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Applications of Normal Distributions

Using computational skill and technology to sketch and shade appropriate area under the normal curve

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Area Under Density Curve of Normal Distribution

A normal density curve is an idealized representation of a normal distribution in which the area under the curve is defined to be 1.

  • The points at which the curve changes from being concave up to being concave down are called the inflection points 
  • On a normal density curve, these inflection points are always exactly one standard deviation away from the mean

The Empirical Rule for Normal Distributions

One way to calculate the area under a density curve up to a certain point is by using the Empirical Rule.

The Empirical Rule states that 

  • 50% of all data points are above the mean and 50% are below
  • Approximately 68% of all data points are within 1 standard deviation of the mean
  • Approximately 95% of all data points are within 2 standard deviations of the mean
  • Approximately 99.7% of all data points are within 3 standard deviations of the mean

License: CC BY-NC 3.0

[Figure1]

Z-Score

If the data point you are interested in is not a whole number standard deviation away from the mean, you use a z-score.

A z-score measures how many standard deviations a score is away from the mean.  The z-score of the term \begin{align*}x\end{align*} in a population distribution whose mean is \begin{align*}\mu\end{align*} and whose standard deviation is \begin{align*}\sigma\end{align*} is given by: \begin{align*}z=\frac{x-\mu}{\sigma}\end{align*} .

Use a z-score probability table to find a decimal expression of the percentage of values that are less than \begin{align*}x\end{align*}.


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