density curves and normal distribution
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Density Curves and Normal Distribution. Mr. Markwalter. Notagoras ’ Theorem. 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 - PowerPoint PPT PresentationTRANSCRIPT
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DensityCurves and
Normal DistributionMr. Markwalter
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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
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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…
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947 students tested Distribution of scores is below
Let’s See What I Mean
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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
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If we shade all the scores less than 6, what percentage of scores do you think we shaded?
Let’s See What I Mean
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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%
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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%
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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%
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The curve is a pretty good model for the bars!
Let’s See What I Mean
30.3% 29.3%
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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
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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
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Examples of Density Curves
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Examples of Density Curves
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Examples of Density Curves
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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
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What percentage of observations are to the left of the line in the curve below?
Test
0.40
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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…
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Describe Normal Distributions They are defined by two numbers Mean: μ Standard Deviation: σ (the average distance
from the mean) Bell Shaped
Normal Curves
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They are defined by two numbers Mean: μ Standard Deviation: σ Bell Shaped
Normal Curves
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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
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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
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The 68-95-99.7 Rule
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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
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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
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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
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Spend 15 minutes doing this worksheet. You may work with those around you. If you do not finish it is homework.
Your Turn
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1, 4, 5, 5, 6, 9 Find the standard deviation
Standard Deviation Practice?