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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