Download Standard Normal Distribution Curves and Probability

GRAPHING OF NORMAL DISTRIBUTION CURVE AND
GRAPHICAL DETERMINATION OF PROBABILITY
ON THE TI-83 OR TI-84 PLUS CALCULATOR
We have previously demonstrated that frequency and relative frequency
histograms provide an excellent visual representation of the distribution of data. If data is
normally distributed about the data mean, then when the profile of the histogram bars is
“smoothed,” a bell shaped curve is created that is symmetrical about the mean value.
This is called a normal curve. In the special case when the data mean is 0 and the
standard deviation from the mean is 1, the curve is referred to as a standard normal
curve:
NORMAL CURVES
 = 0.5
=1



=2
Symmetrical about the mean.
Amplitude and “spread” of curve depends on standard deviation and graph scale
STANDARD Normal Curve - Mean ( x )= 0; Standard Deviation
As with the relative frequency histogram, the area under the normal curve directly
represents the probability that the modeled event or measurement will occur between
given lower and upper data values.
Both normal and standard normal curves may be drawn on the TI-83 graphing
calculator using the Normal Probability Density Function (normalpdf) located in the
DISTR menu.
GRAPHING THE STANDARD NORMAL CURVE:
 Clear any functions from the [Y = ] equation editor screen.
 Paste the Probability Density Function into the [Y = ] screen as follows:
 Press [2nd][VARS] to get to the DISTR menu.
 The first choice of normalpdf( will be automatically selected. Press
ENTER to paste into [ Y = ] screen.
 Type in the variable x, the mean of 0 and standard deviation of 1, all
separated by commas. Your entry should look as follows:

Set the graph window dimensions in the [WINDOW] menu to the values
shown below:

Press the [GRAPH] button. The standard normal curve should be displayed.
GRAPHING OF NORMAL DISTRIBUTION CURVE AND
GRAPHICAL DETERMINATION OF PROBABILITY
ON THE TI-83 OR TI-84 PLUS CALCULATOR
GRAPHING A NORMAL CURVE:
If you choose to work with data that is not “normalized” to standard normal
values (i.e. mean is not 0 and standard deviation is not 1), then a normal curve may also
be graphed on the TI-83. The procedure varies slightly from that of the Standard Normal
curve since the Window dimensions will change depending on the value of the mean and
standard deviation for the data set. Graph a normal curve as follows:
 Paste the Probability Density Function into the [Y = ] screen as described
earlier and enter the actual data mean and standard deviation values.
 In the [WINDOW] menu, set the Xmin and Xmax values so that the mean
falls between them.
 Press the [ZOOM] key and select option 0 [0:ZOOMFIT]. Refine your
minimum and maximum scale as necessary.
PROBABILITY:
Once you have graphed the normal curve, the probability that the modeled event
will occur between given minimum and maximum values may be graphically determined
via the area under the curve as follows (NOTE: Instructions are based on a standard normal curve):
 With the normal curve displayed, press [2nd][TRACE] to get to [CALC]
menu.
 Select option 7 [7:  f(x)dx ]. This option represents the calculus concept of


integration where the area under the curve of f(x) is calculated with respect to
a lower (starting) and upper (ending) limit on the value of x.
Enter a value for the lower limit and press [ENTER]. Repeat for upper limit.
As soon as you enter the upper limit, the area under the curve between the two
limits is shaded and the calculated area is displayed below as  f(x)dx . This
area therefore represents the probability of the event occurring between the
lower and upper limit.
Shown for standard normal curve with lower limit of –1.3 and upper limit of 0.75



To clear the shading and integration calculation, press [2nd][PRGM] to get to
the [DRAW] menu and then select option 1 [1:ClrDraw].
To complete integration of the upper “tail” of a standard normal curve for a
probability calculation such as P ( x  1.234) , enter the lower limit as 1.234 and
the upper limit as 4 (the upper end of the display window). Since the area
under the normal curve at greater than  3 standard deviations is very small,
this upper limit will yield a fairly accurate probability value.
Likewise for integration of the lower tail on a standard normal curve, set the
minimum value to –4 (the lower end of the display window).
F.Y.I. The probability density function is given by f ( x) 
1
2
 ( x  x )2
e
2 2
where   0