Transcript Density Curves and Empirical Rule

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.

Remember…
When we explore data on a single
quantitative variable:
 Plot
your data (usually a histogram or
stemplot)
 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.

Quartiles
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.

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0.6<X<0.8
0<X<0.4
0<X<0.2

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
Symmetric
 Single-peaked
(also called
unimodal)
 Bell-shaped

μ
σ
The mean, μ, is located at the center of the
curve.
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
deviation.
 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%
fall?
 The tallest 2.5% of women are taller than what?
 What is the percentile for a woman who is 64.5
inches tall?
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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?
Homework
Chapter 2 # 15a, 25, 41-45