3.3 Density Curves and Normal Distributions

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3.3 Density Curves and Normal Distributions

• Density Curves
• Measuring Center and Spread for Density Curves
• Normal Distributions
• The 68-95-99.7 Rule
• Standardizing Observations
• Using the Standard Normal Table
• Inverse Normal Calculations
• Normal Quantile Plots

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Exploring Quantitative Data
We now have a kit of graphical and numerical
tools for describing distributions. We also have
a strategy for exploring data on a single
quantitative variable. Now, well add one more
step to the strategy.
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Exploring Quantitative Data

1. Always plot your data Make a graph.
2. Look for the overall pattern (shape, center, and
   spread) and for striking departures such as
   outliers.
3. Calculate a numerical summary to briefly describe
   center and spread.
4. Sometimes, the overall pattern of a large number
   of observations is so regular that we can
   describe it by a smooth curve.

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Density curves
A density curve is a mathematical model of a
distribution The total area under the curve, by
definition, is equal to 1, or 100. The area
under the curve for a range of values is the
proportion of all observations for that range.
Area under Density Curve Relative Frequency
of Histogram
Histogram of a sample with the smoothed, density
curve describing theoretically the population.
rel. freq of left histogram287/947.303
area .293 under rt. curve
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Density Curves

• A density curve is a curve that
• Is always on or above the horizontal axis
• Has an area of exactly 1 underneath it
• A density curve describes the overall pattern of
  a distribution. The area under the curve and
  above any range of values on the horizontal axis
  is the proportion of all observations that fall
  in that range.

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Density curves come in many shapes. Some are well
known mathematically and others arent but they
all lie above the horizontal axis and have total
area 1.
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Density Curves

• Our measures of center and spread apply to
  density curves as well as to actual sets of
  observations.

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Distinguishing the Median and Mean of a Density
Curve

• The median of a density curve is the equal-areas
  point?the point that divides the area under the
  curve in half.
• The mean of a density curve is the balance point,
  at which the curve would balance if made of solid
  material.
• The median and the mean are the same for a
  symmetric density curve. They both lie at the
  center of the curve. The mean of a skewed curve
  is pulled away from the median in the direction
  of the long tail.

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Density Curves

• The mean and standard deviation computed from
  actual observations (data) are denoted by and
  s, respectively, and are called the sample mean
  and sample standard deviation.
• The mean and standard deviation of the idealized
  distribution represented by the density curve are
  denoted by µ (mu) and ? (sigma),
  respectively, and are sometimes called the
  population mean and population standard
  deviation.

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Normal Distributions

• One particularly important class of density
  curves are the Normal curves, which describe
  Normal distributions.
• All Normal curves are symmetric, single-peaked,
  and bell-shaped.
• A specific Normal curve is described by giving
  its mean µ and standard deviation s.

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Normal Distributions

• A Normal distribution is described by a Normal
  density curve. Any particular Normal
  distribution is completely specified by two
  numbers its mean µ and standard deviation s.
• The mean of a Normal distribution is the center
  of the symmetric Normal curve.
• The standard deviation is the distance from the
  center to the change-of-curvature points on
  either side, the points of inflection of the
  density.
• We abbreviate the Normal distribution with mean µ
  and standard deviation s as N(µ,s).

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Normal distributions
Normal or Gaussian distributions are a family
of symmetrical, bell-shaped density curves
defined by a mean m (mu) and a standard deviation
s (sigma) N(m,s).
x
x
e 2.71828 The base of the natural logarithm p
pi 3.14159
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A family of density curves
Here, means are the same (m 15) while standard
deviations are different (s 2, 4, and 6).
Here, means are different (m 10, 15, and 20)
while standard deviations are the same (s 3).
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The 68-95-99.7 Rule

• The 68-95-99.7 Rule
• In the Normal distribution with mean µ and
  standard deviation s
• Approximately 68 of the observations fall within
  s of µ.
• Approximately 95 of the observations fall within
  2s of µ.
• Approximately 99.7 of the observations fall
  within 3s of µ.

Heres a N(64.5, 2.5) distribution of heights
of college-aged females.
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The 68-95-99.7 Rule

• The distribution of Iowa Test of Basic Skills
  (ITBS) vocabulary scores for 7th-grade students
  in Gary, Indiana, is close to Normal. Suppose
  the distribution is N(6.84, 1.55).
• Sketch the Normal density curve for this
  distribution.
• What percent of ITBS vocabulary scores are less
  than 3.74?
• What percent of the scores are between 5.29 and
  9.94?

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Standardizing Observations
All Normal distributions are the same if we
measure in units of size s from the mean µ as
center.
The standard Normal distribution is the Normal
distribution with mean 0 and standard deviation
1. That is, the standard Normal distribution is
N(0,1) it is represented by Z and we write Z
N(0,1)
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The Standard Normal Table
Because all Normal distributions are the same
when we standardize, we can find areas under any
Normal curve from a single table.
The Standard Normal Table Table A is a table of
areas under the standard Normal curve. The table
entry for each value z is the area under the
curve to the left of z.
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The Standard Normal Table
Suppose we want to find the proportion of
observations from the standard Normal
distribution that are less than 0.81. We can
use Table A
P(z lt 0.81)
0.7910
Z 0.00 0.01 0.02
0.7 0.7580 0.7611 0.7642
0.8 0.7881 0.7910 0.7939
0.9 0.8159 0.8186 0.8212
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Tips on using Table A
To calculate the area between 2 z- values, first
get the area under N(0,1) to the left for each
z-value from Table A.
Then subtract the smaller area from the larger
area.
A common mistake made by students is to subtract
both z values - it is the areas that are
subtracted, not the z-scores!
area between z1 and z2 area left of z1 area
left of z2
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Normal Calculations
How to Solve Problems Involving Normal
Distributions

• Express the problem in terms of the observed
  variable X.
• Draw a picture of the distribution of X and shade
  the area of interest under the curve.
• Perform calculations.
• Standardize X to restate the problem in terms of
  a standard Normal variable Z.
• Use Table A and the fact that the total area
  under the curve is 1 to find the required area
  under the standard Normal curve.
• Write your conclusion in the context of the
  problem.

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Inverse Normal Calculations

• According to the Health and Nutrition Examination
  Study of 19761980, the heights X (in inches) of
  adult men aged 1824 are N(70, 2.8).

How tall must a man be (? below) to be in the
lower 10 for men aged 1824?
N(70, 2.8)
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Inverse Normal Calculations
How tall must a man be in the lower 10 for men
aged 1824?
Look up the closest probability (closest to 0.10)
in the table. Find the corresponding z the
standardized score. The value you seek is that
many standard deviations from the mean.
z 0.07 0.08 0.09
1.3 0.0853 0.0838 0.0823
1.2 .1020 0.1003 0.0985
1.1 0.1210 0.1190 0.1170
Z 1.28
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Normal Calculations
How tall must a man be in the lower 10 for men
aged 1824?
Z 1.28
We need to unstandardize the z-score to find
the observed value (x)
x 70 z(2.8) 70 (-1.28 ) ? (2.8)
70 (3.58) 66.42
A man would have to be approximately 66.42 inches
tall or less to place in the lower 10 of all men
in the population.
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Normal Quantile Plots

• One way to assess if a distribution is indeed
  approximately Normal is to plot the data on a
  Normal quantile plot.
• The data points are ordered from smallest to
  largest and their percentile ranks are converted
  to z-scores with Table A. These z-scores are then
  plotted against the data to create a Normal
  quantile plot.
• If the distribution is indeed Normal, the plot
  will show a straight line, indicating a good
  match between the data and a Normal distribution
  in JMP the points fall within the dotted lines.
• Systematic deviations from a straight line
  indicate a non-Normal distribution. Outliers
  appear as points that are far away from the
  overall pattern of the plot some points fall
  outside the dotted lines in JMP.

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Normal Quantile Plots
Good fit to a straight line the distribution of
rainwater pH values is close to normal. The
intercept of the line mean of the data and the
slope of the line s.d. of the data
Curved pattern The data are not Normally
distributed. Instead, it shows a right skew A
few individuals have particularly long survival
times.
Normal quantile plots are complex to do by hand,
but they are easy to do in JMP under the red
triangle, choose Normal Quantile Plot but
notice the difference when compared to the above
plots
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• HW
• Finish reading section 3.3
• Work over all the examples!
• Ive put up some videos on computing Normal
  Probabilities as an online assignment due 9/19
  9/22 at 900am
• Work on 3.82-3.93, 3.95, 3.100-3.102,
• 3.104, 3.106-3.109, 3.110-3.128. Do as many
• of these problems in order to really
• understand whats going on here!
• Quiz in class on Monday 9/22
• Test 1 on October 1, covering Chapts. 1-4