Statistics for Business and Economics 7th Edition Chapter Describing Data: Numerical Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-1 Chapter Goals After completing this chapter, you should be able to:  Compute and interpret the mean, median, and mode for a set of data  Find the range, variance, standard deviation, and coefficient of variation and know what these values mean  Apply the empirical rule to describe the variation of population values around the mean  Explain the weighted mean and when to use it  Explain how a least squares regression line estimates a linear relationship between two variables Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-2 Chapter Topics  Measures of central tendency, variation, and shape      Mean, median, mode, geometric mean Quartiles Range, interquartile range, variance and standard deviation, coefficient of variation Symmetric and skewed distributions Population summary measures   Mean, variance, and standard deviation The empirical rule and Bienaymé-Chebyshev rule Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-3 Chapter Topics (continued)  Five number summary and box-and-whisker plots  Covariance and coefficient of correlation  Pitfalls in numerical descriptive measures and ethical considerations Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-4 Describing Data Numerically Describing Data Numerically Central Tendency Variation Arithmetic Mean Range Median Interquartile Range Mode Variance Standard Deviation Coefficient of Variation Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-5 2.1 Measures of Central Tendency Overview Central Tendency Mean Median Mode Midpoint of ranked values Most frequently observed value n x i x  i1 n Arithmetic average Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-6 Arithmetic Mean  The arithmetic mean (mean) is the most common measure of central tendency  For a population of N values: N x i x1  x    x N μ  N N i1 Population values Population size  For a samplen of size n: x x i1 n i x1  x    x n  n Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Observed values Sample size Ch 2-7 Arithmetic Mean (continued)    The most common measure of central tendency Mean = sum of values divided by the number of values Affected by extreme values (outliers) 10 Mean =     15  3 5 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall 10 Mean =     10 20  4 5 Ch 2-8 Median  In an ordered list, the median is the “middle” number (50% above, 50% below) 10 10 Median = Median =  Not affected by extreme values Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-9 Finding the Median  The location of the median: n 1 Median position  position in the ordered data    If the number of values is odd, the median is the middle number If the number of values is even, the median is the average of the two middle numbers n 1 Note that is not the value of the median, only the position of the median in the ranked data Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-10 Chebychev’s Theorem (continued)  Regardless of how the data are distributed, at least (1 - 1/k2) of the values will fall within k standard deviations of the mean (for k > 1)  Examples: At least within (1 - 1/1.52) = 55.6% …… k = 1.5 (μ ± 1.5σ) (1 - 1/22) = 75% … k = (μ ± 2σ) (1 - 1/32) = 89% …….… k = (μ ± 3σ) Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-42 The Empirical Rule   If the data distribution is bell-shaped, then the interval: μ 1σ contains about 68% of the values in the population or the sample 68% μ μ 1σ Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-43 The Empirical Rule   μ 2σ contains about 95% of the values in the population or the sample μ 3σ contains almost all (about 99.7%) of the values in the population or the sample 95% 99.7% μ 2σ μ 3σ Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-44 2.3  Weighted Mean The weighted mean of a set of data is n w x i x  i n Where wi is the weight of the ith observation w and n   w 1x1  w x    w n x n  n i1 i Use when data is already grouped into n classes, with wi values in the ith class Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-45 Approximations for Grouped Data Suppose data are grouped into K classes, with frequencies f1, f2, fK, and the midpoints of the classes are m1, m2, , mK  For a sample of n observations, the mean is K  fm i x i1 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall n i K where n  fi i1 Ch 2-46 Approximations for Grouped Data Suppose data are grouped into K classes, with frequencies f1, f2, fK, and the midpoints of the classes are m1, m2, , mK  For a sample of n observations, the variance is K f (m  x ) i i s2  i1 Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall n Ch 2-47 2.4 The Sample Covariance  The covariance measures the strength of the linear relationship between two variables  The population covariance: N  (x   i Cov (x , y)  xy  i1  x )(yi   y ) N The sample covariance: n  (x  x)(y  y) i Cov (x , y) s xy  i1   i n Only concerned with the strength of the relationship No causal effect is implied Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-48 Interpreting Covariance  Covariance between two variables: Cov(x,y) > x and y tend to move in the same direction Cov(x,y) < x and y tend to move in opposite directions Cov(x,y) = x and y are independent Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-49 Coefficient of Correlation  Measures the relative strength of the linear relationship between two variables  Population correlation coefficient: Cov (x , y) ρ σXσY  Sample correlation coefficient: Cov (x , y) r sX sY Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-50 Features of Correlation Coefficient, r  Unit free  Ranges between –1 and  The closer to –1, the stronger the negative linear relationship  The closer to 1, the stronger the positive linear relationship  The closer to 0, the weaker any positive linear relationship Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-51 Scatter Plots of Data with Various Correlation Coefficients Y Y r = -1 X Y Y r = -.6 X Y Y r = +1 X Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall r=0 X r = +.3 X r=0 X Ch 2-52 Using Excel to Find the Correlation Coefficient  Select Data / Data Analysis  Choose Correlation from the selection menu  Click OK Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-53 Using Excel to Find the Correlation Coefficient (continued)   Input data range and select appropriate options Click OK to get output Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-54 Interpreting the Result  r = 733  There is a relatively strong positive linear relationship between test score #1 and test score #2  Students who scored high on the first test tended to score high on second test Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-55 Chapter Summary  Described measures of central tendency   Illustrated the shape of the distribution    Symmetric, skewed Described measures of variation   Mean, median, mode Range, interquartile range, variance and standard deviation, coefficient of variation Discussed measures of grouped data Calculated measures of relationships between variables  covariance and correlation coefficient Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-56 .. .Chapter Goals After completing this chapter, you should be able to:  Compute and interpret the mean, median, and mode for a set of data  Find the range,... standard deviation The empirical rule and Bienaymé-Chebyshev rule Copyright © 2010 Pearson Education, Inc Publishing as Prentice Hall Ch 2-3 Chapter Topics (continued)  Five number summary and... the most common measure of central tendency  For a population of N values: N x i x1  x    x N μ  N N i1 Population values Population size  For a samplen of size n: x x i1 n i x1 