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 abellshaped curve known as a normal distribution. When appropriate,using a normal distribution model to represent distributions thatoccur in reallife situations can be extremely useful in statisticalanalysis.
Density curve:
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 teetertotter.)
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 teetertotter) is 6
  
 




 

 
             


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A special type of density curves form normal distributions.These distributions are bellshaped, 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
In the normal distribution,
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 TI83, 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 TI83, 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 testtakers scored higher than you did.
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