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Understanding Descriptive Statistics
1.
© 2002 Thomson
/ South-Western Slide 3-1 Chapter 3 Descriptive Statistics
2.
© 2002 Thomson
/ South-Western Slide 3-2 Learning Objectives • Distinguish between measures of central tendency, measures of variability, and measures of shape • 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
3.
© 2002 Thomson
/ South-Western Slide 3-3 Learning Objectives -- Continued • Differentiate between sample and population variance and standard deviation • Understand the meaning of standard deviation as it is applied by using the empirical rule • Understand box and whisker plots, skewness, and kurtosis
4.
© 2002 Thomson
/ South-Western Slide 3-4 Measures of Central Tendency • Measures of central tendency yield information about “particular places or locations in a group of numbers.” • Common Measures of Location –Mode –Median –Mean –Percentiles –Quartiles
5.
© 2002 Thomson
/ South-Western Slide 3-5 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.
© 2002 Thomson
/ South-Western Slide 3-6 • The mode is 44. • There are more 44s than any other value. 35 37 37 39 40 40 41 41 43 43 43 43 44 44 44 44 44 45 45 46 46 46 46 48 Mode -- Example
7.
© 2002 Thomson
/ South-Western Slide 3-7 Median (ΔΙΑΜΕΣΟΣ) • 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.
© 2002 Thomson
/ South-Western Slide 3-8 Median: Computational Procedure • First Procedure – Arrange observations in an ordered array. – If number of terms is odd, the median is the middle term of the ordered array. – If number of terms is even, 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)/2.
9.
© 2002 Thomson
/ South-Western Slide 3-9 Median: Example with an Odd Number of Terms Ordered Array includes: 3 4 5 7 8 9 11 14 15 16 16 17 19 19 20 21 22 • There are 17 terms in the ordered array. • Position of median = (n+1)/2 = (17+1)/2 = 9 • The median is the 9th term, 15. • If the 22 is replaced by 100, the median remains at 15. • If the 3 is replaced by -103, the median remains at 15.
10.
© 2002 Thomson
/ South-Western Slide 3-10 Mean (ΜΕΣΟΣ) • 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
11.
© 2002 Thomson
/ South-Western Slide 3-11 Population Mean X N N X X X XN 1 2 3 24 13 19 26 11 5 93 5 18 6 ... .
12.
© 2002 Thomson
/ South-Western Slide 3-12 Sample Mean X X n n X X X Xn 1 2 3 57 86 42 38 90 66 6 379 6 63167 ... .
13.
© 2002 Thomson
/ South-Western Slide 3-13 Quartiles Measures of central tendency that divide a group of data into four subgroups • Q1: 25% of the data set is below the first quartile • Q2: 50% of the data set is below the second quartile • Q3: 75% of the data set is below the third quartile
14.
© 2002 Thomson
/ South-Western Slide 3-14 Quartiles, continued • Q1 is equal to the 25th percentile • Q2 is located at 50th percentile and equals the median • Q3 is equal to the 75th percentile Quartile values are not necessarily members of the data set
15.
© 2002 Thomson
/ South-Western Slide 3-15 Quartiles 25% 25% 25% 25% Q3 Q2 Q1
16.
© 2002 Thomson
/ South-Western Slide 3-16 • Ordered array: 106, 109, 114, 116, 121, 122, 125, 129 • Q1: • Q2: • Q3: Quartiles: Example i Q 25 100 8 2 109 114 2 1115 1 ( ) . i Q 50 100 8 4 116 121 2 1185 2 ( ) . i Q 75 100 8 6 122 125 2 1235 3 ( ) .
17.
© 2002 Thomson
/ South-Western Slide 3-17 Measures of Variability • 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
18.
© 2002 Thomson
/ South-Western Slide 3-18 Variability Mean Mean Mean No Variability in Cash Flow Variability in Cash Flow Mean
19.
© 2002 Thomson
/ South-Western Slide 3-19 Variability No Variability Variability
20.
© 2002 Thomson
/ South-Western Slide 3-20 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 = Largest - Smallest = 48 - 35 = 13 35 37 37 39 40 40 41 41 43 43 43 43 44 44 44 44 44 45 45 46 46 46 46 48
21.
© 2002 Thomson
/ South-Western Slide 3-21 Interquartile Range • Range of values between the first and third quartiles • Range of the “middle half” • Less influenced by extremes Interquartile Range Q Q 3 1
22.
© 2002 Thomson
/ South-Western Slide 3-22 Deviation from the Mean • Data set: 5, 9, 16, 17, 18 • Mean: • Deviations from the mean: -8, -4, 3, 4, 5 X N 65 5 13 0 5 10 15 20 -8 -4 +3 +4 +5
23.
© 2002 Thomson
/ South-Western Slide 3-23 Mean Absolute Deviation • Average of the absolute deviations from the mean 5 9 16 17 18 -8 -4 +3 +4 +5 0 +8 +4 +3 +4 +5 24 X X X M A D X N . . . . 24 5 4 8
24.
© 2002 Thomson
/ South-Western Slide 3-24 Population Variance • Average of the squared deviations from the arithmetic mean 5 9 16 17 18 -8 -4 +3 +4 +5 0 64 16 9 16 25 130 X X 2 X 2 2 130 5 26 0 X N .
25.
© 2002 Thomson
/ South-Western Slide 3-25 Population Standard Deviation • Square root of the variance 2 2 2 130 5 26 0 26 0 5 1 X N . . . 5 9 16 17 18 -8 -4 +3 +4 +5 0 64 16 9 16 25 130 X X 2 X
26.
© 2002 Thomson
/ South-Western Slide 3-26 Empirical Rule • Data are normally distributed (or approximately normal) 1 2 3 95 99.7 68 Distance from the Mean Percentage of Values Falling Within Distance
27.
© 2002 Thomson
/ South-Western Slide 3-27 Sample Variance • Average of the squared deviations from the arithmetic mean 2,398 1,844 1,539 1,311 7,092 625 71 -234 -462 0 390,625 5,041 54,756 213,444 663,866 X X X 2 X X 2 2 1 663 866 3 221 288 67 S X X n , , .
28.
© 2002 Thomson
/ South-Western Slide 3-28 Sample Standard Deviation • Square root of the sample variance 2 2 2 1 663 866 3 221 288 67 221 288 67 470 41 S X X S n S , , . , . . 2,398 1,844 1,539 1,311 7,092 625 71 -234 -462 0 390,625 5,041 54,756 213,444 663,866 X X X 2 X X
29.
© 2002 Thomson
/ South-Western Slide 3-29 Coefficient of Variation • Ratio of the standard deviation to the mean, expressed as a percentage • Measurement of relative dispersion C V . . 100
30.
© 2002 Thomson
/ South-Western Slide 3-30 Coefficient of Variation 1 29 4 6 100 4 6 29 100 1586 1 1 1 1 . . . . . CV 2 84 10 100 10 84 100 1190 2 2 2 2 CV . . .
31.
© 2002 Thomson
/ South-Western Slide 3-31 Measures of Shape • Skewness – Absence of symmetry – Extreme values in one side of a distribution • Kurtosis – Peakedness of a distribution • Box and Whisker Plots – Graphic display of a distribution – Reveals skewness
32.
© 2002 Thomson
/ South-Western Slide 3-32 Skewness Negatively Skewed Positively Skewed Symmetric (Not Skewed)
33.
© 2002 Thomson
/ South-Western Slide 3-33 Skewness Negatively Skewed Mode Median Mean Symmetric (Not Skewed) Mean Median Mode Positively Skewed Mode Median Mean
34.
© 2002 Thomson
/ South-Western Slide 3-34 Coefficient of Skewness • Summary measure for skewness • If S < 0, the distribution is negatively skewed (skewed to the left). • If S = 0, the distribution is symmetric (not skewed). • If S > 0, the distribution is positively skewed (skewed to the right). S Md 3
35.
© 2002 Thomson
/ South-Western Slide 3-35 Coefficient of Skewness 1 1 1 1 1 1 1 23 26 12 3 3 3 23 26 12 3 073 M S M d d . . . 2 2 2 2 2 2 2 26 26 12 3 3 3 26 26 12 3 0 M S M d d . . 3 3 3 3 3 3 3 29 26 12 3 3 3 29 26 12 3 073 M S M d d . . .
36.
© 2002 Thomson
/ South-Western Slide 3-36 Kurtosis • Peakedness of a distribution – Leptokurtic: high and thin – Mesokurtic: normal in shape – Platykurtic: flat and spread out Leptokurtic Mesokurtic Platykurtic
37.
© 2002 Thomson
/ South-Western Slide 3-37 Box and Whisker Plot • Five specific values are used: –Median, Q2 –First quartile, Q1 –Third quartile, Q3 –Minimum value in the data set –Maximum value in the data set
38.
© 2002 Thomson
/ South-Western Slide 3-38 Box and Whisker Plot, continued • Inner Fences – IQR = Q3 - Q1 – Lower inner fence = Q1 - 1.5 IQR – Upper inner fence = Q3 + 1.5 IQR • Outer Fences – Lower outer fence = Q1 - 3.0 IQR – Upper outer fence = Q3 + 3.0 IQR
39.
© 2002 Thomson
/ South-Western Slide 3-39 Box and Whisker Plot Q1 Q3 Q2 Minimum Maximum
40.
© 2002 Thomson
/ South-Western Slide 3-40 Skewness: Box and Whisker Plots, and Coefficient of Skewness Negatively Skewed Positively Skewed Symmetric (Not Skewed) S < 0 S = 0 S > 0
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