A density curve follows 3 rules:
- Always on or above the horizontal axis
- Has an area of exactly 1 underneath it
- Not necessarily a curved shape
Describing Density Curves
The measures of center and spread apply to density curves as well as to actual sets of observations. For density curves, the area under the curve represents proportions of the total number of observations.
The median of a density curve is the point with half the area to the left and right.
Because density curves are idealized patterns, a symmetric density curve is exactly symmetric with the mean and median as the same.
The median of a density curve is still much more resistant to outliers and skewness than the mean.
Because a density curve is idealized, not actual observations, the notation used for the mean is μ and for the standard deviation is σ.
Normal Distributions
All Normal curves have the same overall shape: symmetric, single-peaked, and bell-shaped. Specific Normal distributions are described using their mean and standard deviation in this form: N(μ, σ)
The mean is located at the center of the symmetric curve and is the same as median. Changing μ without changing σ moves the normal curve along the horizontal axis without changing its spread.
σ controls the spread of a Normal curve. Curves with larger standard deviations are more spread out.
68-95-99.7 rule
In a Normal curve, 68% of the data lies 1 standard deviation away from the mean. 95% of the data lies 2 standard deviations away from the mean. 99.7% of the data lies 3 standard deviations away from the mean.
How to use the calculator to find z-score/percentile
Use normalCDF to find the area from z-score
Use invNorm to find z-score from area or percentile
Finding areas in any Normal distribution
- State distribution and values of interest, draw normal curve, and shade and label values
- Perform calculations
- Answer question