Statistical Measures of Variability Range: The difference between the highest and lowest values Interquartile Range (IQR): Middle 50% of the data
- 1. Chapter No. 3 variability in data.
- 2. 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
- 3. Variability Mean Mean Mean No Variability in Cash Flow Variability in Cash Flow Mean
- 5. 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
- 6. 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
- 7. 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
- 8. 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
- 9. 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 .
- 10. Population Standard Deviation • Square root of the variance 2 2 2 130 5 26 0 26 0 51 X N . . . 5 9 16 17 18 -8 -4 +3 +4 +5 0 64 16 9 16 25 130 X X 2 X
- 11. 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
- 12. • The Empirical Rule is a statement about normal distributions. Normal Distribution A specific type of symmetrical distribution, also known as a bell- shaped distribution Empirical Rule On a normal distribution about 68% of data will be within one standard deviation of the mean, about 95% will be within two standard deviations of the mean, and about 99.7% will be within three standard deviations of the mean
- 13. Example: Pulse Rates • Suppose the pulse rates of 200 college men are bell-shaped with a mean of 72 and standard deviation of 6. • About 68% of the men have pulse rates in the interval • About 95% of the men have pulse rates in the interval • About 99.7% of the men have pulse rates in the interval
- 14. Example: IQ scores • IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. • About 68% of individuals have IQ scores in the interval • About 95% of individuals have IQ scores in the interval • About 99.7% of individuals have IQ scores in the interval
- 15. Answers
- 16. 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 , , .
- 17. 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
- 18. Coefficient of Variation • Ratio of the standard deviation to the mean, expressed as a percentage • Measurement of relative dispersion C V . . 100
- 19. 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 . . .
- 20. 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
- 23. 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
- 24. 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 . . .
- 25. Kurtosis • Peakedness of a distribution – Leptokurtic: high and thin – Mesokurtic: normal in shape – Platykurtic: flat and spread out Leptokurtic Mesokurtic Platykurtic
- 26. 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
- 27. 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
- 28. Box and Whisker Plot Q1 Q3 Q2 Minimum Maximum
- 29. Skewness: Box and Whisker Plots, and Coefficient of Skewness Negatively Skewed Positively Skewed Symmetric (Not Skewed) S < 0 S = 0 S > 0
- 31. mean Add everything up and divide by the number of data points. Ex. Find the mean of the data set: 3, 4, 6, 8, 3
- 32. median • Middle point of the data • To calculate: – Put the values in order from smallest to largest – If there is an odd number of data points, value in the middle – If there is an even number of data points, average the two middle values • Ex. Find the median of the data set: 6, 3, 9, 1, 7, 8, 3, 5, 3, 2, 7, 3, 8, 4, 9, 10
- 33. mode • Value that occurs most frequently • If no value occurs more than once, then the data set has no mode • Data set can have more than one mode • Ex. Find the mode of the data set: 7, 4, 9, 2, 4, 3, 9, 1, 7, 8, 3, 2, 3, 3, 3, 5, 2
- 34. Example 1 • Data Set I Mean is ____ Median is ____ Mode is ____ • Data Set II Mean is ____ Median is ____ Mode is ____ Data Set I Data Set II
- 35. Solution • Data Set I Mean is 483.8461538 Median is 400 Mode is 300 Data Set II Mean is 474 Median is 350 Mode is 300 Data Set II Data Set I
- 36. Five Number Summary Used to construct a box-and-whisker plot 1.Minimum 2.Quartile 1 (Q1) – median of the lower data points 3.Median 4.Quartile 3 (Q3) – median of the upper data points 5.Maximum
- 37. Example 2 Use this set of data to make a box-and-whisker plot: 59, 27, 18, 78, 61, 91, 52, 34, 54, 93, 100, 87, 85, 82, 68 STEP 1: Write the numbers in numerical order, from smallest to greatest.
- 38. STEP 2: Create a five number summary. Minimum: Quartile 1: Median: Quartile 3: Max:
- 39. STEP 3: Construct and box-and-whisker plot. Minimum: Quartile 1: Median: Quartile 3: Max:
- 40. Example 3: You try! Use the following data set to create a five number summary and construct the box-and- whisker plot. 87, 7, 41, 50, 15, 220, 23, 99, 11, 45, 11, 61, 3, 39, 21
- 41. Solution Minimum: 3 Q1: 11 Median: 39 Q3: 61 Maximum: 220