Chapter 3 Descriptive Statistics: Numerical Methods

Chapter 3Descriptive Statistics: Numerical MethodsCopyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/IrwinDescriptive Statistics3.1 Describing Central Tendency3.2 Measures of Variation3.3 Percentiles, Quartiles and Box-and-Whiskers Displays3.4 Covariance, Correlation, and the Least Square Line (Optional)3.5 Weighted Means and Grouped Data (Optional)3.6 The Geometric Mean (Optional)3-*3.1 Describing Central TendencyIn addition to describing the shape of a distribution, want to describe the data set’s central tendencyA measure of central tendency represents the center or middle of the dataPopulation mean (μ) is average of the population measurementsPopulation parameter: a number calculated from all the population measurements that describes some aspect of the populationSample statistic: a number calculated using the sample measurements that describes some aspect of the sampleLO3-1: Compute and interpret the mean, median, and mode.3-*Measures of Central TendencyMean,  The average or expected value Median, Md The value of the middle point of the ordered measurementsMode, Mo The most frequent valueLO3-13-*3.2 Measures of VariationKnowing the measures of central tendency is not enoughBoth of the distributions below have identical measures of central tendencyLO3-2: Compute and interpret the range, variance, and standard deviation.Figure 3.133-*Measures of VariationRange Largest minus the smallest measurementVariance The average of the squared deviations of all the population measurements from the population meanStandard The square root of the populationDeviation varianceLO3-23-*The Empirical Rule for Normal PopulationsIf a population has mean µ and standard deviation σ and is described by a normal curve, then68.26% of the population measurements lie within one standard deviation of the mean: [µ-σ, µ+σ]95.44% lie within two standard deviations of the mean: [µ-2σ, µ+2σ]99.73% lie within three standard deviations of the mean: [µ-3σ, µ+3σ]LO3-3: Use the EmpiricalRule and Chebyshev’s Theorem to describe variation.3-*Chebyshev’s TheoremLet µ and σ be a population’s mean and standard deviation, then for any value k > 1At least 100(1 - 1/k2)% of the population measurements lie in the interval [µ-kσ, µ+kσ]Only practical for non-mound-shaped distribution population that is not very skewedLO3-33-*z ScoresFor any x in a population or sample, the associated z score isThe z score is the number of standard deviations that x is from the meanA positive z score is for x above (greater than) the meanA negative z score is for x below (less than) the meanLO3-33-*3.3 Percentiles, Quartiles, and Box-and-Whiskers Displays For a set of measurements arranged in increasing order, the pth percentile is a value such that p percent of the measurements fall at or below the value and (100-p) percent of the measurements fall at or above the valueThe first quartile Q1 is the 25th percentile The second quartile (median) is the 50th percentileThe third quartile Q3 is the 75th percentileThe interquartile range IQR is Q3 - Q1LO3-4: Compute and interpret percentiles, quartiles, and box-and-whiskers displays.3-*3.4 Covariance, Correlation, and the Least Squares Line (Optional)When points on a scatter plot seem to fluctuate around a straight line, there is a linear relationship between x and yA measure of the strength of a linear relationship is the covariance sxyLO3-5: Compute and interpret covariance, correlation, and the least squares line (Optional).3-*3.5 Weighted Means and Grouped Data (Optional)Sometimes, some measurements are more important than othersAssign numerical “weights” to the dataWeights measure relative importance of the valueCalculate weighted mean as where wi is the weight assigned to the ith measurement xiLO3-6: Compute and interpret weighted means and the mean and standard deviation of grouped data (Optional).3-*3.6 The Geometric Mean (Optional)For rates of return of an investment, use the geometric mean to give the correct wealth at the end of the investmentSuppose the rates of return (expressed as decimal fractions) are R1, R2, , Rn for periods 1, 2, , nThe mean of all these returns is the calculated as the geometric mean:LO3-7: Compute and interpret the geometric mean (Optional).3-*