Chapter 131 Normal Distributions

Chapter 132 Thought Question 2 What does it mean if a person’s SAT score falls at the 20th percentile for all people who took the test? Birth weights of babies born in the United States follow, at least approximately, a bell- shaped curve. What does that mean? Thought Question 1

Chapter 133 Thought Question 3 A study in found that males (ages 18-24) have a mean height of 70 inches and a standard deviation of 2.8 in., while females (ages 18-24) have a mean height of 65 in. and a standard deviation of 2.5 in. A “standardized score” is the number of standard deviations an individual falls above or below the mean for the whole group (is positive for values above the mean, and negative for those below the mean). Thus, a man who is 72.8 inches tall has a standardized score of 1. What is the standardized score for your height?

Chapter 134 Thought Question 4 Many measurements in nature tend to follow a similar pattern. The pattern is that most of the individual measurements take on values that are near the average, with fewer and fewer measurements taking on values that are farther from the average in either direction. Describe what shape the distribution of such measurements would have.

Chapter 135 Bell-Shaped Curve: The Normal Distribution of Population Values

Chapter 136 Asymmetric Distributions of the Population Values

Chapter 137 The Normal Distribution standard deviation mean

Chapter 138 With the Mean and Standard Deviation of the Normal Distribution We Can Determine: u What proportion of individuals fall into any range of values u At what percentile a given individual falls, if you know their value u What value corresponds to a given percentile

Chapter 139 Empirical Rule for Any Normal Curve u 68% of the values fall within one standard deviation of the mean u 95% of the values fall within two standard deviations of the mean u 99.7% of the values fall within three standard deviations of the mean “ Rule”

Chapter 1310 Empirical Rule for Any Normal Curve +1sd-1sd 68% +2 sd-2 sd 95% +3 sd-3 sd 99.7%

Chapter 1311 Health and Nutrition Examination Study of (HANES) u Heights of adults, ages –women v mean: 65.0 inches v standard deviation: 2.5 inches –men v mean: 70.0 inches v standard deviation: 2.8 inches

Chapter 1312 Health and Nutrition Examination Study of (HANES) u Empirical Rule –women v 68% are between 62.5 and 67.5 inches [mean  1 std dev = 65.0  2.5] v 95% are between 60.0 and 70.0 inches v 99.7% are between 57.5 and 72.5 inches –men v 68% are between 67.2 and 72.8 inches v 95% are between 64.4 and 75.6 inches v 99.7% are between 61.6 and 78.4 inches

Chapter standard deviations: height values: Health and Nutrition Examination Study of (HANES) u What proportion of men are less than 72.8 inches tall? ? (height values) +1 68% (by Empirical Rule) ? = 84% 16% 32% / 2 =

Chapter 1314 Health and Nutrition Examination Study of (HANES) u What proportion of men are less than 68 inches tall? ? (height values) standard deviations: height values:

Chapter 1315 Standardized Scores u How many standard deviations is 68 from 70? u standardized score = (observed value minus mean) / (std dev) [ = (68  70) / 2.8 =  0.71 ] u The value 68 is 0.71 standard deviations below the mean 70.

Chapter 1316 Standardized Scores u standardized score = (observed value minus mean) / (std dev)  z is the standardized score  x is the observed value   is the population mean   is the population standard deviation

Chapter 1317 Health and Nutrition Examination Study of (HANES) u What proportion of men are less than 68 inches tall? (standardized values) (height values) ?

Chapter 1318 Table B: Percentiles of the Standardized Normal Distribution u See Table B (the “Standard Normal Table”) in back of the text (or back of the supplement). u Look up the closest standardized score in the table. u Find the percentile corresponding to the standardized score (this is the percent of values below the corresponding standardized score or z-value).

Chapter 1319 Table B

Chapter 1320 Table B: Percentiles of the Standardized Normal Distribution

Chapter (standardized values) (height values) Health and Nutrition Examination Study of (HANES) u What proportion of men are less than 68 inches tall? 24.20%

Chapter 1322 Health and Nutrition Examination Study of (HANES) u What height value is the 10th percentile for men ages 18 to 24? 10% ? 70 (height values)

Chapter 1323 Table B: Percentiles of the Standardized Normal Distribution u See Table B (the “Standard Normal Table”) in back of the text (or back of the supplement). u Look up the closest percentile in the table. u Find the corresponding standardized score. u The value you seek is that many standard deviations from the mean.

Chapter 1324 Table B: Percentiles of the Standardized Normal Distribution

Chapter 1325 Health and Nutrition Examination Study of (HANES) u What height value is the 10th percentile for men ages 18 to 24? 10% ? 70 (height values) (standardized values)

Chapter 1326 Observed Value for a Standardized Score u What height value is the 10th percentile for men ages 18 to 24? u observed value = mean plus [(standardized score)  (std dev)] = 70 + [(  1.3 )  (2.8)] = 70 + (  3.64) = u The value is approximately the 10th percentile of the population.

Chapter 1327 Observed Value for a Standardized Score u observed value = mean plus [(standardized score)  (std dev)]  x is the observed value   is the population mean  z is the standardized score   is the population standard deviation

Chapter (standardized values) (height values) Health and Nutrition Examination Study of (HANES) u RECALL: What proportion of men are less than 68 inches tall? 24.20% NOW: what proportion of men are greater than 68 inches tall? 100%  24.2% = 75.8% ?

Chapter 1329 Health and Nutrition Examination Study of (HANES) u The average height of males ages 18– 24 years old was 70.0 inches with a standard deviation of 2.8 inches. u It is also known that this distribution of heights follows a normal or bell-shaped curve. u What proportion of men are between 68 inches tall and 74 inches tall?

Chapter 1330 Health and Nutrition Examination Study of (HANES) u First, draw and label a normal curve standard deviations: height values:  = 70 in.  = 2.8 in.

Chapter 1331 Health and Nutrition Examination Study of (HANES) u Shade on the graph the range of heights between 68 and 74 inches. (height values) ? (standardized values) ? 0 ?

Chapter 1332 Standardized Scores u How many standard deviations is 68 from 70? u standardized score = (observed value minus mean) / (std dev) [ = (68  70) / 2.8 = –0.71 ] u The value 68 is 0.71 standard deviations below the mean 70.

Chapter 1333 Health and Nutrition Examination Study of (HANES) u What proportion of men are less than 68 inches tall? ?% (height values) (standardized values) ?

Chapter 1334 Table B: Percentiles of the Standardized Normal Distribution

Chapter 1335 Health and Nutrition Examination Study of (HANES) u What proportion of men are less than 68 inches tall? 24.20% (height values) (standardized values) ?

Chapter 1336 Health and Nutrition Examination Study of (HANES) u What proportion of men are less than 74 inches tall? 24.20% (height values) (standardized values) ? ?%

Chapter 1337 Standardized Scores u How many standard deviations is 74 from 70? u standardized score = (observed value minus mean) / (std dev) [ = (74  70) / 2.8 = 1.43 ] u The value 74 is 1.43 standard deviations above the mean 70.

Chapter 1338 Health and Nutrition Examination Study of (HANES) u What proportion of men are less than 74 inches tall? (standardized values) (height values) ?% 24.20%

Chapter 1339 Table B: Percentiles of the Standardized Normal Distribution

Chapter 1340 Health and Nutrition Examination Study of (HANES) u What proportion of men are less than 74 inches tall? (height values) (standardized values) % 24.20%

Chapter 1341 Health and Nutrition Examination Study of (HANES) u What proportion of men are between 68 inches tall and 74 inches tall? 24.20% (height values) (standardized values) % 91.92% – 24.20% = 67.72% 67.72%

Chapter 1342 Key Concepts u Population values are distributed with differing shapes, some normal, some non-normal. u Empirical Rule (“ Rule”) u Standardized Score u Percentile u Standard Normal Table