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Statistics: Part III

Statistics: Part III. SOL A.9. Objectives. Review vocabulary Variance Normal distributions Applications of statistics. Mean absolute deviation. Mean Absolute Deviation is a measure of the dispersion of data (how spread out the numbers are from the mean)

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Statistics: Part III

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  1. Statistics: Part III SOL A.9

  2. Objectives • Review vocabulary • Variance • Normal distributions • Applications of statistics

  3. 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

  4. 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:

  5. 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.

  6. 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.

  7. Z-Score • The Z-Score Formula is:

  8. 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:

  9. What’s Normal? • A normal curve is symmetric about the mean and has a bell shape.

  10. What’s Normal? Mean ( ) -1 Standard Deviation ( ) +1 Standard Deviation ( ) -2 +2 +3 -3

  11. 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.

  12. 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

  13. Application #1, continued • Which brand of light bulb has the greater average life span? • 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. • Which brand of light bulb do you think that Mr. Smith should choose. Explain.

  14. 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:

  15. Application Question #2, Continued • Find the following for each team: • Chicago Bulls Toronto Raptors • =____________ = ____________ • = ___________ = ____________ • = ____________ = _____________ 79.8 78.4 12.96 14.6 3.6 3.83

  16. 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. Thez-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.

  17. 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 isbetween -0.4 and 0.6 standard deviations from the mean? 6’0” Toronto Raptors 5

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