A normal density curve is an idealized representation of a normal distribution in which the area under the curve is defined to be 1.
- The points at which the curve changes from being concave up to being concave down are called the inflection points
- On a normal density curve, these inflection points are always exactly one standard deviation away from the mean
The Empirical Rule for Normal Distributions
One way to calculate the area under a density curve up to a certain point is by using the Empirical Rule.
The Empirical Rule states that
- 50% of all data points are above the mean and 50% are below
- Approximately 68% of all data points are within 1 standard deviation of the mean
- Approximately 95% of all data points are within 2 standard deviations of the mean
- Approximately 99.7% of all data points are within 3 standard deviations of the mean
If the data point you are interested in is not a whole number standard deviation away from the mean, you use a z-score.
A z-score measures how many standard deviations a score is away from the mean. The z-score of the term in a population distribution whose mean is and whose standard deviation is is given by: .
Use a z-score probability table to find a decimal expression of the percentage of values that are less than .