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AP STATISTICS NOTES – CHAPTER 2
2.1 DENSITY CURVES AND THE NORMAL DISTRIBTION
The histogram to the right represents the final grades for Mr. L’s
Algebra classes.
1. What proportion of students had scores below 70?
2. What proportion of students had scores between 75 and 100?
As data sets become larger, we can draw a curve which describes
the overall pattern of the distribution.
1. About what proportion of this curve lies below 70?
2. About what proportion of this curve lies between 70 and 75?
3. If the proportions represented by each bar in the historgram were equal to the area of each bar, then what is the
total area under this curve?
This curve, which represents
is called a DENSITY CURVE.
A DENSITY CURVE is a curve that
and
.
SYMMETRIC AND SKEWED DISTRIBUTIONS
Since the median tells us the 50th percentile, the median of a density curve is
.
Locate the mean and median in these density curves:
In a symmetric density curve,
The mean represents
.
.
NOTATION FOR MEAN AND STANDARD DEVIATION
Mean
Standard Deviation
NORMAL DISTRIBUTIONS
Words which describe a normal distribution:
The density curve for a normal distribution is described by giving
The Empirical Rule: also known as
Example: The scores on the math portion of the SAT exam are normally distributed with a mean score (  ) of 500
points and a standard deviation (  ) of 50 points.
2.2 STANDARD NORMAL CALCULATIONS
Recall the standard normal distribution:
What if we know that data is distributed normally,
but has a mean and standard deviation other than 0
and 1?
allows us to
change the units so that they can be compared with
the standard normal distribution.
Given: 1)
2)
3)
The standardized value of x is given by:
This new value is called the
.
Ex: Former NBA star Michael Jordan is 78 in. tall, while WNBA player Rebecca Lobo is 76 tall. Men’s heights
have a mean of 69 in, and a standard deviation of 2.8 in. Women’s heights have a mean of 63.6 in, and a standard
deviation of 2.5 in. Which player is relatively taller? In other words, which player is taller in comparison to their
gender?
Former NBA player Mugsy Bogues is 63 in. tall. Whose height is more “unusual”, Mugsy’s or Michael’s?
NORMAL DISTRIBUTION CALCULATIONS
Consider babies born in the “normal” range of 37 to 43 weeks of gestation. Extensive data supports the assumption
that for such babies born in the U.S., birth weight is normally distributed with a mean of 3432 g, and a standard
deviation of 482 g.
Notation:
we call the distribution of scores
, and say that
has the
distribution.
In the following problems, and problems of this type:
1)
2)
3)
Ex 1: What is the probability that the birth weight of a randomly chosen baby of this type is less than 4000 g?
Ex 2: What is the probability that the birth weight of a randomly chosen baby of this type is more than 3000 g?
Ex 3: What is the probability that the birth weight will be between 3000 and 4000 g?
Ex 4: What is the probability that the birth weight will be between 2500 and 3500 g?
Ex 5: What is the 30% percentile of babies’ weights? What is the interquartile range?
2.2 NORMAL PROBABILITY PLOTS
A normal probability plot is a special graph which allows us to assess the normality of a given data set.
Steps for constructing a normal probability plot:
Weight
56
65
78
79
83
Percentile
Z-score
1.
2.
85
88
90
91
105
Constructing the plot:
x-axis:
What to look for:
y-axis:
For each data point, compute the corresponding
percentile.
Use these percentiles to find the corresponding zscore for each data point.
SOME SAMPLE PLOTS:
From www.basic.northwestern.edu
Suspected outlier(s): For data sampled from a normal
distribution, the X-Y values in the normality plot will lie
along a hypothetical straight line passing through the main
body of the X-Y values. If this is generally true, with a few
points lying off that hypothetical line, those points are likely
outliers, as with the smallest data value and, perhaps, the
largest two data values in the hypothetical example shown
here:
Skewness to the right: If both ends of the normality plot
bend above a hypothetical straight line passing through the
main body of the X-Y values of the probability plot, then the
population distribution from which the data were sampled
may be skewed to the right. Here is a hypothetical example
of a normal probability plot for data sampled from a
distribution that is skewed to the right:
Skewness to the left: If both ends of the normality plot
bend below a hypothetical straight line passing through the
main body of the X-Y values of the probability plot, then the
population distribution from which the data were sampled
may be skewed to the left. Here is a hypothetical example
of a normal probability plot for data sampled from a
distribution that is skewed to the left:
Light-tailedness: If the right (upper) end of the normality
plot bends below a hypothetical straight line passing
through the main body of the X-Y values of the probability
plot, while the left (lower) end bends above that line (an S
curve), then the population distribution from which the data
were sampled may be light-tailed. Here is a hypothetical
example of a normal probability plot for data sampled from
a distribution that is light-tailed:
Heavy-tailedness: If the right (upper) end of the normality
plot bends above a hypothetical straight line passing
through the main body of the X-Y values of the probability
plot, while the left (lower) end bends below it, then the
population distribution from which the data were sampled
may be heavy-tailed. Here is a hypothetical example of a
normal probability plot for data sampled from a distribution
that is heavy-tailed: