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Chapter 5: Exploring Data: Distributions Normal Distributions (5.8)

Chapter 5: Exploring Data: Distributions Normal Distributions (5.8). Normal Distributions When the overall pattern of a large number of observations is so regular, we can describe it as a smooth curve. Normal curves are symmetric and bell-shaped, smoothed-out histograms.

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Chapter 5: Exploring Data: Distributions Normal Distributions (5.8)

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  1. Chapter 5: Exploring Data: DistributionsNormal Distributions (5.8) • Normal Distributions • When the overall pattern of a large number of observations is so regular, we can describe it as a smooth curve. • Normal curves are symmetric and bell-shaped, smoothed-out histograms. • The total area under the Normal curve is exactly 1 (specific areas under the curve actually are proportions of the observations). Histogram of the vocabulary scores of all seventh-grade students. The smooth curve shows the overall shape of the distribution. 1

  2. Standard Deviation of a Normal Curve • The shape of a Normal distribution is completely described by two numbers, the mean and its standard deviation. • The mean is at the center of symmetry of the Normal curve. • The standard deviation is the distance from the center to the change-of-curvature points on either side. 2

  3. Calculating Quartiles • The first quartile of any Normal distribution is located 0.67 standard deviation below the mean. Q1 = Mean − (0.67)(Stand. dev.) • The third quartile is 0.67 standard deviation above the mean. Q3 = Mean + (0.67)(Stand. dev.) Example: Mean = 64.5, Stand. dev.= 2.5 Q3 = 64.5 + 0.67(2.5) = 64.5 + 1.7 = 66.2 3

  4. Example: The scores on a marketing exam were normally distributed with a mean of 68 and a standard deviation 0f 4.5. a) Find the 1st and 3rd quartile for the exam scores. Q1 = Mean − (0.67)(Stand. dev.) Q3= Mean + (0.67)(Stand. dev.) Q1 = 68 − (0.67)(4.5) Q3= 68 + (0.67)(4.5) Q1 = 65 Q3= 71 b) Find a range containing exactly the middle 50% of the students’ scores. Since 25% of the data lie BELOW the 1st quartile and 25% of the data lie ABOVE the 3rd quartile, 50% of the data will fall between the 1st and 3rd quartiles. Between 65 and 71

  5. Chapter 5: Exploring Data: DistributionsThe 68-95-99.7 Rule (5.9) • Normal Distributions 68-95-99.7 Rule • 68% of the observations fall within 1 standard deviation of the mean. • 95% of the observations fall within 2 standard deviations of the mean. • 99.7% of the observations fall within 3 standard deviations of the mean. 5

  6. Example SAT scores are close to a Normal distribution, with a mean = 500 and a standard deviation = 100. What percent of scores are above 700? Answer: Score of 700 is +2 stand. dev. Since 95% of data is between +2 and −2 stand. dev., then above 700 is in top 2.5%. mean 95% of the data 5% ÷ 2 = 2.5% SAT scores have Normal distribution

  7. Example: The scores on a marketing exam were normally distributed with a mean of 71.3 and a standard deviation of 5.5. a) Almost all (99.7%) scores fall within what range? The range of scores is 54.8 to 87.8. If the scores are understood to be whole numbers, then the range is [55, 87]. b) What percent of scores are more than 82? 64 ÷ 2 2.5% c) What percent of scores fall in the interval [66,82]? 95 ÷ 2 34% + 47.5% = 81% of scores fall in the interval [66, 82]

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