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Chapter 14-4 Normal Distribution

The normal probability distribution, depicted in this Interactive Diagram, is an example of a frequency distribution. If you know, or assume, that a population is normally distributed for a measure of a particular trait or behavior, you can predict the probability that members of the population express a particular value of that trait. The value of a trait is recorded on the graph's horizontal axis; the frequency of occurrence of the trait's values is recorded on the vertical axis.

By default, the normal distribution is typically shown partitioned in multiples of the standard deviation .

Initially, this Interactive Diagram has a mean of 0 and a standard deviation of 1.0. You can change the mean and standard deviation to fit any particular situation, by adjusting the entry fields at the upper left of the diagram.

If you select the "Draggable partitions" button above the graph, you can choose to divide the display of the frequency distribution using a) a single partition, b) paired independently-draggable partitions, or c) paired symmetrically-draggable partitions. Try each of these to examine the behavior of the diagram. The flags on the partitions display the partitions' positions relative to the value of the particular population (top flag) and the Z-score for the distribution (bottom flag.)

The colored areas on the pie chart correspond to the colored portions on the normal distribution. The pie chart wedges correspond to the probability of a data point lying in the same-color region on the distribution.



Questions
1. If a frequency distribution can be regarded as a "measure of central tendency," what is the central tendency in a normal distribution?

2. You may find it useful to select a single draggable partition, and position it to help you answer the following questions.

  1. What percent of the normal distribution has values less than the mean?

  2. What percent of the distribution has values less than one standard deviation below the mean?

  3. What percent of the distribution has values greater than one standard deviation above the mean?

3. You may find it useful to select a paired partitions to help you answer the following questions:

  1. Determine the probability that the trait of a randomly-selected individual from a normally-distributed population is greater than two standard deviations below the mean, but less than the mean.

  2. Determine the probability that a member of the population has a trait value between one s.d. below the mean and one s.d. above the mean.

4. If you change the mean or standard deviation in this Interactive Diagram, what is the effect on the graph? Explain.

5. Consider the use of Z-scores in the following questions:

  1. What is the relation between Z-score and standard deviation?

  2. For a population with mean of 30 and standard deviation of 10, demonstrate with this diagram the probabilities that a randomly-selected individual from the population will exhibit the following Z-scores:
a) Z < -3
b) Z > 1
c) -0.5 < Z < 0.5
d) 2.5 < Z < 3