Normal Distribution and The Empirical

Rule

Get out your notes.Open Note Unit 3 Test on

Friday.

Learning Objectives

• Be able to recognize a normal distribution curve

• Be able to use the Empirical Rule to predict measurements that are normally distributed.

• Apply the Empirical Rule to predicting the quality of measured blocks.

A distribution of all possible values of a variable with an indication of the likelihood that each will occur–A probability distribution can be represented by a probability density function

• Normal Distribution – most commonly used probability distribution

Probability Distribution Distribution

http://en.wikipedia.org/wiki/File:Normal_Distribution_PDF.svg

“Is the data distribution normal?”

1.Translation: Is the histogram/dot plot bell-shaped?2.Does the greatest frequency of the data values occur at about the mean value?3.Does the curve decrease on both sides away from the mean?4.Is the curve symmetric about the mean?

Normal Distribution Quick Check Noteworthy

Freq

uenc

y

Data Elements0 1 2 3 4 5 6-

1-2

-3

-4

-5

-6

Bell shaped curve

Normal Distribution? Distribution

Freq

uenc

y

Data Elements0 1 2 3 4 5 6-

1-2

-3

-4

-5

-6

Mean Value

Normal Distribution? Distribution

Does the greatest frequency of the data values occur at about the mean value?

Freq

uenc

y

Data Elements0 1 2 3 4 5 6-

1-2

-3

-4

-5

-6

Mean Value

Normal Distribution ? Distribution

Does the curve decrease on both sides away from the mean?

Freq

uenc

y

Data Elements0 1 2 3 4 5 6-

1-2

-3

-4

-5

-6

Mean Value

Normal Distribution?

Is the curve symmetric about the mean?

What if the data is not symmetric?

Histogram Interpretation: Skewed (Non-Normal) Right

What if the data is not symmetric?

A normal distribution is a reasonable assumption.

Review

1. How can you determine if it is a Normal Distribution?

2. Bell Shaped Curve3. Peaks at the mean4. Curve decreases on both sides away from the

mean5. Curve is symmetric. Looks the same on both

side.

Learning Objectives

• Be able to recognize a normal distribution curve

• Be able to use the Empirical Rule to predict measurements that are normally distributed.

• Apply the Empirical Rule to predicting the quality of measured blocks.

• Applies to normal distributions• Almost all data will fall within three standard

deviations of the mean

Empirical Rule

• 68% of the observations fall within 1 standard deviation of the mean.

• 95% of the observations fall within 2 standard deviations of the mean.

• 99.7% of the observations fall within 3 standard deviations of the mean.

Empirical Rule

If the data are normally distributed:

Empirical Rule ExampleData from a sample of a larger population

Data Elements

Normal Distribution

s +1.77

s -1.77

0.08

+ 1

.77

= 1.

88

0.08

+ -

1.77

= -1

.69

68 %

Data Elements

Normal Distribution

2s - 3.54

2s + 3.54

0.08

+ 3

.54

= 3.

62

0.08

+ -3

.54

= - 3

.46

95 %

Your TurnRevisit the data you collected during the Fling Machine Instant Challenge. •Assume that you repeated launch cotton balls with your device. Using the mean and sample standard deviation of your data:

– Predict the range of travel distances within which 68% of cotton balls would fall

– Predict the range of travel distances within which 95% of cotton balls would fall

Example

Example

Review• What is the Empirical Rule?• Given a mean value of 10 with a standard

deviation of 2:– What range of values would 68% of the data fall in?– What range of values would 95% of the data fall in?– What range of values would 99.7% of the data fall

in?• Can you apply the Empirical rule to all sets of

data?

Tasks• Complete Worksheet 3.3 (the cube measuring worksheet)• When finished you can check past PowerPoints and add

information to your notes to be used on Friday’s exam.• Test Topics

– US/SI Measurements– Unit Conversions– Using Dial Calipers– Statistics: Mean, Median, Mode, Standard Deviation,

Histogram– Using Excel to calculate statistics– Normal Curves– The Empirical Rule

Friday’s Test is Open Notes (Your notebooks), not internet/cell phone/Smith’s Website