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1 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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 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 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 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 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 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 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 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 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 Addison Wesley Longman no exact middle -- shared by two numbers (even number of values) MEDIAN is 5.02

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

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12 Chapter 2. Section 2-4. Triola, Elementary Statistics, Eighth Edition. Copyright 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 Addison Wesley Longman a b c 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 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 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 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 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 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 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 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 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 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 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 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 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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