2379-section9-1(Part2)

2379-section9-1(Part2) - MAT 2379 Introduction to...

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MAT 2379, Introduction to Biostatistics, Section 9.1 (Part 2) 1 MAT 2379, Introduction to Biostatistics Section 9.1. Random Sampling and Data Description Part 2: Descriptive Statistics Suppose we have a quantitative variable X whose observed values are x 1 , . . . , x n . A statistic is a function of these observed value. Descriptive statistics give a useful summary of the data. The most commonly used descriptive statistics are: a) the mean and the median b) minimum and maximum values, the quartiles Q 1 , Q 3 , and the interquartile range c) the standard deviation a) Measures of Center The sample mean ¯ x is the average of the n observations: ¯ x = x 1 + . . . + x n n = 1 n n X i =1 x i . If we put a mass of 1 /n on each observation x i , then ¯ x is the center of the mass. Example 1. (weight) Let X be the weight (in kg) of a randomly chosen 12-year girl. The following observations represent the weights of n = 10 girls: x 1 = 27 . 7 , x 2 = 31 . 5 x 3 = 30 . 9 x 4 = 29 . 6 x 5 = 27 . 0 x 6 = 38 . 1 x 7 = 32 . 4 x 8 = 31 . 1 x 9 = 36 . 7 x 10 = 28 . 4 The sample mean is: ¯ x = 27 . 7 + 31 . 5 + 30 . 9 + 29 . 6 + 27 . 0 + 38 . 1 + 32 . 4 + 31 . 1 + 36 . 7 + 28 . 4 10 = 31 . 34 The difference between a data point and its mean is called a deviation . The deviation of the i -th observation is x i - ¯ x . Note that: n X i =1 ( x i - ¯ x ) = 0 . . Home Exercise: Calculate the deviations in the previous example. Check that their sum is equal to 0. The median ˜ x is the sample value which divides the sample into two approximately equal-sized data subsets. To obtain the median, arrange the sample values x 1 , x 2 , . . . , x n in ascending order y 1 y 2 . . . y n . The median is given by: ˜ x = y { ( n +1) / 2 } , if n is odd ( y { n/ 2 } + y { n/ 2+1 } ) / 2 , if n is even Note that the median is a sample value if n is odd, but not necessarily a sample value if n is even. (When is a median a sample value if n is even?) Example 1.
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