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Unformatted text preview: Chapter 3 Numerically Summarizing Data Fall 2008 1 Overview ● How could we describe or summarize the distribution of data in numeric ways? ● How could we describe or summarize the distribution of data in numeric ways? ● In this chapter, we will discuss Measures of the “center” of a set of data Measures of the “spread” of a set of data Ways of using these numeric measures to analyze data sets ● How could we describe or summarize the distribution of data in numeric ways? ● In this chapter, we will discuss Measures of the “center” of a set of data Measures of the “spread” of a set of data Ways of using these numeric measures to analyze data sets ● This complements what we did in Chapter 2 where we organized and summarized data in more visual ways Fall 2008 2 Chapter 3 Section 1 Measures of Central Tendency Fall 2008 3 Populations vs Samples ● Analyzing populations versus analyzing samples ● Analyzing populations versus analyzing samples ● For populations We know all of the data Descriptive measures of populations are called parameters Parameters are often written using Greek letters ( μ ) ● Analyzing populations versus analyzing samples ● For populations We know all of the data Descriptive measures of populations are called parameters Parameters are often written using Greek letters ( μ ) ● For samples We know only part of the entire data Descriptive measures of samples are called statistics Statistics are often written using Roman letters ( ) x Fall 2008 4 Central Tendency Calculating the Arithmetic Mean ● The arithmetic mean of a variable is often what people mean by the “average” … add up all the values and divide by the number of measurements in the data set ● Compute the arithmetic mean of 6, 1, 5 ● Add up the three numbers and divide by 3 (6 + 1 + 5) / 3 = 4.0 ● The arithmetic mean is 4.0, one more decimal place than the data Fall 2008 5 Summation Notation Calculating the Arithmetic Mean Used to simplify summation instructions Each observation in a data set is identified by a subscript x 1 , x 2 , x 3 , x 4 , x 5 , …. x n Notation used to sum the above numbers together is n n i i x x x x x x + + + + + = = ∑ 4 3 2 1 1 Fall 2008 6 Summation Notation Calculating the Arithmetic Mean Data set: 1, 2, 3, 4 Are these the same? and ∑ = 4 1 2 i i x 2 4 1 ∑ = i i x 2 2 2 2 2 1 2 3 4 4 1 1 4 9 16 30 i i x x x x x + + + = = = + + + = ( 29 1 2 3 4 2 4 2 2 2 1 1 2 3 4 10 100 i i x x x x x + + + = = = + + + = = & Fall 2008 7 Central Tendency The mean is an arithmetic average of the elements of the data set The mean of a sample of n measurements is denoted by and equals If the data are from a population , the mean is denoted by μ (mu) and equals n x x n i i ∑ = = 1 N x N i i ∑ = = 1 μ x Fall 2008 8 Central Tendency ● One interpretation ● The arithmetic mean can be thought of as the center of gravity … where the yardstick balances Fall 2008 9 Central Tendency...
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