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Skewness & Kurtosis: Reference

Skewness & Kurtosis: Reference. Source: http://mathworld.wolfram.com/NormalDistribution.html. Further Moments – Skewness. Skewness measures the degree of asymmetry exhibited by the data If skewness equals zero, the histogram is symmetric about the mean

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Skewness & Kurtosis: Reference

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  1. Skewness & Kurtosis: Reference Source: http://mathworld.wolfram.com/NormalDistribution.html

  2. Further Moments – Skewness • Skewness measures the degree of asymmetry exhibited by the data • If skewness equals zero, the histogram is symmetric about the mean • Positive skewness vs negative skewness • Skewness measured in this way is sometimes referred to as “Fisher’s skewness”

  3. Further Moments – Skewness Source: http://library.thinkquest.org/10030/3smodsas.htm

  4. Mode Median Mean A B

  5. Median Mean n = 26 mean = 4.23 median = 3.5 mode = 8

  6. Value Occurrences Deviation Cubed deviation Occur*Cubed 1 1 (1 – 4.23) = -3.23 (-3.23)3 = -33.70 -33.70 2 4 (2 – 4.23) = -2.23 (-2.23)3 = -11.09 -44.36 3 8 (3 – 4.23) = -1.23 (-1.13)3 = -1.86 -14.89 4 4 (4 – 4.23) = -0.23 (-0.23)3 = -0.01 -0.05 5 3 (5 – 4.23) = 0.77 (+0.77)3 = 0.46 1.37 6 2 (6 – 4.23) = 1.77 (+1.77)3 = 5.54 11.09 7 1 (7 – 4.23) = 2.77 (+2.77)3 = 21.25 21.25 8 1 (8 – 4.23) = 3.77 (+3.77)3 = 53.58 53.58 9 1 (9 – 4.23) = 4.77 (+4.77)3 = 108.53 108.53 10 1 (10 - 4.23)= 5.77 (+5.77)3 = 192.10 192.10 Sum = 294.94 Mean = 4.23 s = 2.27 Skewness = 0.97

  7. Mode Median Mean Skewness > 0 (Positively skewed)

  8. Mode Median Mean A B Skewness < 0 (Negatively skewed)

  9. Source: http://mathworld.wolfram.com/NormalDistribution.html Skewness = 0 (symmetric distribution)

  10. Skewness – Review • Positive skewness • There are more observations below the mean than above it • When the mean is greater than the median • Negative skewness • There are a small number of low observations and a large number of high ones • When the median is greater than the mean

  11. Kurtosis – Review • Kurtosis measures how peaked the histogram is (Karl Pearson, 1905) • The kurtosis of a normal distribution is 0 • Kurtosis characterizes the relative peakedness or flatness of a distribution compared to the normal distribution

  12. Kurtosis – Review • Platykurtic– When the kurtosis < 0, the frequencies throughout the curve are closer to be equal (i.e., the curve is more flat and wide) • Thus, negative kurtosis indicates a relatively flat distribution • Leptokurtic– When the kurtosis > 0, there are high frequencies in only a small part of the curve (i.e, the curve is more peaked) • Thus, positive kurtosis indicates a relatively peaked distribution

  13. Source: http://espse.ed.psu.edu/Statistics/Chapters/Chapter3/Chap3.html

  14. Measures of central tendency – Review • Measures of the location of the middle or the center of a distribution • Mean • Median • Mode

  15. Mean – Review • Mean – Average value of a distribution; Most commonly used measure of central tendency • Median – This is the value of a variable such that half of the observations are above and half are below this value, i.e., this value divides the distribution into two groups of equal size • Mode - This is the most frequently occurring value in the distribution

  16. An Example Data Set • Daily low temperatures recorded in Chapel Hill (01/18-01/31, 2005, °F) Jan. 18 – 11 Jan. 25 – 25 Jan. 19 – 11 Jan. 26 – 33 Jan. 20 – 25 Jan. 27 – 22 Jan. 21 – 29 Jan. 28 – 18 Jan. 22 – 27 Jan. 29 – 19 Jan. 23 – 14 Jan. 30 – 30 Jan. 24 – 11 Jan. 31 – 27 • For these 14 values, we will calculate all three measures of central tendency - the mean, median, and mode

  17. Mean – Review • Mean –Most commonly used measure of central tendency • Procedures • (1) Sum all the values in the data set • (2) Divide the sum by the number of values in the data set • Watch for outliers

  18. Mean – Review • (1) Sum all the values in the data set  11 + 11 + 11 + 14 + 18 + 19 + 22 + 25 + 25 + 27 + 27 + 29 + 30 + 33 = 302 • (2) Divide the sum by the number of values in the data set  Mean= 302/14 = 21.57 • Is this a good measure of central tendency for this data set?

  19. Median – Review • Median -1/2of the values are above it & 1/2 below • (1) Sort the data in ascending order • (2) Find the value with an equal number of values above and below it • (3) Odd number of observations  [(n-1)/2]+1 value from the lowest • (4) Even number of observations  average (n/2) and [(n/2)+1] values • (5) Use the median with asymmetric distributions, particularly with outliers

  20. Median – Review • (1) Sort the data in ascending order:  11, 11, 11, 14, 18, 19, 22, 25, 25, 27, 27, 29, 30, 33 • (2) Find the value with an equal number of values above and below it Evennumber of observations  average the (n/2) and [(n/2)+1] values  (14/2) = 7; [(14/2)+1] = 8  (22+25)/2 = 23.5 (°F) • Is this a good measure of central tendency for this data?

  21. Mode – Review • Mode – This is the most frequently occurring value in the distribution • (1) Sort the data in ascending order • (2) Count the instances of each value • (3) Find the value that has the most occurrences • If more than one value occurs an equal number of times and these exceed all other counts, we have multiple modes • Use the mode for multi-modal data

  22. Mode – Review • (1) Sort the data in ascending order:  11, 11, 11, 14, 18, 19, 22, 25, 25, 27, 27, 29, 30, 33 • (2) Count the instances of each value:  11, 11, 11, 14, 18, 19, 22, 25, 25, 27, 27, 29, 30, 33 3x 1x 1x 1x 1x 2x 2x 1x 1x 1x • (3) Find the value that has the most occurrences mode = 11 (°F) • Is this a good measure of the central tendency of this data set?

  23. Measures of Dispersion – Review • In addition to measures of central tendency, we can also summarize data by characterizing its variability • Measures of dispersion are concerned with the distribution of values around the mean in data: • Range • Interquartile range • Variance • Standard deviation • z-scores • Coefficient of Variation (CV)

  24. An Example Data Set • Daily low temperatures recorded in Chapel Hill (01/18-01/31, 2005, °F) Jan. 18 – 11 Jan. 25 – 25 Jan. 19 – 11 Jan. 26 – 33 Jan. 20 – 25 Jan. 27 – 22 Jan. 21 – 29 Jan. 28 – 18 Jan. 22 – 27 Jan. 29 – 19 Jan. 23 – 14 Jan. 30 – 30 Jan. 24 – 11 Jan. 31 – 27 • For these 14 values, we will calculate all measures of dispersion

  25. Range – Review • Range – The difference between the largest and the smallest values • (1) Sort the data in ascending order • (2) Find the largest value  max • (3) Find the smallest value  min • (4) Calculate the range  range = max - min • Vulnerable to the influence of outliers

  26. Range – Review • Range – The difference between the largest and the smallest values • (1) Sort the data in ascending order  11, 11, 11, 14, 18, 19, 22, 25, 25, 27, 27, 29, 30, 33 • (2) Find the largest value  max = 33 • (3) Find the smallest value  min = 11 • (4) Calculate the range  range = 33 – 11 =22

  27. Interquartile Range – Review • Interquartile range – The difference between the 25th and 75th percentiles • (1) Sort the data in ascending order • (2) Find the 25th percentile – (n+1)/4 observation • (3) Find the 75th percentile – 3(n+1)/4 observation • (4) Interquartile range is the difference between these two percentiles

  28. Interquartile Range – Review • (1) Sort the data in ascending order  11, 11, 11, 14, 18, 19, 22, 25, 25, 27, 27, 29, 30, 33 • (2) Find the 25th percentile – (n+1)/4 observation  (14+1)/4 = 3.75  11+(14-11)*0.75 = 13.265 • (3) Find the 75th percentile– 3(n+1)/4 observation  3(14+1)/4 = 11.25  27+(29-27)*0.25 = 27.5 • (4) Interquartile range is the difference between these two percentiles  27.5 – 13.265 = 14.235

  29. Variance – Review • Variance is formulated as the sum of squares of statistical distances (or deviation) divided by the population size or the sample size minus one:

  30. Variance – Review • (1) Calculate the mean  • (2) Calculate the deviation for each value  • (3) Square each of the deviations  • (4) Sum the squared deviations  • (5) Divide the sum of squares by (n-1) for a sample 

  31. Variance – Review • (1) Calculate the mean  • (2) Calculate the deviation for each value  Jan. 18 (11 – 25.7) = -10.57 Jan. 25 (25 – 25.7) = 3.43 Jan. 19 (11 – 25.7) = -10.57 Jan. 26 (33 – 25.7) = 11.43 Jan. 20 (25 – 25.7) = 3.43 Jan. 27 (22 – 25.7) = 0.43 Jan. 21 (29 – 25.7) = 7.43 Jan. 28 (18 – 25.7) = -3.57 Jan. 22 (27 – 25.7) = 5.43 Jan. 29 (19 – 25.7) = -2.57 Jan. 23 (14 – 25.7) = -7.57 Jan. 30 (30 – 25.7) = 8.42 Jan. 24 (11 – 25.7) = -10.57 Jan. 31 (27 – 25.7) = 5.42

  32. Variance – Review • (3) Square each of the deviations  Jan. 18 (-10.57)^2 = 111.76 Jan. 25 (3.43)^2 = 11.76 Jan. 19 (-10.57)^2 = 111.76 Jan. 26 (11.43)^2 = 130.61 Jan. 20 (3.43)^2 = 11.76 Jan. 27 (0.43)^2 = 0.18 Jan. 21 (7.43)^2 = 55.18 Jan. 28 (-3.57)^2 = 12.76 Jan. 22 (5.43)^2 = 29.57 Jan. 29 (-2.57)^2 = 6.61 Jan. 23 (7.57)^2 = 57.33 Jan. 30 (8.43)^2 = 71.04 Jan. 24 (-10.57)^2 = 111.76 Jan. 31 (5.43)^2 = 29.57 • (4) Sum the squared deviations  = 751.43

  33. Variance – Review • (5) Divide the sum of squares by (n-1) for a sample  = 751.43 / (14-1) = 57.8 • The variance of the Tmin data set (Chapel Hill) is 57.8

  34. Standard Deviation – Review • Standard deviation is equal to the square root of the variance • Compared with variance, standard deviation has a scale closer to that used for the mean and the original data

  35. Standard Deviation – Review • (1) Calculate the mean  • (2) Calculate the deviation for each value  • (3) Square each of the deviations  • (4) Sum the squared deviations  • (5) Divide the sum of squares by (n-1) for a sample  • (6) Take the square root of the resulting variance 

  36. Standard Deviation – Review • (1) – (5)  s2 = 57.8 • (6) Take the square root of the variance  • The standard deviation (s) of the Tmin data set (Chapel Hill) is 7.6 (°F)

  37. z-score – Review • Since data come from distributions with different means and difference degrees of variability, it is common to standardize observations • One way to do this is to transform each observation into a z-score • May be interpreted as the number of standard deviations an observation is away from the mean

  38. z-scores – Review • z-score is the number of standard deviations an observation is away from the mean • (1) Calculate the mean  • (2) Calculate the deviation  • (3) Calculate the standard deviation  • (4) Divide the deviation by standard deviation 

  39. Z-score for maximum Tmin value (33 °F) (1) Calculate the mean  (2) Calculate the deviation  (3) Calculate the standard deviation (SD)  (4) Divide the deviation by standard deviation  z-scores – Review

  40. Coefficient of Variation – Review • Coefficient of variation (CV) measures the spread of a set of data as a proportion of its mean. • It is the ratio of the sample standard deviation to the sample mean • It is sometimes expressed as a percentage • There is an equivalent definition for the coefficient of variation of a population

  41. Coefficient of Variation – Review • (1) Calculate mean  • (2) Calculatestandard deviation  • (3) Dividestandard deviation by mean  CV =

  42. Coefficient of Variation – Review • (1) Calculate mean  • (2) Calculatestandard deviation  • (3) Dividestandard deviation by mean  CV =

  43. Histograms – Review • We may also summarize our data by constructing histograms, which are vertical bar graphs • A histogram is used to graphically summarize the distribution of a data set • A histogram divides the range of values in a data set into intervals • Over each interval is placed a bar whose height represents the percentage of data values in the interval.

  44. Building a Histogram – Review • (1)Develop an ungrouped frequency table  11, 11, 11, 14, 18, 19, 22, 25, 25, 27, 27, 29, 30, 33 

  45. Building a Histogram – Review • 2. Construct a grouped frequency table  Select a set of classes 

  46. Building a Histogram – Review • 3. Plot the frequencies of each class

  47. We can also use a box plot to graphically summarize a data set A box plot represents a graphical summary of what is sometimes called a “five-number summary” of the distribution Minimum Maximum 25th percentile 75th percentile Median Interquartile Range (IQR) 75th %-ile max. median 25th %-ile min. Rogerson, p. 8. Box Plots – Review

  48. Boxplot – Review

  49. Source: Earickson, RJ, and Harlin, JM. 1994. Geographic Measurement and Quantitative Analysis. USA: Macmillan College Publishing Co., p. 91. Further Moments of the Distribution • While measures of dispersion are useful for helping us describe the width of the distribution, they tell us nothing about the shape of the distribution

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