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The Normal Distribution

Density Curves. Example: here is a histogram of vocabulary scores of 947 seventh graders.. BPS - 5th Ed.. Chapter 3. 2. The smooth curve drawn over the histogram is a mathematical model (curve) for the distribution.. Density Curves. Example: the areas of the shaded bars in this histogram represent the proportion of scores in the observed data that are less than 6.0. This proportion is equal to 0.303..

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The Normal Distribution

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    1. Chapter 3 The Normal Distribution Chapter 3 1

    2. Density Curves Example: here is a histogram of vocabulary scores of 947 seventh graders. BPS - 5th Ed. Chapter 3 2

    3. Density Curves Example: the areas of the shaded bars in this histogram represent the proportion of scores in the observed data that are less than 6.0. This proportion is equal to 0.303. BPS - 5th Ed. Chapter 3 3

    4. Density Curves Example: now the area under the smooth curve to the left of 6.0 is shaded. If the total area under the curve is exactly 1 (or 100%), then this curve is called a density curve. The proportion of the area to the left of 6.0 is now equal to 0.293, not far from what was calculated in the previous slide. BPS - 5th Ed. Chapter 3 4

    5. Always on or above the horizontal axis Have area exactly 1 (or 100%) under the curve Area under the curve and above any range of values on the horizontal axis is the proportion of all observations that fall in that range Chapter 3 5 Density Curves

    6. The median of a density curve is the equal-areas point - the point that divides the area under the curve in two equal halves The mean of a density curve is the balance point, at which the curve would balance if made of solid material. When we do this we treat the values as though they were weights. Chapter 3 6 Density Curves

    7. Density Curves The mean and standard deviation computed from actual observations (data) are denoted by ?? ands, respectively. BPS - 5th Ed. Chapter 3 7

    8. BPS - 5th Ed. Chapter 3 8 Question From Seeing Through Statistics, 2nd Edition by Jessica M. Utts. From Seeing Through Statistics, 2nd Edition by Jessica M. Utts.

    9. Bell-Shaped Curve: The Normal Distribution BPS - 5th Ed. Chapter 3 9

    10. The Normal Distribution Knowing the mean µ and standard deviation ? allows us to make various conclusions about Normal distributions. Notation: N(µ,?). This means that we have a Normal distribution with mean µ and standard deviation?. BPS - 5th Ed. Chapter 3 10

    11. 68% of the observations fall within one standard deviation of the mean 95% of the observations fall within two standard deviations of the mean 99.7% of the observations fall within three standard deviations of the mean Chapter 3 11 The 68-95-99.7 Rule for Any Normal Curve

    12. Chapter 3 12 The 68-95-99.7 Rule for Any Normal Curve

    13. Chapter 3 13 68-95-99.7 Rule for Any Normal Curve A Different Picture

    14. Heights of adult men, aged 18-24 mean: ??= 70.0 inches standard deviation: ??= 2.8 inches heights follow a normal distribution, so we have that heights of men 18 - 24 are N(70, 2.8). Chapter 3 14 Health and Nutrition Examination Study of 1976-1980

    15. 68-95-99.7 Rule for men’s heights (inches) 68% are between 67.2 and 72.8 inches µ ? ? =70.0 ? 2.8 95% are between 64.4 and 75.6 inches µ ? 2? = 70.0 ? 2(2.8) =70.0 ? 5.6 99.7% are between 61.6 and 78.4 inches µ ? 3? = 70.0 ? 3(2.8) = 70.0 ? 8.4 Chapter 3 15 Health and Nutrition Examination Study of 1976-1980

    16. BPS - 5th Ed. Chapter 3 16

    17. The 68-95-99.7 rule can be used ONLY when we are looking at values that are EXACTLY 1, 2, or 3 standard deviations above or below the mean. If we are looking at a value that is, for example, 1.25 standard deviations above the mean, we cannot use the 68-95-99.7 rule. 68-95-99.7 Rule for Any Normal Curve BPS - 5th Ed. Chapter 3 17

    18. What proportion of men are less than 68 inches tall? Chapter 3 18 Health and Nutrition Examination Study of 1976-1980

    19. Standard Normal Distribution The standard Normal distribution is the Normal distribution with mean 0 and standard deviation 1 or N(0,1). If a variable x has any Normal distribution with mean µ and standard deviation ? [ x ~ N(µ,?) ], then the standardized value (or z-score) for the variable x is: ??= ??-?? ?? = ???????????????? ?????????? -???????? ?????? ?????????????????? z is the “z-score” for the value x BPS - 5th Ed. Chapter 3 19

    20. The z-score (or standardized score) tells us how many standard deviations a value is from the mean. How many standard deviations is 68 from 70? standardized score, z = ??= ?? - ?? ?? = ???? -???? ??.?? = -0.71 The value 68 is 0.71 standard deviations below the mean 70. Negative z-scores indicate that the value, x, is below the mean. Positive z-scores indicate that x is above the mean. Chapter 3 20 Standardized Scores

    21. What proportion of men are less than 68 inches tall? Chapter 3 21 Health and Nutrition Examination Study of 1976-1980

    22. See pages 690-691 in text for Table A. (the “Standard Normal Table”) Look up the closest standardized score (z) in the table. Find the probability (area) to the left of the standardized score. Chapter 3 22 Table A: Standard Normal Probabilities

    23. Chapter 3 23 Table A: Standard Normal Probabilities

    24. Table A: Standard Normal Probabilities BPS - 5th Ed. Chapter 3 24

    25. What proportion of men are less than 68 inches tall? Chapter 3 25 Health and Nutrition Examination Study of 1976-1980

    26. What proportion of men are greater than 68 inches tall? Chapter 3 26 Health and Nutrition Examination Study of 1976-1980

    27. How tall must a man be to place in the lower 10% for men aged 18 to 24? Chapter 3 27 Health and Nutrition Examination Study of 1976-1980

    28. See pages 690-691 in text for Table A. Look up the closest probability (to .10 here) in the table. Find the corresponding standardized score. The value you seek is that many standard deviations from the mean. Chapter 3 28 Table A: Standard Normal Probabilities

    29. Table A: Standard Normal Probabilities BPS - 5th Ed. Chapter 3 29

    30. How tall must a man be to place in the lower 10% for men aged 18 to 24? Chapter 3 30 Health and Nutrition Examination Study of 1976-1980

    31. Find the Observed Value for a Standardized Score Need to “unstandardize” the z-score to find the observed value (x) : Solving for x in our equation for the z-score: BPS - 5th Ed. Chapter 3 31

    32. observed value = mean plus [(standardized score) ? (std dev)] = 70 + [(-1.28 ) ? (2.8)] = 70 + (?3.58) = 66.42 A man would have to be approximately 66.42 inches tall or less to place in the lower 10% of all men in the population. Chapter 3 32 Observed Value for a Standardized Score

    33. Example problem An instructor in a large lecture class found at the end of the semester that total points scored by students in his class was approximately normal with mean 530 and standard deviation 80. Study Plan question: About what percent of students will score between 370 and 690? 68-95-99.7% Rule BPS - 5th Ed. Chapter 3 33

    34. 68-95-99.7% Rule BPS - 5th Ed. Chapter 3 34

    35. 68-95-99.7% Rule BPS - 5th Ed. Chapter 3 35

    36. BPS - 5th Ed. Chapter 3 36

    37. BPS - 5th Ed. Chapter 3 37

    38. BPS - 5th Ed. Chapter 3 38

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