The Normal distributionChapter 5

Assessing Normality (5.4)

• You have a bunch of data. • Question: Is it Normal?

Assessing Normality (5.4)

• You have a bunch of data. • Question: Is it Normal?– Check if your data has the same properties that

Normal data does

1. Construct a histogram or stem and leaf plot. Does it look Normal-ish? (moundshaped and symmetric)

1. Construct a histogram or stem and leaf plot. Does it look Normal-ish? (moundshaped and symmetric)

2. Find the % of data within 1 standard deviation of the mean, 2 standard deviations and 3 standard deviations (should be 68%, 95%, 99.6%)

1. Construct a histogram or stem and leaf plot. Does it look Normal-ish? (moundshaped and symmetric)

2. Find the % of data within 1 standard deviation of the mean, 2 standard deviations and 3 standard deviations (should be 68%, 95%, 99.6%)

3. IQR/s=1.3

1. Construct a histogram or stem and leaf plot. Does it look Normal-ish? (moundshaped and symmetric)

2. Find the % of data within 1 standard deviation of the mean, 2 standard deviations and 3 standard deviations (should be 68%, 95%, 99.6%)

3. IQR/s=1.34. Construct a Normal Probability Plot

1. Construct a histogram or stem and leaf plot. Does it look Normal-ish? (moundshaped and symmetric)

2. Find the % of data within 1 standard deviation of the mean, 2 standard deviations and 3 standard deviations (should be 68%, 95%, 99.6%)

3. IQR/s=1.34. Construct a Normal Probability Plot

Note: 3 and 4 are true and good checks for Normality, but we will not be covering them in this class.

Toilet Flush Example

Is this data normal??

Toilet Flush Example

Is this data normal??

1. Construct a histogram or stem and leaf plot. Does it look Normal-ish? (moundshaped and symmetric)

Housefly Wings Example

Is this data set normal??

Housefly Wings Example

Is this data normal??

1. Construct a histogram or stem and leaf plot. Does it look Normal-ish? (moundshaped and symmetric)

Housefly Wings Example

Is this data normal??Stem Leaf Plot

3 | 6 7 8 8 9 94 | 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 5 | 0 0 0 0 0 0 1 1 1 1 2 2 3 3 4 5

n = 100Mean = 45.5St. Dev = 3.92

Housefly Wings Example

Is this data normal??Stem Leaf Plot

3 | 6 7 8 8 9 94 | 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 5 | 0 0 0 0 0 0 1 1 1 1 2 2 3 3 4 5

n = 100Mean = 45.5St. Dev = 3.92

2. Find % of data within 1,2,3 standard deviations of the mean

Housefly Wings ExampleStem Leaf Plot

3 | 6 7 8 8 9 94 | 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 5 | 0 0 0 0 0 0 1 1 1 1 2 2 3 3 4 5

n = 100Mean = 45.5St. Dev = 3.92

2. Find % of data within 1,2,3 standard deviations of the mean

(mean-s,mean+s)=(41.6,49.4)(mean-2s,mean+2s)=(37.7,53.3)(mean-3s,mean+3s)=(?,?)

What percent of the fly data is in these intervals?

Approximating a Binomial with a Normal

• Flip a fair coin 100 times. How many heads do we get?

Approximating a Binomial with a Normal

• Flip a fair coin 100 times. How many heads do we get?– Repeat this process 1000 times. What do we

expect our results to look like?

Approximating a Binomial with a Normal

• Flip a fair coin 100 times. How many heads do we get?– Repeat this process 1000 times. What do we

expect our results to look like?

Does this shape remind you of anything??

We can (often) approximate a Binomial Distribution with a Normal!

• Ex: Given the above flips, let X be the number of Heads. – Find the P(X<=45)=

Could do the binomial probabilities. But we’d have to find: P(X=0)+P(X=1)+…+P(X=45)

Or we could convert it to a normal. Find z-score and use a table!!

Example of Normal Approximation of Binomial Random Variable

Do on chalkboard!

Is this approximation valid?

• Note that the interval (x-3s,x+3s) contains nearly all the data (see Empirical Rule of Thumb or Chebyshev’s Rule)

• For the approximation to be valid we need that interval to be in the middle of our binomial possibilities. The range of binomial possibilities is (0,n)

• Therefore approximation is valid if: (x-3s) > 0 (x+3s) < n