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Section 2.1
Density Curves
Get out a coin and flip it 5 times. Count
how many heads you get.
 Repeat this trial 10 times.
 Create a histogram for your data
(frequency of how many heads you got in
each of the 10 trials).
 Put your histogram on the board.
When we explore data on a single
quantitative variable:
 Plot
your data (usually a histogram or
 Look for the overall pattern (center, shape,
and spread) and for outliers
 Calculate a numerical summary to describe
center (median or mean) and spread (IQR or
standard deviation)
From Histograms to Curves
Sometimes, the overall pattern from a large
number of observations is so regular that we
can overlay a smooth curve.
Mathematical Models
This curve is a mathematical model, or an
idealized description, for the distribution.
 It is easier to work with the smooth curve
than with the histogram.
Density Curves
Density curves are always positive (meaning it’s
always on or above the horizontal axis).
 Areas under the curve represent proportions of
the observations. The area under a density
curve always equals 1.
 The density curve describes the overall pattern
of a distribution. The area under the curve,
within a range of values, is the proportion of all
observations that fall in that range.
How much area would be to the left of the
first quartile?
 How much area would be to the right of
the first quartile?
 How much area would be between the
first and third quartiles?
What Does All of This Mean?
When a density curve is a geometric
shape (rectangle, trapezoid, or a
combination of shapes) we can use
geometry to find areas. Those areas help
us find the median and the quartiles.
Verify that the graph is a valid density curve.
 For each of the following, use areas under
density curve to find the proportion of
observations within the given interval.
The median of this density curve is a point
between X = 0.2 and X = 0.4. Explain why.
I’m seeing Greek!
In a distribution, mean is x-bar and the standard
deviation is s.
 When looking at a density curve, the mean is μ
(pronounced mu) and the standard deviation is σ
(pronounced sigma).
Normal Distributions
Density curves have an area = 1 and are
always positive.
 Normal curves are a special type of
density curves. Normal curves are
symmetrical density curves.
 T/F
All density curves are normal curves.
 T/F All normal curves are density curves.
Characteristics of Normal Curves
 Single-peaked
(also called
 Bell-shaped
The mean, μ, is located at the center of the
The standard deviation, σ, is located at the
inflection points of the curve.
Parameters of the Normal Curve
A normal curve is
defined by its mean
and standard
 Notation for a normal
curve is N(μ, σ).
Why Be Normal?
Normal curves are good descriptions for lots of
real data: SAT test scores, IQ, heights, length
of cockroaches (yum!).
 Normal curves approximate random
experiments, like tossing a coin many times.
 Not all data is normal (or even approximately
normal). Income data is skewed right.
The Empirical Rule
a.k.a. 68-95-99.7 Rule
All normal distributions follow this rule:
 68%
of the observations are within one
standard deviation of the mean
 95% of the observations are within two
standard deviations of the mean
 99.7% of the observations are within three
standard deviations of the mean
Yay, Math!
IQ scores on the WISC-IV are normally
distributed with a mean of 100 and a standard
deviation of 15.
 Going up one σ and down one σ from 100 gives
us the range from 85 to 115. 68% of people
have an IQ between 85 and 115.
 95% of people have an IQ between ____ and
 99.7% of people have an IQ between ____ and
Try This
The heights of women aged 18 – 24 are
approximately normally distributed with a mean
μ = 64.5 inches and a standard deviation σ =
2.5 inches.
 Between what two heights do the middle 95%
 The tallest 2.5% of women are taller than what?
 What is the percentile for a woman who is 64.5
inches tall?
The army reports that the distribution of
head circumference among male soldiers
is approximately normal with mean 22.8
inches and standard deviation 1.1 inches.
What percent of soldiers have a head
circumference greater than 23.9 inches?
What percentile is this?
What percent of soldiers have a head
circumference between 21.7 inches and
23.9 inches?
Chapter 2 # 15a, 25, 41-45