Business Statistics, 5 th ed. by Ken Black

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1 Business Statistics, 5 th ed. by Ken Black Chapter 3 Discrete Distributions Descriptive Statistics PowerPoint presentations prepared by Lloyd Jaisingh, Morehead State University

2 Learning Objectives Distinguish between measures of central tendency, measures of variability, measures of shape, and measures of association. Understand the meanings of mean, median, mode, quartile, percentile, and range. Compute mean, median, mode, percentile, quartile, range, variance, standard deviation, and mean absolute deviation on ungrouped data. Differentiate between sample and population variance and standard deviation.

3 Learning Objectives -- Continued Understand the meaning of standard deviation as it is applied by using the empirical rule and Chebyshev s theorem. Compute the mean, mode, standard deviation, and variance on grouped data. Understand skewness, kurtosis, and box and whisker plots. Compute a coefficient of correlation and interpret it.

4 Measures of Central Tendency: Ungrouped Data Measures of central tendency yield information about the center, or middle part, of a group of numbers. Common Measures of central tendency Mode Median Mean Percentiles Quartiles

5 Mode The most frequently occurring value in a data set Applicable to all levels of data measurement (nominal, ordinal, interval, and ratio) Bimodal -- Data sets that have two modes Multimodal -- Data sets that contain more than two modes

6 Mode -- Example The mode is is the most frequently occurring data value

7 Median Middle value in an ordered array of numbers Applicable for ordinal, interval, and ratio data Not applicable for nominal data Unaffected by extremely large and extremely small values

8 Median: Computational Procedure First Procedure Arrange the observations in an ordered array. If there is an odd number of terms, the median is the middle term of the ordered array. If there is an even number of terms, the median is the average of the middle two terms. Second Procedure The median s position in an ordered array is given by (n+1)/.

9 Median: Example with an Odd Number of Terms Ordered Array There are 17 terms in the ordered array. Position of median = (n+1)/ = (17+1)/ = 9 The median is the 9th term, which is 15. If the is replaced by 100, the median is 15. If the 3 is replaced by -103, the median is 15.

10 Median: Example with an Even Number of Terms Ordered Array There are 16 terms in the ordered array. Position of median = (n+1)/ = (16+1)/ = 8.5 The median is between the 8th and 9th terms, If the 1 is replaced by 100, the median is If the 3 is replaced by -88, the median is 14.5.

11 Arithmetic Mean Commonly called the mean Is the average of a group of numbers Applicable for interval and ratio data Not applicable for nominal or ordinal data Affected by each value in the data set, including extreme values Computed by summing all values in the data set and dividing the sum by the number of values in the data set

12 Population Mean X N N X X X XN

13 Sample Mean X X n n X X X Xn

14 Percentiles Measures of central tendency that divide a group of data into 100 parts At least n% of the data lie below the nth percentile, and at most (100 - n)% of the data lie above the nth percentile Example: 90th percentile indicates that at least 90% of the data lie below it, and at most 10% of the data lie above it The median and the 50th percentile have the same value. Applicable for ordinal, interval, and ratio data Not applicable for nominal data

15 Percentiles: Computational Procedure Organize the data into an ascending ordered array. Calculate the percentile location: P i () n 100 Where P = percentile i= percentile location n= sample size Determine the percentile s location and its value. If i is a whole number, the percentile is the average of the values at the i and (i + 1) positions. If i is not a whole number, the percentile is at the whole number part of (i + 1) in the ordered array.

16 Percentiles: Example Raw Data: 14, 1, 19, 3, 5, 13, 8, 17 Ordered Array: 5, 1, 13, 14, 17, 19, 3, 8 Location of 30th percentile: 30 i 100 () 8 4. The location index, i, is not a whole number; i + 1 = = 3.4; the whole number portion is 3; the 30th percentile is at the 3rd location of the array; the 30th percentile is 13.

17 Quartiles Measures of central tendency that divide a group of data into four subgroups Q 1 : 5% of the data set is below the first quartile Q : 50% of the data set is below the second quartile Q 3 : 75% of the data set is below the third quartile Q 1 is equal to the 5th percentile Q is located at 50th percentile and equals the median Q 3 is equal to the 75th percentile Quartile values are not necessarily members of the data set

18 Ordered array: 106, 109, 114, 116, 11, 1, 15, 19 Q 1 5 Q : Q 3 : Quartiles: Example i i i Q () Q () Q ()

19 Variability No Variability in Cash Flow (same amounts) Mean Mean Variability in Cash Flow (different amounts) Mean Mean

20 Variability Variability No Variability

21 Measures of Variability: Ungrouped Data Measures of variability describe the spread or the dispersion of a set of data. Common Measures of Variability Range Interquartile Range Mean Absolute Deviation Variance Standard Deviation Z scores Coefficient of Variation

22 Range The difference between the largest and the smallest values in a set of data Simple to compute Ignores all data points except the two extremes Example: Range = 44 Largest - Smallest = =

23 Interquartile Range Range of values between the first and third quartiles Range of the middle 50% of the ordered data set Less influenced by extremes Interquartile Range Q3Q1

24 Deviation from the Mean Data set: 5, 9, 16, 17, 18 Mean: = 13 Deviations (x - ) from the mean: -8, -4, 3, 4,

25 Mean Absolute Deviation Average of the absolute deviations from the mean X X X M. A. D X N

26 Population Variance Average of the squared deviations from the arithmetic mean X X X X N

27 Population Standard Deviation Square root of the variance X N

28 Sample Variance Average of the squared deviations from the arithmetic mean X X X X X,398 1,844 1,539 1,311 7, X X 390,65 5,041 54,756 13, ,866 S n 1 663, ,

29 Sample Standard Deviation Square root of the sample variance X X S n 1 663, , S S 1,

30 Uses of Standard Deviation Indicator of financial risk Quality Control construction of quality control charts process capability studies Comparing populations household incomes in two cities employee absenteeism at two plants

31 Standard Deviation as an Indicator of Financial Risk Annualized Rate of Return Financial Security A 15% 3% B 15% 7%

32 Coefficient of Variation Ratio of the standard deviation to the mean, expressed as a percentage Measurement of relative dispersion CV 100

33 Coefficient of Variation CV CV

34 Empirical Rule Data are normally distributed (or approximately normal) Distance from the Mean 1 Percentage of Values Falling Within Distance

35 Chebyshev s Theorem Applies to all distributions P( k X k) 1 1 k for k>1

36 Chebyshev s Theorem Applies to all distributions Number of Standard Deviations K = Distance from the Mean Minimum Proportion of Values Falling Within Distance 1-1/ = 0.75 K = /3 = K = 4 1-1/4 = 0.94

37 Measures of Central Tendency and Variability: Grouped Data Measures of Central Tendency Mean Median Mode Measures of Variability Variance Standard Deviation

38 Mean of Grouped Data Weighted average of class midpoints Class frequencies are the weights fm f fm N f 1M 1 f M f 3M 3 fim f 1 f f 3 fi i

39 Calculation of Grouped Mean Class Interval Frequency Class Midpoint fm 0-under under under under under under fm 150 f

40 Median of Grouped Data N cf Median L fmed p W Where: L the lower limit of the median class cf p = cumulative frequency of class preceding the median class f med = frequency of the median class W = width of the median class N = total of frequencies

41 Median of Grouped Data -- Example Cumulative Class Interval Frequency Frequency 0-under under under under under under N = 50 Md L N cf fmed p W 10

42 Mode of Grouped Data Midpoint of the modal class Modal class has the greatest frequency Class Interval Frequency Mode 0-under under under under under under

43 Variance and Standard Deviation of Grouped Data Population Sample f M N S S S f M n 1 X

44 Population Variance and Standard Deviation of Grouped Data Class Interval 0-under under under under under under 80 f M fm M M f M f 700 N M

45 Measures of Shape Skewness Absence of symmetry Extreme values in one side of a distribution Kurtosis Peakedness of a distribution Leptokurtic: high and thin Mesokurtic: normal shape Platykurtic: flat and spread out Box and Whisker Plots Graphic display of a distribution Reveals skewness

46 Symmetrical and Skewness Symmetrical Right or Positively Skewed Left or Negatively Skewed

47 Relationship of Mean, Median and Mode

48 Relationship of Mean, Median and Mode

49 Relationship of Mean, Median and Mode

50 Coefficient of Skewness Summary measure for skewness S k 33 M d If S k < 0, the distribution is negatively skewed (skewed to the left). If S k = 0, the distribution is symmetric (not skewed). If S k > 0, the distribution is positively skewed (skewed to the right).

51 Coefficient of Skewness M 1 d1 3 6 M d 6 6 M 3 d S d1 M S 3 d M S d3 M

52 Types of Kurtosis Platykurtic Distribution Leptokurtic Distribution Mesokurtic Distribution

53 Box and Whisker Plot Five specific values are used: Median, Q First quartile, Q 1 Third quartile, Q 3 Minimum value in the data set Maximum value in the data set Inner Fences IQR = Q 3 -Q 1 Lower inner fence = Q IQR Upper inner fence = Q IQR Outer Fences Lower outer fence = Q IQR Upper outer fence = Q IQR

54 Box and Whisker Plot Minimum Q 1 Q Q 3 Maximum

55 Measures of Association Measures of association are statistics that yield information about the relatedness of numerical variables. Correlation is a measure of the degree of relatedness of variables.

56 Pearson Product-Moment Correlation Coefficient r SSXY SSX SSY X X Y Y X X Y Y XY X X Y n X n Y Y n 1r 1

57 Three Degrees of Correlation r < 0 r > 0 r = 0

58 Computation of r for the Economics Example (Part 1) Futures Day Interest X Index Y X Y XY ,841 1, ,84 1, ,076 1, ,65 1, ,176 1, ,79 1, ,79 1, ,076 1, ,076 1, ,5 1, ,89 1, ,081 1,98.00 Summations 9.93, ,07 1,115.07

59 Computation of r for the Economics Example (Part ) r X X Y Y XY X Y n n n ,...,..

60 Scatter Plot and Correlation Matrix for the Economics Example Futures Index Interest Interest Futures Index Interest 1 Futures Index

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