AP Stats BW 9/171) Which set has the largest standard deviation? The smallest?

a. 0 10 20 30 40b. 0 0 20 40 40c. 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?

Section2.4 – Normal Distributions

SWBAT: Identify and analyze patterns of distributions using shape, center and spread.

Source: www.stat119review.com

Normal DistributionsMany 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.

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

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.

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.

Emperical Rule (68 – 95 – 99.7 Rule)

50% of the data will fall above the mean / 50% below16% of the data falls more than 1 std deviation

above/below the mean99.85% of the data falls above -3 std deviations.

Emperical Rule EXAMPLE 1The 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?

500 600 700 800400300200

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) = 84TH percentile

Emperical Rule EXAMPLE 2Battery 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

40 41.58 43.16 44.7438.4236.8435.26

68% of data lies between 38.42 and 41.58 which is pretty close to the estimated 39 – 41.

Emperical Rule EXAMPLE 2, cont’dWe 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.

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?

HOMEWORK:Normal Curve Worksheet and Catch UP