1 / 7

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.

akando
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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]

More Related