Density curves and normal distributions

"Statisticalthinking will one day be as necessary for efficient citizenship asthe ability to read and write." -- H.G. Wells, 1866-1946

2.1 DENSITY CURVES AND NORMALDISTRIBUTIONS (Pages 64-79)

OVERVIEW: Sometimes the overall pattern of adistribution is such that we can describe it with a smooth curve. Itis remarkable how many natural phenomena appear to be related to abell-shaped curve known as a normal distribution. When appropriate,using a normal distribution model to represent distributions thatoccur in real-life situations can be extremely useful in statisticalanalysis.

Density curve:

Displays the overall pattern (shape) of a distribution.
Has an area of exactly 1 sq. unit underneath it.
Is on or above the horizontal axis.
A histogram becomes a density curve if the scale is adjusted so that the total area of the bars is 1 sq. unit.

The median of a density curve is the point that divides thearea under the curve into halves.
The mean of a density curve is the "balance point" of thecurve. (Think of a teeter-totter.)

As an illustration, consider the set {1,2,3,5,11,14}.If each box with an X has an area of 1/6, then the total area of thesix boxes would be 1. The median of this set is 4, and the mean (thebalance point of the teeter-totter) is 6

X	X	X		X						X			X
1	2	3	4	5	6	7	8	9	10	11	12	13	14
			M E D I A N		M E A N

A special type of density curves form normal distributions.These distributions are bell-shaped, and a normal curve is determinedby the mean (m) and standard deviation(s) of the data set. While it will not beused directly in this course, the formula for the normal distributionfunction, which involves the two amazing numbers pi ande, is

y = [ssqrt(2pi)]^-1e^{[-.5(x-m)(x-m)]/(ss)}

In the normal distribution,

68% of the observations fall within 1 standard deviation of the mean.
95% of the observations fall within 2 standard deviations of the mean.
99.7% of the observations fall within 3 standard deviations of the mean.

A normal distribution curve has two points where curvaturechanges. These are called points of inflection, and they arelocated 1 standard deviation on either side of the mean.

On the TI-83, normalcdf(lowerbound,upperbound,mean,standarddeviation) can be very useful in statistical analysis. Note that if anormal distribution has mean = 0 and st.dev. = 1, then

normalcdf(-1,1,0,1) = .6826894809
normalcdf(-2,2,0,1) = .954499876
normalcdf(-3,3,0,1) = .9973000656

Also, on the TI-83, if you let y1 = normalpdf(x,0,1), setXmin = -4, Xmax = 4, and use ZoomFit, you will see a graphicrepresentation of the normal distribution curve with mean = 0 andstandard deviation = 1.

An observation's percentile is the percent of thedistribution that is at or to the left of the observation. If, forinstance, if you have a test score representing the 90th percentile,then only 10% of the test-takers scored higher than you did.

Extremely important for success of the Advanced PlacementStatistics Examination:
If you are given a numerical data set, always (I repeat,always) display the shape of the distribution.
Using the TI-83, this can be done very easily with a histogram or aboxplot.

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