2 would tend to underestimate the unknown population variance  2. DispersionDispersion  Variance

4A-39 The square root of the variance.The square root of the variance. Units of measure are the same as X.Units of measure are the same as X. Population standard deviation Sample standard deviation Explains how individual values in a data set vary from the mean.Explains how individual values in a data set vary from the mean. DispersionDispersion  Standard Deviation

4A-40 Excel’s built in functions areExcel’s built in functions are Statistic Excel population formula Excel sample formula Variance=VARP(Array)=VAR(Array) Standard deviation =STDEVP(Array)=STDEV(Array) DispersionDispersion  Standard Deviation

4A-41 Consider the following five quiz scores for Stephanie.Consider the following five quiz scores for Stephanie. DispersionDispersion  Calculating a Standard Deviation

4A-42 Now, calculate the sample standard deviation:Now, calculate the sample standard deviation: Somewhat easier, the two-sum formula can also be used:Somewhat easier, the two-sum formula can also be used: DispersionDispersion  Calculating a Standard Deviation

4A-43 The standard deviation is nonnegative because deviations around the mean are squared.The standard deviation is nonnegative because deviations around the mean are squared. When every observation is exactly equal to the mean, the standard deviation is zero.When every observation is exactly equal to the mean, the standard deviation is zero. Standard deviations can be large or small, depending on the units of measure.Standard deviations can be large or small, depending on the units of measure. Compare standard deviations only for data sets measured in the same units and only if the means do not differ substantially.Compare standard deviations only for data sets measured in the same units and only if the means do not differ substantially. DispersionDispersion  Calculating a Standard Deviation