Download Empirical Rule Notes and Example

DENSITY CURVES AND THE NORMAL DISTRIBUTIONS
Quarter 1 we learned graphical and numerical tools for describing distributions.
We have a clear strategy for exploring data on a single quantitative variable:
• Always plot your data: make a graph, usually a histogram or a stemplot.
• Look for the overall pattern (shape, center, spread) and for striking deviations
such as outliers.
• Calculate a numerical summary to briefly describe the center and spread.
Here is one more step to add to the strategy:
• Sometimes the overall pattern of a large number of observations is so regular
that we can describe it by a smooth curve.
Density curves
The figure below is a histogram of the scores of all 947 seventh-grade students in
Gary, Indiana, on the vocabulary part of the Iowa Test of Basic Skills1. Data from
Gary Community School Corporation, courtesy of Celeste Foster, Department of
Education, Purdue University. Scores of many students on this national test have a
quite regular distribution. The histogram is symmetric, and both tails fall off quite
smoothly from a single center peak. There are no large gaps or obvious outliers.
The smooth curve drawn through the tops of the histogram bars is a good
description of the overall pattern of the data.
The curve is a mathematical model for the distribution. A mathematical model is an
idealized description. It gives a compact picture of the overall pattern of the data
but ignores minor irregularities as well as any outliers. We will see that it is easier
to work with the smooth curve than with the histogram. The reason is that the
histogram depends on our choice of classes, while with a little care we can use a
curve that does not depend on any choices we make.
A density curve is a curve that



Is always on the horizontal axis.
Has an area equal to 1 underneath it.
Idealized Histogram (continuous instead of discrete).
A density curve describes the overall pattern of the distribution.
The area under the curve and between any range of values = relative
frequency of all observations that fall in that range.
Therefore: AREA = RELATIVE FREQUENCY Total Area = 1
We use different symbols when we’re measuring population characteristics
versus sample characteristics.
Population:
Mean = µ (Greek Letter mu)
Standard Deviation =
Sample:

(Greek Letter sigma)
Mean =
Standard Deviation = s
(We will typically use Greek letters for populations and English notation for
samples.)
Normal Distributions
Normal Curves:
Characteristics:
 Symmetric
 Unimodal
 Bell-shaped
 Mean = Median = Center of curve
 The higher the standard deviation, the more spread out (flatter) the curve
 The center of the curve is determined by µ



It’s height and width are determined by
The points at which the change of curvature take place are called the
inflection points. They are located at distance = µ ±

The 68-95-99.7 Rule (Empirical Rule)
In a normal distribution with mean µ and standard deviation
68% of the observations fall within 1 •


:
of the mean, µ
 of µ
99.7% of the observations fall within 3 •  of µ
95% of the observations fall within 2 •
Normal Distribution with mean µ and standard deviation


is
represented: N(µ,
)
Example: a Normal Distribution with a mean of 3 and a standard deviation of
1 would be represented as: N(3, 1).
Use the Empirical Rule to Describe Data That Are Bell Shaped
If data have a distribution that is bell shaped, the Empirical Rule can be used to
determine the percentage of data that will lie within k standard deviations of the
mean.
Empirical Rule
Using the Empirical Rule
Problem: Use the data from University A in the table below.
(a) Determine the percentage of students who have IQ scores within 3 standard
deviations of the mean according to the Empirical Rule.
(b) Determine the percentage of students who have IQ scores between 67.8 and
132.2 according to the Empirical Rule.
(c) Determine the actual percentage of students who have IQ scores between 67.8
and 132.2.
(d) According to the Empirical Rule, what percentage of students will have IQ
scores above 132.2?
Approach: To use the Empirical Rule, a histogram of the data must be roughly bell
shaped. Figure below shows the histogram of the data from University A.
Solution: The histogram of the data drawn in the figure above is roughly bell
shaped. We know that the mean IQ score of the students enrolled in University A is
100 and the standard deviation is 16.1. To help organize our thoughts and make
the analysis easier, we draw a bell-shaped curve like the one in figure below, with
the x = 100 and s = 16.1.
See the figure on the top of the next page:
(a) Determine the percentage of students who have IQ scores within 3 standard
deviations of the mean according to the Empirical Rule.
(b) Determine the percentage of students who have IQ scores between 67.8 and
132.2 according to the Empirical Rule.
(c) Determine the actual percentage of students who have IQ scores between 67.8
and 132.2.
(d) According to the Empirical Rule, what percentage of students will have IQ
scores above 132.2?
Answers
(a) According to the Empirical Rule, approximately 99.7% of the IQ scores will be
within 3 standard deviations of the mean. That is, approximately 99.7% of the data
will be greater than or equal to 100 – 3(16.1) = 51.7 and less than or equal to
100 + 3(16.1) = 148.3
(b) Since 67.8 is exactly 2 standard deviations below the mean
[100 – 2(16.1) = 67.8] and 132.2 is exactly 2 standard deviations above the mean
[100 + 2(16.1) = 132.2], we use the Empirical Rule to determine that
approximately 95% of all IQ scores lie between 67.8 and 132.2.
(c) Of the 100 IQ scores listed in Table 7, 96, or 96%, are between 67.8 and 132.2.
This is very close to the approximation given by the Empirical Rule.
(d) Based on the figure above, approximately 2.35% + 0.15% = 2.5% of students
at University A will have IQ scores above 132.2.