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Wize AP Statistics Textbook > The Normal Distribution

Normal Distribution

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Normal Distribution

A continuous random variable XX follows a normal distribution if its probability distribution curve has the following properties:

  1. It is symmetric about the mean
  2. It has a bell-shape (mean=median=mode)
  3. The total area under the curve is equal to 1
  4. P(a<X<b)P\left(a<X<b\right) is the same as the area underneath the bell-curve, between the values aa and bb
  5. The horizontal axis measures the possible value of the continuous random variable
  6. The vertical axis measures the frequency or %’s

Note
We denote this as X ∼ N(μ, σ)\boxed{X\ \sim\ N\left(\mu,\ \sigma\right)}

Wize Tip
Suppose you plot continuous data in a histogram and draw a smooth curve connecting the tops of the bars. If this curve forms a symmetrical bell-shape, then we say that the data is normally distributed or follows the normal model.

Examples
  • The height of a randomly selected student from a particular University
  • The weight of an apple that is picked from an apple tree
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Shape of a Normal Distribution

Two normal distributions can have the same mean but different standard deviations:

Example:



Two normal distributions can have different means and the same standard deviation:

Example:


Portions of information contained in this publication/book are printed with permission of Minitab, LLC. All such material remains the exclusive property and copyright of Minitab, LLC. All rights reserved.





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Empirical Rule (68 - 95 - 99.7% Rule)

The Empirical Rule states that almost all of the data fall within three standard deviations for a normal distribution.

68% Rule
  • The probability that XX takes on a value between μ−σ\mu-\sigma and μ+σ\mu+\sigma is approximately 68%
  • 68% of the data lies within 1 standard deviation away from the mean

95% Rule
  • The probability that XX takes on a value between μ−2σ\mu-2\sigma and μ+2σ\mu+2\sigma is approximately 95%
  • 95% of the data lies within 2 standard deviations away from the mean

99.7% Rule
  • The probability that XX takes on a value between μ−3σ\mu-3\sigma and μ+3σ\mu+3\sigma is approximately 99.7%
  • 99.7% of the data lies within 3 standard deviations away from the mean

Car batteries are designed to last on average 5 years with standard deviation of 1.3 years. Assume that battery life is normally distributed.

a) Using the 68-95-99.7% rule, what percentage of batteries will last between 2.4 and 3.7 years?



b) Using the 68-95-99.7% rule, what proportion of batteries will last under 2.4 years or more than 6.3 years?