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STATISTICS: PART III SOL A.9

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OBJECTIVES Review vocabulary Variance Normal distributions Applications of statistics

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MEAN ABSOLUTE DEVIATION Mean Absolute Deviation is a measure of the dispersion of data (how spread out the numbers are from the mean) Mean Absolute Deviation Formula: Mean absolute deviation

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STANDARD DEVIATION Standard Deviation is a measure of the dispersion of data (how spread out numbers are from the mean) The larger the standard deviation is, the more spread out the numbers tend to be from the mean. Standard Deviation Formula:

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MEAN ABSOLUTE DEVIATION AND STANDARD DEVIATION Mean Absolute Deviation (MAD) and Standard Deviation are statistics that are used to measure the dispersion (spread) of the data Standard Deviation is a more traditional way to measure the spread of data. Mean Absolute Deviation may be a better way to measure the dispersion of data when there are outliers since it is less affected by outliers.

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Z-SCORE A z-score (or standard score) indicates how many standard deviations a data point is above or below the mean of a data set. A positive z-score indicates that a data point is above the mean. A negative z-score indicates that a data point is below the mean. A z-score of zero indicates that a data point is equal to the mean.

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Z-SCORE The Z-Score Formula is:

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VARIANCE Variance is a measure of the dispersion of the data. It is used to find the standard deviation and is equal to the standard deviation squared. Variance formula:

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WHAT’S NORMAL? A normal curve is symmetric about the mean and has a bell shape.

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WHAT’S NORMAL? Mean ( ) +1 Standard Deviation ( ) +2 +3 -1 Standard Deviation ( ) -3 -2

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EMPIRICAL RULE In a normal distribution with mean μ and standard deviation σ: 68% of the data fall within σ of the mean μ. 95% of the data fall within 2σ of the mean μ. 99.7% of the data fall within 3σ of the mean μ.

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APPLICATION #1 Mr. Smith is planning to purchase new light bulbs for his art studio. He tested a sample of 10 Power-Up bulbs and found they lasted 4,356 hours on the average (mean) with a standard deviation of 211 hours. Then, he tested 10 Lights-A-Lot bulbs and found the following results. 5,066 4,130 4,568 4,884 4,730 5,122 4,910 4,866 4,779 4,721

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APPLICATION #1, CONTINUED A)Which brand of light bulb has the greater average life span? B)Mr. Smith bought 2 Lights-A-Lot bulbs. One bulb lasted for 5,150 hours and the other bulb lasted for 4,700 hours. Give the z-score for each bulb. Explain what each z-score represents. C)Which brand of light bulb do you think that Mr. Smith should choose. Explain.

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APPLICATION #2 The chart below lists the height (in inches) of 10 players from two NBA teams. Use the data to answer the following questions: Chicago Bulls 77”81”79”77”81”79”76”82”83”69” Toronto Raptors 79”78”84”79” 80”84”81”83”71”

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APPLICATION QUESTION #2, CONTINUED A)Find the following for each team: Chicago Bulls Toronto Raptors =____________ = ____________ = ___________ = ____________ = ____________ = _____________ 78.4 3.83 14.6 79.8 12.96 3.6

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APPLICATION QUESTION #2, CONTINUED B) Find the z-score for a height of 6’11” for each team: Chicago Bulls: _____________ Toronto Raptors: ___________ C) Is it more likely that a player for the Bulls or the Raptors would be 6’11”? Explain. 1.201 0.889 He is more likely to play for the Raptors. The z- score for that height for the Raptors is 0.889 and for the Bulls is 1.201. Therefore, a player who is 6’11” tall is more unusual (farther away from the mean) if he plays for the Bulls.

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APPLICATION QUESTION #2, CONTINUED D) A player who is drafted by the Chicago Bulls has a z- score of -1.671. How tall is the player? Round to the nearest inch. E) A player who is 6’7” tall has a z-score of -0.222. For which team does he play? F) How many players on the Raptors are a height that is between -0.4 and 0.6 standard deviations from the mean? 5 6’0” Toronto Raptors

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Math 20: Foundations FM20.6 Demonstrate an understanding of normal distribution, including standard deviation and z-scores. FM20.7 Demonstrate understanding.

Math 20: Foundations FM20.6 Demonstrate an understanding of normal distribution, including standard deviation and z-scores. FM20.7 Demonstrate understanding.

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