- Represent the shape of a histogram if a huge amount of data had been collected
- Are always above the axis
- Have a total area between them and the axis equal to one
- In density curves, as in histograms, area is proportional to the number of values
- Since the total area is 1 the area under a density curve above a range of values equals the proportion of values in that range
Random number density curves
The Bell Curves
- Are density curves that represent many data sets
- Also are called density curves of normal distributions
- Are all the same shape except for stretching or shrinking
- The mean and standard deviation can be read off the curve
- If you know just the mean and standard deviation you can sketch the density curve
All bell curves obey the 68% - 95% - 99.7% rule which says that
- 68% of all values are within one standard deviation of the center (i.e. the mean)
- 95% of all values are within two standard deviations of the mean
- 99.7% of all values are within three standard deviations of the mean
Exact precentages, and percentages for other ranges of values, can be found by converting the values in question into standard deviation units = Z , and then consulting tables + -
The conversion formula into standard deviations from the mean is
called a z-score
Example calculations of standard deviatons from the mean ( Z values)
We can use these Z-values along with the standard normal tables to find the proportion of values below a given value, above a given value or between two values in any normal distribution.
Example normal proportion calculations
Finding positions in a nornal distribution for a given a proportion
Long and short tailed
distributions
The tails of a somewhat bell shaped distribution are often compared to those of the standard bell curve.
Distributions where the data values in the tails are less spread out than in the standard bell curve are referred to as short tailed distributions.
Distributions where the data values in the tails are more spread out are referred to as long tailed or heavy tailed distributions.
- The red line represents a frequency curve of a long tailed distribution.
- The blue line represents a frequency curve of a short tailed distribution.
- The black line is the standard bell curve..