Subject: Biostatistics

Topic: Kurtosis Skewness Question:

COMPUTATION (10 pts final answer: 10 pts Solution) Calculate Sample Skewness, Sample Kurtosis from the following grouped data Class Frequency 1-2 4 2-4 4 4- 6 N 6-8 3Minimum o This number is the data value that is less than or equal to Day Height all other values in the ascending order. o the first number in the list 12 6.5 13 6.2 The Maximum 14 6.6 o This number is the data value that is greater than or equal to all other values in our set of data in the ascending order 15 7.1 o the last number in the list listed. 16 72 17 6.8 Average Height = 6.68 Limitations of the Maximum and Minimum 18 62 o The maximum and minimum are very sensitive to outliers. Minimum Height = 6.2 This is for the simple reason that if any value is added to a 19 6.4 data set that is less than the minimum, then the minimum 20 7.3 changes and it is this new value. In a similar way, if any Maximum Height = 7.3 value that exceeds the maximum is included in a data set, 21 7.1 then the maximum will change. 6.3 Many times the maximum or minimum are outliers of our ZZ data set. To determine if they indeed are outliers, we can 23 68 use the interquartile range rule. 24 64 https:/ /www.mathsisfun.com/definitions/maximum.html http://on.wikipedia.org/wil/Skewness Skewness a measure of the asymmetry of the probability distribution of a Mode real-valued random variable about its mean. Mode Mean- Mode o Two kinds negative skew: The left tail is longer; the mass of the distribution is concentrated on the right of the figure. o positive skew: The right tail is longer; the mass of the distribution is concentrated on the left of the figure. Positive Symmetrical Negative o A symmetrical distribution have a skewness of 0. Skew Distribution Skew If the skewness is between -0.5 and 0.5, the data are fairly Mean, Median >Mode Mean, Median (x-x) 3 Kurtosis =[(x-x)4 Sample Kurtosis from the Mean r=If SV_(x-x) 2 (n-1)-53 ("-1).54 following grouped data Sf 7-1 Class Frequency 34.56 299.792 V35.6 10 9.(1.9889) 3 9.(1.9889) 4 9 2 - 4 3 34.56 299.792 14 - 6 4 -5.2 =13.9556 9.7.867 9-15.6464 6 - 8 N =1.9889 =0.4881 =2.1289 8 - 10 lass Mid value (x) (x-x) S-(x-x)2 S.(x-x)3 (1) (2) (3 ) (4)=(2)*(3) (5) (6) (3)*(5) (7)=(5)* (6) 2 - 4 3 -2.2 14.52 -31.944 4 - 6 5 20 -0.2 0.16 -0.032 6 - 8 7 2 14 1.8 6.48 11.664 8 - 10 9 3.8 14.44 54.872 #-10 Sfox-52 ... -35.6 -34.56 Examples 1. Calculate Sample Skewness, Sample Standard deviation Skewness =>(x-x) 3 Kurtosis =[(xx)4 Sample Kurtosis from the Mean x=Efx S=VS(x-x) 2 (n-1).53 following grouped data 7-1 Class Frequency =V35.6 34.56 299.792 10 9.(1.9889) 9.(1.988 2 -4 3 34.56 299.792 4 - 6 4 -5.2 =V3.9556 9.7.867 9.15.6464 = term in data set 6 - 8 2 =1.9889 X = Sample meen =0.4881 =2.1289 I - Sum 8 - 10 (frequency x mid value) (mid value-mean) n = Sample size Class Mid value (x) f.x (x-X f-(x-x)2 f.(x-x)3 (1) (2) (3) (4)=(2)*(3) (5) (6)-(3)x(5) (7)=(5)*(6) 2 - 4 3 =2+4/2 9=3x3 -2.2= 3-5.21 14.52=3x-2.2x-2.2 -31.944=14.52x-2.2 (4)=(2)*(3) x-5.21 (6) (2)x(5) (7) (2) * (6) 4 - 6 5 =4+6/2 20 =4x5 -0.2= 15-5.21 0.16=4x-0.2x-0.2 -0.032 =0.16*-0.2 (4)=(2)*(3) |x-5.21 (6)=(2) x(5) (7)=(2) (6) 6 - 8 7 =6+8/2 14=2x7 2 1.8= 17-5.2 6.48=2x1.8x1.8 11.664=6.48x1.8 (4)=(2)x(3) x-5.2 (6)-(2)x(5) (7) (2)*(6)\f