Download Stat6_Normal_Curve.doc Ch 5 - The Normal Curve

Stat6_Normal_Curve.doc College Math
Natural Phenomena many times result in "Normal" Curves
Line Graph
Ch 5 - The Normal Curve
55
46
37
28
19
10
1
Prob. of x Heads
Probabilities
0.6
Many phenomena, especially natural phenomena, result in
0.4
probability histograms that are bell shaped, symmetric, and
Series1
would "fit" the equation for a "normal distribution". By
0.2
repeatedly decreasing the class sizes of a histogram, and
0
then connecting the midpoints of the pedestals, we can
obtain (in some cases) curves that are very close to the
Lengths of Gobies normal distribution.
Centimeters
The example graphs below and
Microsoft Excel 97 Line Graph
to the right show the results of
Graph A
tossing n = 6 to 200 coins with
the relative frequency of the
The sum of the areas of the rectangles is one.
number of heads as the tops of
The area under the curve is close to one. As the
numbers of rectangles increase, the area under
the pedestals. These relative
.2
the curve gets closer and closer to one. We can
frequencies are, as usual, the
then think of probabilities as the same as the
empirical probabilities that a
areas under the curve.
certain number of heads will
result from n tosses of the coin.
In the lower right examples, only
the tops of the pedestals were
0
1
2
3
4
5
6->x
graphed (as little boxes for
simplicity) and a smooth curve was graphed through them. In all of the graphs, the "pedestals" would have widths of one
(1). The heights of the pedestals (boxes) are the probabilities P(x) - see graph A to the right. When we use the formula A
= W x L for the area of
The experiments below consist of flipping increasing numbers of coins and determining the probabilities of
a rectangle to calculate
getting 1, 2, 3, etc. heads. As the number of flips increases, the curve becomes "more" normal.
the areas, with W = 1
and L = P(x), each
rectangle then has an
area equal to the
probability of the class.
Adding all of the areas
will then give us one
The vertical scales (probabilities) vary from box to box (as do the "Windows").
(1), the sum of the
The horizontal scales represent the number of heads…from 0 to n All the areas are 1.0.
probabilities (which is
always one (1) for the sums of all the probabilities for a sample space). In this manner, we associate the areas of the
rectangles with the probabilities of the numbers of heads. Then we can find any probability geometrically by adding the
appropriate areas and we can consider the area as equal (or synonymous) to the probability.
The Normal distributions at the left are both normal. They have been
superimposed on each other. They are both bell-shaped but have
different means and standard deviations. The wide one has a standard
deviation of 5.
The narrow one is a STANDARD NORMAL
DISTRIBUTION. It has a mean of 0 and a standard deviation of 1. The
areas between one standard deviation on either side of the means are
.6827 for both of them. Because of the complexity of calculating the areas
(and hence probabilities) under the curves, we "standardize" all normal
distributions with a simple formula, and then use one table to look up the
probabilities. For the same number of standard deviations, the areas
will be the same for all normal distributions.
Total area under any
probability curve is 1,
regardless of the shape.
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Created on 7/19/98 8:53 AM Last printed 3/29/02 7:00 AM R Mower, Instructor