Download Notes: 3.1 Normal Density Curves

Notes: 3.1 Normal Density Curves
Learning Target  Use the properties of a normal curve to calculate areas and probabilities.
♦ I can use the properties of a normal density curve to sketch a normal distribution♦
Density Curve: Any curve (graph) where the area under the curve = 1 (100%).
We use density curves to simplify histograms.
Example 1: Simplify the following histograms by drawing density curves.
Normal Density Curve: A special density curve that has the following properties:
 Density Curve (total area under curve = 1)
 Symmetrical (mean = median)
 Bell shaped
 Most of the data is within + 3 standard deviations of the mean (99.7%)
 The distribution of data follows the Empirical Rule (68-95-99.7)
 The inflection point on the curve corresponds to + 1 standard deviation
Empirical Rule: Also called the 68-95-99.7 Rule. It is used to estimate the area (determine probabilities) under
a normal density curve within +1, +2, and +3 standard deviations from the mean.
In a normal distribution:
+ 1 standard deviation away from the mean contains 68% of the data
+ 2 standard deviations away from the mean contains 95% of the data
+ 3 standard deviations away from the mean contains 99.7% of the data
Intro to Stats
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Notes: 3.1 Normal Density Curves
♦ I can use the Empirical Rule to calculate the probability of an event based on a normal distribution ♦
Example 1: Use the curve below to show the percent of data within 1, 2, and 3 standard deviations away from
the mean
Example 2: Use the Empirical Rule (68-95-99.7) to determine the area under the curve within each standard
deviation.
The mean = median so half of the data is above the mean and half is below.
68% of data is within 1 standard deviation, so 34% (half) will be below the mean and 34% will be above.
95% of data is within 2 standard deviations, 47.5% on each side of the mean. Since 34% is within 1 standard
deviation, then 47.5 – 34 = 13.5. The area between 1 and 2 standard deviations is 13.5%
99.7% is within 3 standard deviations, 49.85% on each side. Since 47.5 is within 2 standard deviations, then
49.85 – 47.5 = 2.35. The area between 2 and 3 standard deviations is 2.35%
99.7% of the data is within 3 standard deviations (not quite 100%). That leaves 0.3% outside of 3 standard
deviations or 0.15% on either side
Intro to Stats
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