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1 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Section 2-4 Measures of Center

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2 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman a value at the center or middle of a data set Measures of Center

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3 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Mean (Arithmetic Mean) AVERAGE the number obtained by adding the values and dividing the total by the number of values Definitions

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4 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Notation denotes the addition of a set of values x is the variable usually used to represent the individual data values n represents the number of data values in a sample N represents the number of data values in a population

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5 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Notation is pronounced ‘x-bar’ and denotes the mean of a set of sample values x = n x x x

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6 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Notation µ is pronounced ‘mu’ and denotes the mean of all values in a population is pronounced ‘x-bar’ and denotes the mean of a set of sample values Calculators can calculate the mean of data x = n x x x N µ = x x

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7 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definitions Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude

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8 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definitions Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude often denoted by x (pronounced ‘x-tilde’) ~

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9 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definitions Median the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude often denoted by x (pronounced ‘x-tilde’) is not affected by an extreme value ~

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10 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 6.72 3.46 3.606.44 3.46 3.60 6.446.72 no exact middle -- shared by two numbers 3.60 + 6.44 2 (even number of values) MEDIAN is 5.02

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11 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 6.72 3.46 3.606.44 26.70 3.46 3.60 6.446.72 26.70 (in order - odd number of values) exact middle MEDIAN is 6.44 6.72 3.46 3.606.44 3.46 3.60 6.446.72 no exact middle -- shared by two numbers 3.60 + 6.44 2 (even number of values) MEDIAN is 5.02

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12 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definitions Mode the score that occurs most frequently Bimodal Multimodal No Mode denoted by M the only measure of central tendency that can be used with nominal data

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13 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman a. 5 5 5 3 1 5 1 4 3 5 b. 1 2 2 2 3 4 5 6 6 6 7 9 c. 1 2 3 6 7 8 9 10 Examples Mode is 5 Bimodal - 2 and 6 No Mode

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14 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Midrange the value midway between the highest and lowest values in the original data set Definitions

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15 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Midrange the value midway between the highest and lowest values in the original data set Definitions Midrange = highest score + lowest score 2

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16 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Carry one more decimal place than is present in the original set of values Round-off Rule for Measures of Center

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17 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman use class midpoint of classes for variable x Mean from a Frequency Table

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18 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman use class midpoint of classes for variable x Mean from a Frequency Table x = Formula 2-2 f (f x)

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19 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman use class midpoint of classes for variable x Mean from a Frequency Table x = class midpoint f = frequency f = n f = n x = Formula 2-2 f (f x)

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20 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Weighted Mean x = w (w x)

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21 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Advantages - Disadvantages Table 2-13 Best Measure of Center

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22 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Symmetric Data is symmetric if the left half of its histogram is roughly a mirror of its right half. Skewed Data is skewed if it is not symmetric and if it extends more to one side than the other. Definitions

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23 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Skewness Mode = Mean = Median SYMMETRIC Figure 2-13 (b)

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24 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Skewness Mode = Mean = Median SKEWED LEFT (negatively ) SYMMETRIC Mean Mode Median Figure 2-13 (b) Figure 2-13 (a)

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25 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Skewness Mode = Mean = Median SKEWED LEFT (negatively ) SYMMETRIC Mean Mode Median SKEWED RIGHT (positively) Mean Mode Median Figure 2-13 (b) Figure 2-13 (a) Figure 2-13 (c)

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1 a value at the center or middle of a data set Measures of Center.

1 a value at the center or middle of a data set Measures of Center.

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