AP STATS
DENSITY CURVES AND THE NORMAL DISTRIBUTIONS
Chapter 2.1
The histogram displays the Grade equivalent vocabulary scores for 7th graders on
the Iowa Test of Basic Skills. The scores of students on this national test have a
regular distribution.
 This histogram is mostly symmetric
 Both tails fall of smoothly from the center peak
 There are no obvious gaps or outliers.
 THE SMOOTH CURVE IS A GOOD DESCRIPTION OF THE OVERALL
PATTERN
To change from a histogram to a smooth curve –
 Use PROPORTIONS of the observations fall in each range rather than
actual counts of observations.
 Each bar area will represent the proportion of observations in that class
 For the curve, the area under the curve represents the proportions of the
observations.
 Adjust the scale of the graph so that the total area under the curve is equal
to 1.
DENSITY CURVES
 Describe the overall shape of distributions
 Idealized mathematical models for distributions
 Show patterns that are accurate enough for practical
purposes
 Always on or above the horizontal axis
 The total area under the curve is exactly 1
 Areas under the curve represent relative frequencies of
observations
MEASURES OF CENTER ON DENSITY CURVES
The MEDIAN (M) is the point with half the observations on either side. The
QUARTILES divide the area under the curve into quarters.
The MEAN (or arithmetic average) is the point at which the curve would balance
is made of a solid material.
DENSITY CURVES can be symmetrical or skewed. Remember that for
symmetrical distributions, the Median and Mean are equal.
Because a Density Curve is an idealized description of the
distribution of data, we must distinguish between the
mean ( x ) and standard deviation (s) computed from the
actual observations and the mean (  ) and standard
deviation ( ) of the idealized distribution.
EXAMPLES:
Consider the unusual density curve.
Find the % of the data in the
following intervals.
0 < X < 0.6 ?
0.2 < X < 0.4 ?
0 < X < 0.8 ?
What would the density curve for a uniform distribution look like?
HW: Chapter 2 – 2.1 through 2.5