1. AP Stats BW 9/17 1)Which set has the largest standard deviation? The smallest? a. 0 10 20 30 40 b. 0 0 20 40 40 c. 0 19 20 21 40 2)Without calculating, which data set has the greatest standard deviation? Which has the smallest? How are the data sets the same? How do they differ?

2. Section2.4 – Normal Distributions SWBAT: Identify and analyze patterns of distributions using shape, center and spread. Source: www.stat119review.com

3. Normal Distributions Many real-life data sets have distributions that are approximately symmetric and bell-shaped. Sometimes, the overall pattern of a large data set can be described by a smooth curve Many measurements, including heights, weights, and IQ’s will have a normal curve when the data set is large. The distribution whose shape is described by a normal curve is called a normal distribution.

4. Normal Distribution, cont’d Characteristics of Normal Distributions: The mean and the median are the same value. Symmetric and bell-shaped. The area under the curve is exactly 1 or 100%. The spread is completely measured by a single number, the standard deviation. Good approximation of chance outcomes

5. Normal Distribution, cont’d The standard deviation of a normal curve can be found directly from the curve. The point where the normal curve changes “curvature” on either side of the curve is the standard deviation. Same mean - different standard deviations.

6. Emperical Rule (68 – 95 – 99.7 Rule) For data with a bell-shaped distribution, the standard deviation has the following characteristics: About 68% of the data lie within one standard deviation of the mean. About 95% of the data lie within two standard deviations of the mean. About 99.7% of the data lie within three standard deviations of the mean.

7. Emperical Rule (68 – 95 – 99.7 Rule) 50% of the data will fall above the mean / 50% below 16% of the data falls more than 1 std deviation above/below the mean 99.85% of the data falls above -3 std deviations.

8. Emperical Rule EXAMPLE 1 The distribution of scores on tests such as the SAT exam is close to normal. SAT scores are adjusted so that the mean score is about μ = 500 and the standard deviation is about σ = 100. 1.What percent of scores fall between 200 and 800? 2.What percent of scores fall above 700? 3.What percentile is a student that scores a 600 on the SAT? 500600 700800 400300200 1)(2.35+13.5+34+34+13.5+2.35) = 99.7% 2) (2.35+ 0.15) = 2.5% 3) (0.15 + 2.35 + 13.5 + 34 +34) = 84 TH percentile

9. Emperical Rule EXAMPLE 2 Battery A: 37, 38, 38, 39, 39, 39, 40, 40, 40, 40, 40, 41, 41, 41, 42, 42, 43 Using calculator, we find x-bar = 40 and s = 1.58 The Emperical Rule says appx 68% of the data should fall within 1 std dev of the mean. So if we look at 39 to 41, we see there are 11 items out of the 17 So estimating 11/17 = 65% Let’s graph the actual curve: Mean = 40 ± s: 38.42 – 41.58 ± 2s: 36.84 – 43.16 ± 3s: 35.26 – 44.74 4041.58 43.1644.74 38.4236.8435.26 68% of data lies between 38.42 and 41.58 which is pretty close to the estimated 39 – 41.

10. Emperical Rule EXAMPLE 2, cont’d We can also take a look at the frequency histogram for Battery A and see that it has a normal distribution. Battery B is also shown. What can you conclude about Battery B? Both batteries have a mean (and median) of 40, however Battery B has a greater standard deviation than Battery A.

11. You try…. IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. 1) What is the percentile for someone with an IQ of 130?

12. HOMEWORK: Normal Curve Worksheet and Catch UP