2379-section9-1(Part3) - 8 s In general if x 1,x n are...

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MAT 2379, Introduction to Biostatistics, Section 9.1 (Part 3) 1 MAT 2379, Introduction to Biostatistics Section 9.1. Part 3: Transformation of variables Sometimes, it is useful to transform a data set into a new data set, using the same rule for transforming each data point. A linear transformation of a variable X is a new variable of the form: X 0 = aX + b Example 1. The following data set gives the temperatures for 4 women in degrees Celsius: x 1 = 36 . 23 , x 2 = 36 . 41 x 3 = 36 . 77 x 4 = 36 . 15 The sample mean, sample variance, and sample standard deviation are: ¯ x = 36 . 39 , s 2 = 0 . 076 , s = 0 . 2757 We would like to express the data in degrees Fahrenheit. For this, we use the linear transformation: X 0 = 1 . 8 X + 32 The new data points are: x 0 1 = 97 . 214 , x 0 2 = 97 . 538 x 0 3 = 98 . 186 x 0 4 = 97 . 07 The sample mean, sample variance, and sample standard deviation of the new sample are: ¯ x 0 = 97 . 502 , ( s 0 ) 2 = 0 . 246 , s 0 = 0 . 496 Note that: ¯ x 0 = 1 . x + 32 , ( s 0 ) 2 = (1 . 8) 2 s 2 , s 0 = 1 . 8 s In general, if
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Unformatted text preview: 8 s In general, if x 1 ,...,x n are observations from a variable X , and x 1 ,...,x n are the linearly transformed data points: x i = ax i + b, i = 1 ,...,n then ¯ x = a ¯ x + b, ( s ) 2 = a 2 s 2 , s = | a | s Note: In the case of non-linear transformations (e.g. X = √ X , X = e X , etc.), the previous property does not hold. A logarithmic transformation of a variable X is a new variable of the form: Y = ln( X ) Consider the logarithmic measurements y i = ln( x i ) ,i = 1 ,...,n . The geometric mean of the sample x 1 ,...,x n is defined by: g = e ¯ y , where ¯ y = 1 n n X i =1 y i . The geometric standard deviation of the sample x 1 ,...,x n is defined by: e s y , where s y = v u u t 1 n-1 n X i =1 ( y i-¯ y ) 2 ....
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