M119L06 - (Lesson 6: Measures of Spread or Variation; 3-3)...

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(Lesson 6: Measures of Spread or Variation; 3-3) 3.15 LESSON 6: MEASURES OF SPREAD OR VARIATION (SECTION 3-3) PART A: THREE MEASURES Example 1 (same as in Lesson 5 ) The five students in a class take a test. Their scores in points are as follows: 80 76 100 83 100 Let’s look at three possibilities for measuring the spread or variation of a data set. Note: All three of these measures are nonnegative in value. 1) Range Range = Max Min = highest value lowest value in the data set ( ) In Example 1 Range = Max Min = 100 76 = 24 points Pros: The range is quick and easy to find, and it seems like a natural measure of spread. Cons: The range uses only two of the data values (excluding ties), and it is extremely sensitive to outliers.
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(Lesson 6: Measures of Spread or Variation; 3-3) 3.16 We will focus on the following, which use all of the data values and are used in many formulas. They are much harder to compute manually, though: 2) Variance (VAR) 3) Standard Deviation (SD) SD = VAR . This is the most commonly used measure of spread. PART B: NOTATION We typically use Greek letters to denote population parameters. σ is lowercase sigma; remember that the summation operator is uppercase sigma. Mean SD VAR Population (Size N ) μ 2 Sample (Size n ) x s s 2
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(Lesson 6: Measures of Spread or Variation; 3-3) 3.17 PART C: POPULATION DATA What are the population variance, σ 2 , and the population standard deviation, , of a population data set? Idea Focus on the mean as a reference point; envision planting a flag there on the real number line. We want to measure the spread of the data values around the mean. Let’s compare another pair of data sets using the same scale:
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(Lesson 6: Measures of Spread or Variation; 3-3) 3.18 Recipe Given data: x 1 , x 2 , , x N . Steps Notation Step 1) Find the mean. μ = x N Step 2) Find the deviations from the mean by subtracting the mean from all the data values. x ( ) values Step 3) Square the deviations from Step 2). x ( ) 2 values Step 4) VAR, or σ 2 = the average of the squared deviations from Step 3) 2 = x ( ) 2 N Step 5) SD, or = VAR = x ( ) 2 N
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(Lesson 6: Measures of Spread or Variation; 3-3) 3.19 Back to Example 1 Find the population VAR and SD of the given data set. Show all work on exams! Data x ( ) Step 2 Deviations: x μ ( ) values Step 3 Squared Deviations: x ( ) 2 values 80 7.8 60.84 76 11.8 139.24 100 12.2 148.84 83 4.8 23.04 100 12.2 148.84 Step 1 : = 87.8 points See Note 2 below. Sum = 520.8 Do Steps 4, 5 . Note 1: You should fill out the above table row by row. For example, take the “80”, subtract off the mean, and then square the result: 80 Subtract 87.8 ⎯⎯⎯ 7.8 Square ⎯⎯ 60.84 Note 2: Why not use the sum or average of the deviations as a measure of spread? It is always 0 for any data set, so it is meaningless as a measure of spread. The deviations effectively cancel each other out. This reflects the
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This note was uploaded on 09/08/2011 for the course MATH 119 taught by Professor Staff during the Spring '11 term at Mesa CC.

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M119L06 - (Lesson 6: Measures of Spread or Variation; 3-3)...

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