Statistics 11/7/ Statistics: Normal Curve CSCE 115

Statistics 11/7/ Normal distribution l The bell shaped curve l Many physical quantities are distributed in such a way that their histograms can be approximated by a normal curve

Statistics 11/7/ Examples l Examples of normal distributions: »Height of 10 year old girls and most other body measurements »Lengths of rattle snakes »Sizes of oranges l Distributions that are not normal: »Flipping a coin and count of number of heads flips before getting the first tail »Rolling 1 die

Statistics 11/7/ Flipping Coins Experiment l Experiment: Flip a coin 10 times. Count the number of heads l Expected results: If we flip a coin 10 times, on the average we would expect 5 heads. l Because tossing a coin is a random experiment, the number of heads may be more or less than expected.

Statistics 11/7/ Flipping Coins Experiment l Carry out the experiment l But the average was supposed to be 5 l What can we do to improve the results? l How many times do we have to carry out the experiment to get good results? l Faster: Use a computer simulation

Statistics 11/7/ Experiment: Flipping coins

Statistics 11/7/ Experiment: Flipping coins

Statistics 11/7/ Experiment: Flipping coins

Statistics 11/7/ Experiment: Flipping coins

Statistics 11/7/ Theory l If the number of trials of this type increase, the histogram begins to approximate a normal curve

Statistics 11/7/ Normal Curve l One can specify mean and standard deviation l The shape of the curve does not depend on mean. The curve moves so it is always centered on the mean l If the st. dev. is large, the curve is lower and fatter l If the st. dev, is small, the curve is taller and skinnier

Statistics 11/7/ Normal Curve The normal curve with mean = 0 and std. dev. = 1 is often referred to as the z distribution

Statistics 11/7/ Normal Curve

Statistics 11/7/ Normal Curve

Statistics 11/7/ Normal Curve

Statistics 11/7/ Normal Curve l 50% of the area is right of the mean l 50% of the area is left of the mean

Statistics 11/7/ Normal Curve l The total area under the curve is always 1. l 68% (about 2/3) of the area is between 1 st. dev. left of the mean to 1 st. dev. right of the mean. l 95% of the area is between 2 st. dev. left of the mean to 2 st. dev. right of the mean.

Statistics 11/7/ Normal Curve 99.7% 95.4% 68.2% 50%

Statistics 11/7/ Example: Test Scores l Several hundred students take an exam. The average is 70 with a standard deviation of 10. l About 2/3 of the scores are between 60 and 80 l 95% of the scores are between 50 and 90 l About 50% of the students scored above 70

Statistics 11/7/ Example: Test Scores

Statistics 11/7/ Example: Test Scores l Suppose that passing is set at 60. What percent of the students would be expected to pass? l Solution: ( )/10 = -1. Hence passing is 1 standard deviation below the mean. l 34.1% of the scores are between 60 and 70 l 50% of the scores are above 70 l 84.1% of students would expected to pass

Statistics 11/7/ Example: Test Scores Alternate solution l Suppose that passing is set at 60. What percent of the students would be expected to pass? l Solution: According to our charts, 15.9% of the scores will be less than 60 (less than –1 standard deviations below the mean). l 100% – 15.9% = 84.1% of the students pass because they are right of the –1 st. dev. line.

Statistics 11/7/ Grading on the Curve l Assumes scores are normal l Grades are based on how many standard deviations the score is above or below the mean l The grading curve is determined in advanced

Statistics 11/7/ Example: Grading on the Curve l A 1 st. dev. or greater above the mean l B From the mean to one st. dev. above the mean l C From one st. dev. below mean to mean l D From two st. dev below mean to one st. dev. below mean l F More than two st. dev. below mean Selecting these breaks is completely arbitrary. One could assign other grade break downs.

Statistics 11/7/

Statistics 11/7/ Example: Grading on the Curve l Range (in st. dev.) Expected percent Grade from to of scores A +1 50% % = 15.9% B % C % D % F -2 50% % = 2.3%

Statistics 11/7/ Example: Grading on a Curve l Assume that the mean is 70, st. dev. is 10 l Joan scores 82. What is her grade? (82-70)/10 = 12/10 = 1.2 She scored 1.2 st. dev. above the mean. She gets an A

Statistics 11/7/ Example: Grading on a Curve l Assume that the mean is 70, st. dev. 10 l Tom scores 55. What is his grade? ( )/10 = -15/10 = -1.5 He scored 1.5 st. dev. below the mean. He gets a D

Statistics 11/7/ Example: Light bulbs l Assume that the life span of a light bulb is normally distributed with a mean of 1000 hours and a standard deviation of 100. The manufacturer guaranties that the bulbs will last 800 hours. What percent of the light bulbs will fail before the guarantee is up?

Statistics 11/7/ Example: Light bulbs l Solution: ( )/100 = -2 l The percentage of successes is 13.6% % + 50% = 97.7% l The percentage of failures is 100% % = 2.3%

Statistics 11/7/ Example: Mens socks l Assume that the length of men’s feet are normally distributed with a mean of 12 inches and a standard deviation of.5 inch. A certain brand of “one size fits all” strechy socks will fit feet from 11 inches to 13.5 inches. What % of all men cannot wear the socks?

Statistics 11/7/ Example: Mens socks l (11-12)/.5 = -2 l (13.5 – 12)/.5 = 3 l Hence men from 2 standard deviations below the mean and 3 standard deviations above the mean can wear the socks -2 to +2 st. dev. 95.4% +2 to +3 st. dev. 2.2% -2 to + 3 st. dev. 97.6% of men can wear the socks. 100% % = 2.4% cannot wear them.

Statistics 11/7/ Normal Distributions l We might be interested in: »The normal probability (or frequency) »The cumulative area below the curve

Statistics 11/7/ Normal Curves

Statistics 11/7/ Normal Curves

Statistics 11/7/ Excel and the Normal Distribution l Cumulative probabilities are measured from “- infinity”. l NORMDIST(x, mean, st dev, cumulative) cumulative - true cumulative false probability l NORMINV(probability, mean, st dev) returns the x value that gives the specified cumulative probability

Statistics 11/7/ Excel and the Normal Distribution

Statistics 11/7/ Excel and the Normal Distribution l The following functions assume mean = 0 and st. dev. = 1 l NORMSDIST(x) returns cumulative distribution l NORMSINV(probability) returns the x value that obtains the specify cumulative probability

Statistics 11/7/ Example: Excel l Assume mean = 12, st. dev. =.5 and the men’s foot distribution used earlier. l What percent of the men’s feet would be expected to be 13 inches long or shorter? =NORMDIST(13, 12,.5, TRUE) =.977 or 97.7% l What is the probability that a randomly chosen foot would be exactly 13 inches long? =NORMDIST(13, 12,.5, FALSE) =.108

Statistics 11/7/ Example: Excel l Assume mean = 70, st. dev. = 10 and the grade distribution used earlier. l What percent of the students would be expected to score 90 or below? =NORMDIST(90, 70, 10, TRUE) =.977 or 97.7% l What is the probability that a randomly chosen person scores (exactly) 90? Because this is a discrete distribution =NORMDIST(90.5, 70, 10, TRUE) -NORMDIST(89.5, 70, 10, TRUE) =.0054

Statistics 11/7/ Example: Excel l Assume the grade distribution used earlier with a mean of 70 and st. dev. of 10. What percentage of the students are expected to pass with a D or C? (That is, with scores between 50 and 70)? Passing with a D * 10 = 50 Passing with a B- 70 We are looking for 50 <= scores < 70 = NORMDIST(70, 70, 10, TRUE) - NORMDIST(50, 70, 10, TRUE) =.5 - NORMDIST(50, 70, 10, TRUE) =.477 or 47.7%

Statistics 11/7/ Example: Excel

Statistics 11/7/ Example: Excel l What score does one need to be in the top 10%? To be in the top 10%, the student must do better than 100% - 10% = 90% of the students. =NORMINV(90%, 70, 10) = 82.8 (or 83)

Statistics 11/7/ Example: Excel

Statistics 11/7/