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Density Curves and Normal Distributions
Section 2.1
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Sometimes the overall pattern of a distribution can be described by a smooth curve. This histogram shows the distribution of vocab scores. We could use it to see how many students scored at each value, or what percent of students got 4’s, above 10, etc.
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Density CurvesA density curve is an idealized
mathematical model for a set of data.It ignores minor irregularities and
outliers
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Density Curves
Page 79-80
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Density Curves Page 79-80
0.303 0.293
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Density Curve
Always on or above the horizontal axis
Has an area of exactly 1 underneath it
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Types of Density CurvesNormal curvesUniform density curves
Later we’ll see important density curves that are skewed left/right and other curves related to the normal curve
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Density Curve
Area = 1, corresponds to 100% of the data
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What would the results look like if we rolled a fair die 100 times? Press STAT ENTER Choose a list: highlight the name and press
ENTER. Type: MATH PRB 5:randInt(1,6,100)
ENTER Look at a histogram of the results:
2ND Y= ENTER Press WINDOW and change your settings Press GRAPH. Use TRACE button to see
heights.
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What would the results look like if we rolled a fair die 100 times?
1 2 3 4 5 6
Outcomes
30% or 0.3
20% or 0.2
10% or 0.1
Rela
tive F
req
uen
cy
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In a perfect world…
The different outcomes when you roll a die are equally likely, so the ideal distribution would look something like this:
An example of a uniform density curve.
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Other Density Curves
What percent of observations are between 0 and 2? (area between 0 and 2)
Area of rectangle: 2(.2) = .4
Area of triangle: ½ (2)(.2) = .2
Total Area = .4 + .2 = .6 = 60%
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Other Density Curves
What percent of observations are between 3 and 4?
Area: (1)(.2) = .2 = 20%
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Normal curve
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Density Curves: Skewed
Median: the equal-areas point of the curve
Half of the area on each side
M
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Density Curves: Skewed
Mean: the balance point of the curve (if it was made of solid material)
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Mean and Median Of Density CurvesJust remember:Symmetrical distribution
Mean and median are in the centerSkewed distribution
Mean gets pulled towards the skew and away from the median.
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Notation
Since density curves are idealized, the mean
and standard deviation of a density curve will
be slightly different from the actual mean and
standard deviation of the distribution
(histogram) that we’re approximating, and we
want a way to distinguish them
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For actual observations (our sample): use and s.
For idealized (theoretical): use μ (mu) for mean and σ (sigma) for the standard deviation.
Notation
x
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Normal Curves are always:
Described in terms of their mean (µ) and standard deviation (σ)
Symmetric
One peak and two tails
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Normal Curves
Inflection points – points at which this change of curvature takes place.
µ
σ
Inflection point
Concave down
Concave up
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Normal Curves
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The Empirical Rule
The 68-95-99.7 Rule
-3 -2 -1 0 1 2 3
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The Empirical Rule
68% of the observations fall within σ of the mean µ.
-3 -2 -1 0 1 2 3
68 % of data
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The Empirical Rule
95% of the observations fall with 2σ of µ.
-3 -2 -1 0 1 2 3
95% of data
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The Empirical Rule
99.7% of the observations fall within 3σ of µ.
-3 -2 -1 0 1 2 3
99.7% of data
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Heights of Young Women The distribution of heights of young women aged 18 to 24 is approximately
normal with mean µ = 64.5 inches and standard deviation σ = 2.5 inches.
62 64.5 67 Height (in inches)
64.5 – 2.5 = 62
64.5 + 2.5 = 67
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Heights of Young Women The distribution of heights of young women aged 18 to 24 is approximately
normal with mean µ = 64.5 inches and standard deviation σ = 2.5 inches.
62 64.5 67 Height (in inches)
59.5 62 64.5 67 69.5 Height (in inches)
5
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Heights of Young Women The distribution of heights of young women aged 18 to 24 is approximately
normal with mean µ = 64.5 inches and standard deviation σ = 2.5 inches.
62 64.5 67 Height (in inches)
59.5 62 64.5 67 69.5 Height (in inches)
99.7% of data
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Shorthand with Normal Dist.
N(µ,σ)
Ex: The distribution of young women’s heights is N(64.5, 2.5).
What this means:
Normal Distribution centered at µ = 64.5 with a standard deviation σ = 2.5.
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Heights of Young Women What percentile of young women are 64.5 inches or shorter?
57 59.5 62 64.5 67 69.5 72 Height (in inches)
99.7% of data
50%
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Heights of Young Women What percentile of young women are 59.5 inches or shorter?
57 59.5 62 64.5 67 69.5 72 Height (in inches)
99.7% of data
2.5%
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Heights of Young Women What percentile of young women are between 59.5 inches and 64.5 inches?
57 59.5 62 64.5 67 69.5 72 Height (in inches)
99.7% of data
64.5 or less = 50%59.5 or less = 2.5%
50% – 2.5% = 47.5%
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For homework: 2.1, 2.3, 2.4 p. 83 2.6, 2.7, 2.8 p. 89
Practice