Normal Distributions and the Empirical Rule OBJECTIVE: Use the empirical rule (68-95-97 rule) to analyze data

Standard Deviation Standard Deviation shows the variation in data. If the data is close together, the standard deviation will be small. If the data is spread out, the standard deviation will be large. Standard Deviation is often denoted by the lowercase Greek letter sigma, .

Using your Calculator Find mean, standard deviation Given standard deviation find variance Given variance find standard deviation Example

The bell curve which represents a normal distribution of data shows what standard deviation represents. One standard deviation away from the mean ( ) in either direction on the horizontal axis accounts for around 68 percent of the data. Two standard deviations away from the mean accounts for roughly 95 percent of the data with three standard deviations representing about 99 percent of the data.

Normal distributions: N (μ, σ) Symmetric, single peaked and bell shaped. Center of the curve are μ and M. Mean, median, & mode are the same Standard deviation σ controls the spread of the curve.

Normal curves are a good description of some real data: test scores (SAT, ACT, IQ) biological measurements also approximate chance outcomes like tossing coins

The Empirical rule (68-95-99.7 rule) In the normal dist. with mean μ and standard deviation σ. 50% of the observations fall below the mean. 50% of the observations fall above the mean.

The Empirical rule (68-95-99.7 rule) In the normal dist. with mean μ and standard deviation σ. 68% of the observations fall within of the mean. 95% of the observations fall within of the mean. 99.7% of the observations fall within of the mean. 1σ 2σ 3σ

Behold the normal curve 68% 34% 34% 0.15% 95% 0.15% 13.5% 13.5% 2.35% 2.35% 99.7%

Exercise 1: Men’s Heights The distribution of adult American men is approximately normal with mean 69 inches and standard deviation 2.5 inches. Draw the curve and mark points if inflection.

Recall: mean 69 in. and standard deviation 2.5 in. 16%tile 84% tile 2.5%tile 97.5th % 2.35% .15 .15%tile 99.85th % 61.5 64 66.5 69 71.5 74 76.5 a) What percent of men are taller than 74 inches? 74 is two standard dev. above the mean. 2.35 + .15 =2.5 b) Between what heights do the middle 95% of men fall? 69 5 Between 64 and 74 inches

mean 69 in. and standard deviation 2.5 in. 13.5 84% 2.35 .15 61.5 64 66.5 69 71.5 74 76.5 c) What percent of men are shorter than 66.5 inches? 16.0%

Exercise 2: SAT Verbal Scores SAT verbal scores are normally distributed with a mean of 489 and a standard deviation of 93.

Recall: mean 489 in. and standard deviation 93 in. 16%tile 84% tile 13.5% 2.5%tile 97.5th % .15%tile 99.85th % 210 303 396 489 582 675 768 a) What percentage lie between 303 and 582? 303 is two standard dev. below the mean. & 582 is one std. dev above 68 + 13.5 = 81.5%

Recall: mean 489 in. and standard deviation 93 in. 16% 84% 2.35% 2.5% 97.5 % ..15% .15% 99.85 % 210 303 396 489 582 675 768 b) What percentage is above 675? 675 is two standard dev. above the mean. 2.35 +.15 = 2.5% What percentage is below 675 100 - 2.5 = 97.5

Recall: mean 489 in. and standard deviation 93 in. 13.5% 16% 84% 2.5% 97.5 % .15% 99.85 % 210 303 396 489 582 675 768 b) If 3,500 students took the SAT verbal test, about how many received between 396 and 675 points? 68% + 13.5% = 81.5% fall within this range. 3,500 * 81.5% = 2853