DensityCurves and

Normal DistributionMr. Markwalter

People who keep organized notebooks are doing the best

People who copy down my examples are doing the best

People who ask questions are doing the best

∴Take our a notebook. No more loose leaf

I will start putting up models for note-taking

Notagoras’ Theorem

We can make histograms of data. But sometimes we have a lot of data and:THE OVERALL PATTERN OF A LARGE NUMBER

OF OBSERVATIONS IS SO REGULAR WE CAN DESCRIBE IT BY A SMOOT CURVE!

So…

947 students tested Distribution of scores is below

Let’s See What I Mean

We can look at it in the raw numbers OR We can fit a curve (in red) that is a good

model

Let’s See What I Mean

If we shade all the scores less than 6, what percentage of scores do you think we shaded?

Let’s See What I Mean

30.3% or 287 people out of 947 That means the total area of the bars would

be 100% or a proportion of 1!

Let’s See What I Mean

30.3%

If we want to talk about the red curve, we make the total area below the curve 1.

Let’s See What I Mean

30.3%

The area below the curve less than 6 is 0.293.

That is 29.3% of the area which is less than 6.

Let’s See What I Mean

30.3%

The curve is a pretty good model for the bars!

Let’s See What I Mean

30.3% 29.3%

Is always on or above the horizontal axis Has an area of 1 underneath it A density curve describes the overall

pattern of distribution. The area under the curve and above any

interval is the proportion of observations that fall in that interval.

Density Curve

Mean is the balancing point of the curve Median is the marker of equal areas; divides

the area under the curve in half.

Mean and Median Density Curves

Examples of Density Curves

Examples of Density Curves

Examples of Density Curves

If the area to the left line in the density curve shown below is 0.40, what is the area of the other part?

Test

0.40

What percentage of observations are to the left of the line in the curve below?

Test

0.40

There is one kind of curve that trumps them all.

We see it more than anything else It is the basis of 95% of statistics.

Density Curves Are Sweet But…

Describe Normal Distributions They are defined by two numbers Mean: μ Standard Deviation: σ (the average distance

from the mean) Bell Shaped

Normal Curves

They are defined by two numbers Mean: μ Standard Deviation: σ Bell Shaped

Normal Curves

As usual, the area under the curve is 1 Let’s take a look. http://www-stat.stanford.edu/~naras/jsm/No

rmalDensity/NormalDensity.html

Normal Curves

In the Normal distribution with mean μ and standard deviation σ:

About 68% of observations fall within σ of μ.About 95% of observations fall within 2σ of μ.About 99.7% of observations fall within 3σ of

μ.

The 68-95-99.7 Rule

The 68-95-99.7 Rule

Usually we define a Normal curve like this N(μ, σ) N(6, 1) means we have a curve with mean 6

and standard deviation 1. Using our 68-95-99.7 Rule…6-1=5 and 6+1=768% of the observations are between 5 and

7.

Almost Done

I make candies. The mean mass of the candy is 100g and the standard deviation is 5.

Draw a Normal curve for the situation.

What percentage of candies is between 95g and 105g?

What percentage of candies is less than 105g?

Practice

I make throw frisbees. My mean throw is 75 yards with a standard deviation of 5 yards

Draw a Normal curve for the situation.

What percentage of throws is between 65 and 85 yards?

What percentage of throws is less than above 70 yards?

Practice

Spend 15 minutes doing this worksheet. You may work with those around you. If you do not finish it is homework.

Your Turn

1, 4, 5, 5, 6, 9 Find the standard deviation

Standard Deviation Practice?