4.17 The Normal Distribution and the Empirical Rule

Learning Objectives

  • What is a normal distribution?
  • The Empirical Rule (the 68 – 95 – 99.7 Rule)
  • Drawing a normal distribution for a particular scenario
  • Finding the percentage as it pertains to a particular scenario

What is a normal distribution?

Early statisticians noticed the same shape coming up over and over again in different distributions—so they named it the normal distribution.

Features of the Normal Distribution

  • It is bell-shaped
  • It is symmetric about the mean.  In other words, if you divide the graph in half by drawing a vertical line at the mean, there will be two identical halves
  • The mean, median and mode are equal and located at the center of the distribution
  • It has no gaps or holes so we would say the distribution is continuous and extends out along the horizontal axis from  −∞ to +∞
  •  An area under the distribution that falls between two data values is the same as the percentage of the data (in decimal form) between the same data values
  •  The total area under a normal distribution is exactly 1 or 100%

THE EMPIRICAL RULE

For data that is bell-shaped and symmetrical (Normal):

  • Approximately 68% of the data is within +/- 1 standard deviations of the mean.
  • Approximately 95% of the data is within +/- 2 standard deviations of the mean.
  • Approximately 99.7% of the data is within +/- 3 standard deviations of the mean.

Example – Drawing a Normal Distribution

The trunk diameter of a certain variety of pine tree is normally distributed with a mean of μ=150cm and a standard deviation of σ=30cm.  Sketch a normal curve that describes this distribution.

  • Step 1:  Sketch a blank normal distribution
  • Step 2:  The mean of 150 cm goes in the middle
  • Step 3:  Label each standard deviation.  Each standard deviation is a distance of 30 cm

Example – Finding Percentages

A certain variety of pine tree has a mean trunk diameter of μ=150cm and a standard deviation of σ=30cm

a)  Approximately what percent of these trees have a diameter greater than 210?

b)  Approximately what percent of these trees have a diameter between 90 and 210 centimeters?

a)

  • Step 1:  Sketch a normal curve with a mean of 150 cm and a standard deviation of 30 cm
  • Step 2:  Shade the region corresponding to a diameter greater than 210 cm.
  • Step 3:  Add the percentages in the shaded area

2.35%+0.15%=2.5%

About 2.5% of these trees have a diameter greater than 210 cm.

b)

  • Step 1:  Use the same distribution from part a
  • Step 2:  Shade the region between 90 and 210 cm
  • Step 3:  Add the percentages in the shaded area.

13.5% + 34% + 34% + 13.5% = 95%

About 95% of these trees have a diameter between 90 and 210 cm.

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