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Chapter+3 - Chapter 3 Descriptive Statistics: Numerical...

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Unformatted text preview: Chapter 3 Descriptive Statistics: Numerical Measures Part A • Measures of Location • Measures of Variability Measures of Location • • • • • Mean Median Mode Percentiles Quartiles If the measures are computed for data from a sample, they are called sample statistics. If the measures are computed for data from a population, they are called population parameters. 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. x • The sample mean, , is the point estimator of the population mean, µ . Sample Mean x x= ∑x Sum of the values of the n observations i n Number of observations in the sample Population Mean µ µ= ∑x Sum of the values of the N observations i N 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 on the next slide. Sample Mean Apartment Rent Sample Data 445 440 465 450 600 570 510 615 440 450 470 485 515 575 430 440 525 490 580 450 490 590 525 450 472 470 445 435 435 425 450 475 490 525 600 600 445 460 475 500 535 435 460 575 435 500 549 475 445 600 445 460 480 500 550 435 440 450 465 570 500 480 430 615 450 480 465 480 510 440 Sample Mean ∑x x= 34, 356 = = 490.80 n 70 445 440 465 450 600 570 510 615 440 450 470 485 515 575 430 440 525 490 580 450 490 590 525 450 472 470 445 435 i 435 425 450 475 490 525 600 600 445 460 475 500 535 435 460 575 435 500 549 475 445 600 445 460 480 500 550 435 440 450 465 570 500 480 430 615 450 480 465 480 510 440 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 12 14 18 19 26 27 27 the median is the middle value. Median = 19 7 observations in ascending order Median For an even number of observations: 26 18 27 12 14 27 30 19 12 14 18 19 26 27 27 30 8 observations in ascending order the median is the average of the middle two values. Median = (19 + 26)/2 = 22.5 Median Averaging the 35th and 36th data values: Median = (475 + 475)/2 = 475 425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 Note: Data is in ascending order. 440 450 465 480 500 570 615 440 450 465 480 510 570 615 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. Mode 450 occurred most frequently (7 times) Mode = 450 425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 Note: Data is in ascending order. 440 450 465 480 500 570 615 440 450 465 480 510 570 615 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. 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. 80th Percentile i = (p/100)n = (80/100)70 = 56 Averaging the 56th and 57th data values: 80th Percentile = (535 + 549)/2 = 542 425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 Note: Data is in ascending order. 440 450 465 480 500 570 615 440 450 465 480 510 570 615 80 Percentile th “At least 80% of the items take on a value of 542 or less.” 56/70 = .8 or 80% 425 440 450 465 480 510 575 “At least 20% of the items take on a value of 542 or more.” 14/70 = .2 or 20% 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615 Quartiles Quartiles are specific percentiles. First Quartile = 25th Percentile Second Quartile = 50th Percentile = Median Third Quartile = 75th Percentile Third Quartile Third quartile = 75th percentile i = (p/100)n = (75/100)70 = 52.5 = 53 Third quartile = 525 425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 Note: Data is in ascending order. 440 450 465 480 500 570 615 440 450 465 480 510 570 615 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 Range = largest value ­ smallest value Range = 615 ­ 425 = 190 425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 Note: Data is in ascending order. 440 450 465 480 500 570 615 440 450 465 480 510 570 615 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 3rd Quartile (Q3) = 525 1st Quartile (Q1) = 445 Interquartile Range = Q3 ­ Q1 = 525 ­ 445 = 80 425 440 450 465 480 510 575 430 440 450 470 485 515 575 430 440 450 470 490 525 580 435 445 450 472 490 525 590 435 445 450 475 490 525 600 435 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 Note: Data is in ascending order. 440 450 465 480 500 570 615 440 450 465 480 510 570 615 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, x µ 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: ∑ ( xi − x ) 2 s2 = n −1 ∑ ( xi − µ ) σ= N 2 for a sample for a population 2 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: s = s2 σ = σ2 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: s ×100 % x σ ×100 % µ for a sample for a population Variance, Standard Deviation, And Coefficient of Variation • Variance s2 ∑( x = − x )2 i = 2, 996.16 n −1 • Standard Deviation s = s 2 = 2996.16 = 54.74 • Coefficient of Variation the standard deviation is about 11% of the mean s 54.74 × 100 % = × 100 % = 11.15% x 490.80 Chapter 3 Descriptive Statistics: Numerical Measures Part B • Measures of Distribution Shape, Relative Location, and Detecting Outliers • Exploratory Data Analysis • Measures of Association Between Two Variables • The Weighted Mean and Working with Grouped Data • • • • • Measures of Distribution Shape, Relative Location, and Detecting Outliers Distribution Shape z­Scores 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 s Symmetric (not skewed) • Skewness is zero. • Mean and median are equal. Relative Frequency .35 Skewness = 0 .30 .25 .20 .15 .10 .05 0 Distribution Shape: Skewness • Moderately Skewed Left – Skewness is negative. – Mean will usually be less than the median. Relative Frequency .35 Skewness = − .31 .31 .30 .25 .20 .15 .10 .05 0 Distribution Shape: Skewness • Moderately Skewed Right – Skewness is positive. – Mean will usually be more than the median. Relative Frequency .35 Skewness = .31 .30 .25 .20 .15 .10 .05 0 Distribution Shape: Skewness s Highly Skewed Right • Skewness is positive (often above 1.0). • Mean will usually be more than the median. Relative Frequency .35 Skewness = 1.25 .30 .25 .20 .15 .10 .05 0 Distribution Shape: Skewness • Example: Apartment Rents Seventy efficiency apartments were randomly sampled in a small college town. The monthly rent prices for these apartments are listed in ascending order on the next slide. Distribution Shape: Skewness • Example: Apartment 425 430 Rents 430 435 435 435 440 450 465 480 510 575 440 450 470 485 515 575 440 450 470 490 525 580 445 450 472 490 525 590 445 450 475 490 525 600 445 460 475 500 535 600 435 445 460 475 500 549 600 435 445 460 480 500 550 600 440 450 465 480 500 570 615 440 450 465 480 510 570 615 Distribution Shape: Skewness Relative Frequency .35 Skewness = .92 .30 .25 .20 .15 .10 .05 0 z­Scores The z­score is often called the standardized value. It denotes the number of standard deviations a data value xi is from the mean. xi − x zi = s z­Scores An observation’s z­score 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 z­score less than zero. A data value greater than the sample mean will have a z­score greater than zero. A data value equal to the sample mean will have a z­score of zero. z-Scores • z­Score of Smallest Value (425) x i − x 425 − 490.80 z= = = − 1.20 s 54.74 Standardized Values for Apartment Rents ­1.20 ­0.93 ­0.75 ­0.47 ­0.20 0.35 1.54 ­1.11 ­0.93 ­0.75 ­0.38 ­0.11 0.44 1.54 ­1.11 ­0.93 ­0.75 ­0.38 ­0.01 0.62 1.63 ­1.02 ­0.84 ­0.75 ­0.34 ­0.01 0.62 1.81 ­1.02 ­0.84 ­0.75 ­0.29 ­0.01 0.62 1.99 ­1.02 ­0.84 ­0.56 ­0.29 0.17 0.81 1.99 ­1.02 ­0.84 ­0.56 ­0.29 0.17 1.06 1.99 ­1.02 ­0.84 ­0.56 ­0.20 0.17 1.08 1.99 ­0.93 ­0.75 ­0.47 ­0.20 0.17 1.45 2.27 ­0.93 ­0.75 ­0.47 ­0.20 0.35 1.45 2.27 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 75% At least of the data values must be z = 2 standard deviations within of the mean. 89% At least of the data values must be z = 3 standard deviations within of the mean. 94% At least of the data values must be z = 4 standard deviations within of the mean. Chebyshev’s Theorem For example: x 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 x ­ z(s) = 490.80 − 1.5(54.74) = 409 and x + z(s) = 490.80 + 1.5(54.74) = 573 (Actually, 86% of the rent values are between 409 and 573.) Empirical Rule For data having a bell­shaped distribution: 68.26% of the values of a normal random variable +/­ 1 standard deviation are within of its mean. 95.44% of the values of a normal random variable +/­ 2 standard deviations are within of its mean. 99.72% of the values of a normal random variable +/­ 3 standard deviations are within of its mean. Empirical Rule 99.72% 95.44% 68.26% µ – 3σ µ – 1σ µ – 2σ µ µ + 3σ µ + 1σ µ + 2σ x Detecting Outliers An outlier is an unusually small or unusually large value in a data set. A data value with a z­score 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 Detecting Outliers The most extreme z­scores 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 ­1.20 ­0.93 ­0.75 ­0.47 ­0.20 0.35 1.54 ­1.11 ­0.93 ­0.75 ­0.38 ­0.11 0.44 1.54 ­1.11 ­0.93 ­0.75 ­0.38 ­0.01 0.62 1.63 ­1.02 ­0.84 ­0.75 ­0.34 ­0.01 0.62 1.81 ­1.02 ­0.84 ­0.75 ­0.29 ­0.01 0.62 1.99 ­1.02 ­0.84 ­0.56 ­0.29 0.17 0.81 1.99 ­1.02 ­0.84 ­0.56 ­0.29 0.17 1.06 1.99 ­1.02 ­0.84 ­0.56 ­0.20 0.17 1.08 1.99 ­0.93 ­0.75 ­0.47 ­0.20 0.17 1.45 2.27 ­0.93 ­0.75 ­0.47 ­0.20 0.35 1.45 2.27 Measures of Association Between Two Variables • Covariance • Correlation Coefficient Covariance The covariance is a measure of the linear association between two variables. Positive values indicate a positive relationship. Negative values indicate a negative relationship. Covariance The covariance is computed as follows: sxy = ∑ ( xi − x ) ( yi − y ) n −1 for samples σ xy ∑ ( xi − µ x ) ( yi − µ y ) = N for populations Correlation Coefficient The coefficient can take on values between ­1 and +1. Values near ­1 indicate a strong negative linear relationship. Values near +1 indicate a strong positive linear relationship. Correlation Coefficient The correlation coefficient is computed as follows: rxy = sxy ρ xy = sx s y for samples σ xy σ xσ y for populations Correlation Coefficient Correlation is a measure of linear association and not necessarily causation. Just because two variables are highly correlated, it does not mean that one variable is the cause of the other. Covariance and Correlation Coefficient A golfer is interested in investigating the relationship, if any, between driving distance and 18­hole score. Average Driving Distance (yds.) 277.6 259.5 269.1 267.0 255.6 272.9 Average 18­Hole Score 69 71 70 70 71 69 Covariance and Correlation Coefficient x y 277.6 259.5 269.1 267.0 255.6 272.9 69 71 70 70 71 69 ( x i − x ) ( y i − y ) ( x i − x )( y i − y ) 10.65 ­7.45 2.15 0.05 ­11.35 5.95 Average 267.0 70.0 Std. Dev. 8.2192 .8944 ­1.0 1.0 0 0 1.0 ­1.0 ­10.65 ­7.45 0 0 ­11.35 ­5.95 Total ­35.40 Covariance and Correlation Coefficient • Sample Covariance ∑ (x i − x )( y i − y ) = −35.40 = − 7.08 sxy = n−1 6−1 • Sample Correlation Coefficient sxy −7.08 rxy = = = ­.9631 sx sy (8.2192)(.8944) ...
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This note was uploaded on 08/28/2011 for the course BUS 300 taught by Professor White during the Spring '09 term at Rutgers.

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