SECTION 2.1 DENSITY CURVES
When we explore data on a single quantitative variable:

_______________________________________ (usually a histogram or stemplot)

Look for the ___________________________ (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 ________________
___________________ from a large
number of observations is so
__________________ that we can
overlay a smooth curve.
Mathematical Models


This _____________________is a mathematical model, or an idealized description, for the
distribution.
It is easier to work with the _________________________ than with the histogram.
Density Curves

Density curves are ________________________________ (meaning it’s always on or
above the horizontal axis).

_____________________________________ represent proportions of the observations.
______________________________________________________________________!!

The density curve describes the _________________________ of a distribution. The area
under the curve, within a range of values, _______________________________________
_________________________________________________________________________
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?
In a ________________________________, mean is x-bar and the standard deviation is s.
When looking a ___________________________________, the mean is ____________
(pronounced mu) and the standard deviation is __________________ (pronounced sigma).
Mean and Median of a Density Curve

______________________: μ = M

______________________: μ < M

______________________: μ > M

The median of a density curve is the __________________________________, where ½
of the area is to the left and ½ of the area is to the right.

The mean of a density curve is the __________________________________, where the
curve would balance if it were made of solid material.
What Does All of This Mean?


When a density curve is a ________________________________________ (rectangle,
trapezoid, or a combination of shapes) we can use geometry to find areas.
______________________________________________________________________
The mean and standard deviation require more advanced mathematics to find. W e’ll learn
about those later.
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.
 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.
Normal Distributions
 Density curves have an area = 1 and are always ________________________.
 Normal curves are a special type of density curves. Normal curves are _________________
_____________________________________.

T/F All density curves are normal curves.

T/F All normal curves are density curves.
Characteristics of Normal Curve
_________________________, _________________________, ________________________
Parameters of the Normal Curve

The same way a line is defined by its slope
and y-intercept, a normal curve is defined by
its
____________________ and ________________

_____________________.
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 _________________________________________, 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:
o ___________% of the observations are within one standard deviation of the mean
o ___________% of the observations are within two standard deviations of the mean
o ___________% of the observations are within three standard deviations of the mean
IQ scores on the WISC-IV are normally distributed with a mean of 100 and a standard
deviation of 15.
o Going up one σ and down one σ from 100 gives us the range from 85 to 115. ________
of people have an IQ between 85 and 115.
o 95% of people have an IQ between ____ and ____.
o 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.
o Between what two heights does the middle 95% fall?
o The tallest 2.5% of women are taller than what?
o What is the percentile for a woman who is 64.5 inches tall?