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STA 3032 (7661) Engineering Statistics Rob Gordon University of Florida Fall 2011 Rob Gordon (University of Florida) STA 3032 (7661) Fall 2011 1 / 251
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Introduction: Sampling & Descriptive Statistics Definition A population is the entire collection of objects or outcomes about which information is sought. Examples: Entire United States Entire State of Florida All UF students Definition A sample is a subset of the population, containing the objects or outcomes that are actually observed. Rob Gordon (University of Florida) STA 3032 (7661) Fall 2011 2 / 251
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Introduction: Sampling & Descriptive Statistics A common question might be: “ How do I know if a sample is truly representative of its population? Ideally, the best way to accomplish this goal is to select the members of the sample in the most unbiased possible way. Throughout the rest of this course we will assume our samples will follow the definition of a simple random sample: Definition A simple random sample of size n is a sample chosen by a method in which each collection of N population items is equally likely to comprise a sample (as in a lottery). Rob Gordon (University of Florida) STA 3032 (7661) Fall 2011 3 / 251
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Introduction: Sampling & Descriptive Statistics “Summary Statistics” help the important features of a sample stand out. Definition Let x 1 , x 2 , . . . , x n denote numbers in a sample. The value of the sample mean is ¯ x = 1 n n X i =1 x i . Definition The value of the sample variance s 2 = 1 n - 1 n X i =1 ( x i - ¯ x ) 2 = 1 n - 1 n X i =1 x 2 i - n ¯ x 2 ! Definition The value of the sample standard deviation is s = s 2 . Rob Gordon (University of Florida) STA 3032 (7661) Fall 2011 4 / 251
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Introduction: Sampling & Descriptive Statistics Definition Outliers are points in the sample that are much smaller/larger than the rest. Outliers often result from data entry errors (e.g. incorrect decimal place) and can present many problems for statisticians (more on this later). Caution: Only delete an outlier if it exists due to error! Definition The sample median is numerical value separating the higher half of a sample from the lower half. ˜ x = ( x n +1 2 , if n is odd 1 2 ( x n / 2 + x n / 2+1 ) , if n is even. Rob Gordon (University of Florida) STA 3032 (7661) Fall 2011 5 / 251
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Introduction: Sampling & Descriptive Statistics The median divides the sample in halves, while quartiles divide the data into quarters. Let Q 1 = 1st quartile = # greater than 25% of all data points. Q 2 = 2nd quartile = # greater than 50% of all data points. Q 3 = 3rd quartile = # greater than 75% of all data points. Note: Sometimes quartiles are not numbers in the sample. Definition Q 1 = x 0 . 25( n +1) if 0 . 25( n + 1) is an integer avg of values above and below otherwise. Rob Gordon (University of Florida) STA 3032 (7661) Fall 2011 6 / 251
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Introduction: Sampling & Descriptive Statistics Example 1 Sample = { 1, 2, 3, 4, 5, 6, 7 } Q 1 = x 0 . 25(7+1) = x 2 = 2 Example 2 Sample = { 1, 2, 3, 4, 5, 6, 7, 8, 9 } Q 1 = x 0 . 25(10) = x 2 . 5 = 1 2 ( x 2 + x 3 ) = 1 2 (2 + 3) = 2 . 5 Rob Gordon (University of Florida) STA 3032 (7661) Fall 2011 7 / 251
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Introduction: Sampling & Descriptive Statistics Similarly, Definition Q 2 = median Q 3 = x 0 . 75( n +1) if 0 . 75( n + 1) is an integer avg of values above and below otherwise.
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