SP 225 Lecture 9 The Normal Curve

 ‘Bell’ Shaped  Unimodal in center  Tails extend to infinity in either direction

The Ideal Normal Curve

Naturally Normal  IQ tests  Standardized tests  Ergometric Measures  Manufacturing Processes

Normal Curve as an Approximation

Approximation  No data set is perfectly normal  Many are very close  Allows us to ‘sum up’ distributions with two pieces of information: mean and standard deviation  Allows us to draw conclusions based on calculations vs. data inspection

Empirical Rule for Data with a Bell-Shaped Distribution

Empirical Rule for Data with a Bell-Shaped Distribution (2)

Empirical Rule for Data with a Bell-Shaped Distribution (3)

Number of Standard Deviations  Use the z-score  Z-score represents the number of standard deviations from the mean of a single data point  Use it no matter what the mean or standard deviation is. z = x - x s

Properties of Z-score  + if above the mean  - if below the mean  Units are number of standard deviations

Example  Einstein had an IQ of 160 points. Amongst the population, IQ has a mean value of 110 with a standard deviation of 16.  What is the difference between Einstein and the average person?  What is the difference in number of standard deviations?  What is the z-score for Einstein’s IQ?

Areas under the Normal Curve  Areas correspond to proportions of data.  Areas are looked up in the z-score table

Special Properties of the Normal Distribution  Total area under the curve is 1  Partial areas represent proportions  Symmetric

Areas Example  Suppose your weight fluctuates each week with a mean of 0 and a standard deviation of 1 and is normally distributed. What is the probability you lose 1.5 lbs or more? What is the probability you gain less than 1.5 lbs? What is the probability you gain 1.5 lbs or more? What is the probability your weight fluctuates by less than 1.5 lbs?

Areas Example (2)  Find a value, c, where: You gain c or less pounds 75% or the time You lose c or more lbs 20% of the time Your weight fluctuates by c or less with 80% probability

Z-score Example  GPAs of SUNY Oswego freshman biology majors have approximately the normal distribution with mean 2.87 and standard deviation.34. In what range do the middle 90% of all freshman biology majors’ GPAs lie? Students are thrown out of school if their GPA falls below What proportion of all freshman biology majors are thrown out? What proportion of freshman biology majors have GPA above 3.50?

Z-score Example (2)  The length of elephant pregnancies from conception to birth varies according to a distribution that is approximately normal with mean 525 days and standard deviation 32 days. What percent of pregnancies last more than 600 days (that’s about 20 months)? What percent of pregnancies last between 510 and 540 days (that’s between 17 and 18 months)? How short do the shortest 10% of all pregnancies last?

Probabilities  Areas under the curve represent probabilities as well as proportions  Proportions make statements about existing data  Probabilities make statements about future actions