Descriptive Statistics: Numerical Methods Chapter 3 Descriptive Statistics: Numerical Methods McGraw-Hill/Irwin Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.

Descriptive Statistics 3.1 Describing Central Tendency 3.2 Measures of Variation 3.3 Percentiles, Quartiles and Box-and- Whiskers Displays 3.4 Covariance, Correlation, and the Least Square Line (Optional) 3.5 Weighted Means and Grouped Data (Optional) 3.6 The Geometric Mean (Optional)

3.1 Describing Central Tendency LO3-1: Compute and interpret the mean, median, and mode. 3.1 Describing Central Tendency In addition to describing the shape of a distribution, want to describe the data set’s central tendency A measure of central tendency represents the center or middle of the data Population mean (μ) is average of the population measurements Population parameter: a number calculated from all the population measurements that describes some aspect of the population Sample statistic: a number calculated using the sample measurements that describes some aspect of the sample

Measures of Central Tendency LO3-1 Measures of Central Tendency Mean,  The average or expected value Median, Md The value of the middle point of the ordered measurements Mode, Mo The most frequent value

LO3-2: Compute and interpret the range, variance, and standard deviation. 3.2 Measures of Variation Knowing the measures of central tendency is not enough Both of the distributions below have identical measures of central tendency Figure 3.13

Measures of Variation Range Largest minus the smallest measurement LO3-2 Measures of Variation Range Largest minus the smallest measurement Variance The average of the squared deviations of all the population measurements from the population mean Standard The square root of the population Deviation variance

The Empirical Rule for Normal Populations LO3-3: Use the Empirical Rule and Chebyshev’s Theorem to describe variation. The Empirical Rule for Normal Populations If a population has mean µ and standard deviation σ and is described by a normal curve, then 68.26% 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 Chebyshev’s Theorem Let µ and σ be a population’s mean and standard deviation, then for any value k > 1 At 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 skewed

LO3-3 z Scores For any x in a population or sample, the associated z score is The z score is the number of standard deviations that x is from the mean A positive z score is for x above (greater than) the mean A negative z score is for x below (less than) the mean

3.3 Percentiles, Quartiles, and Box-and-Whiskers Displays LO3-4: Compute and interpret percentiles, quartiles, and box-and-whiskers displays. 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 value The first quartile Q1 is the 25th percentile The second quartile (median) is the 50th percentile The third quartile Q3 is the 75th percentile The interquartile range IQR is Q3 - Q1

3.4 Covariance, Correlation, and the Least Squares Line (Optional) LO3-5: Compute and interpret covariance, correlation, and the least squares line (Optional). 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 y A measure of the strength of a linear relationship is the covariance sxy

3.5 Weighted Means and Grouped Data (Optional) LO3-6: Compute and interpret weighted means and the mean and standard deviation of grouped data (Optional). 3.5 Weighted Means and Grouped Data (Optional) Sometimes, some measurements are more important than others Assign numerical “weights” to the data Weights measure relative importance of the value Calculate weighted mean as where wi is the weight assigned to the ith measurement xi

3.6 The Geometric Mean (Optional) LO3-7: Compute and interpret the geometric mean (Optional). 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 investment Suppose the rates of return (expressed as decimal fractions) are R1, R2, …, Rn for periods 1, 2, …, n The mean of all these returns is the calculated as the geometric mean: