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High School Mathematics - 2
11.8 Normal or Gaussian Distributions

Probability Distribution

A probability distribution describes the values and probabilities that a random event can take place. The values must cover all of the possible outcomes of the event, while the total probabilities must sum to exactly 1, or 100%. For example, a single coin flip can take values Heads or Tails with a probability of exactly 1/2 for each; these two values and two probabilities make up the probability distribution of the single coin flipping event.

Normal Distributions

Normal distributions, also called Gaussian distributions, are a family of probability distributions that have the same general shape. All normal distributions are symmetric and have bell-shaped density curves with a single peak.

The normal distributions are a very important class of statistical distributions. Many psychological measurements and physical phenomena (like noise) can be approximated well by the normal distribution.

Characteristics of Normal Distributions

  • Each member of the family may be defined by two parameters, location and scale: the mean (m) and variance or square of standard deviation (s2), respectively. The standard normal distribution is the normal distribution with a mean of zero and a variance of one.

  • In a normal distribution, the mean m (mu) indicates the location of the peak density, and the standard deviation s (sigma), indicates the spread or girth of the curve.

  • Normal distributions differ in how spread out they are, but the area under each curve is the same.

  • Mathematical formula for a normal distribution is:


    • m - mean
    • s - standard deviation
    • p - constant 3.14159
    • e - is the base of natural logarithms = 2.718282
    • X can take on any value from -infinity to + infinity

  • Different values of m and s yield different normal density curves and hence different normal distributions.

  • About 68% of values drawn from a normal distribution are within one standard deviation away from the mean; about 95% of the values are within two standard deviations and about 99.7% lie within three standard deviations (see picture below).

The Standard Normal Distribution or Z Distribution

  • The standard normal distribution is a normal distributions with a mean of 0 and a standard deviation of 1. Normal distributions can be transformed to standard distributions by the formula:

    Z = X - m/s

    Where X is a score from the original normal distribution, m is the mean of the original normal distribution, and s is the standard deviation of original normal distribution.

    The standard normal distribution is sometimes called the Z distribution. A 'Z score' always reflects the number of standard deviations above or below the mean a particular score is. For instance, if a person scored a 70 on a test with a mean of 50 and a standard deviation of 10, then he scored 2 standard deviations above the mean. Converting the test scores to Z scores, an X of 70 would be:

    Z = 70-50/10 = 2

    So, a Z score of 2 means the original score was 2 standard deviations above the mean. Note that the Z distribution will only be a normal distribution if the original distribution (X) is normal.

    Directions: Answer the following questions:
    • What is a probability distribution?
    • Define normal distribution and give its significance.
    • What is a standard normal distribution? Give examples.
    Answer the following objective questions.

  • Q 1: If scores are normally distributed with a mean of 30 and a standard deviation of 5, about what percent of the scores are between 28 and 34?

    Q 2: What proportion of a normal distribution is within one standard deviation of the mean?

    Q 3: If some test scores are normally distributed with a mean of 30 and a standard deviation of 5, what percent of the scores is greater than 30?

    Q 4: What proportion is between 1 and 1.5 standard deviations above the mean?

    Q 5: The standard normal distribution is also called
    Z distribution
    skewed distribution
    discrete distribution
    square distribution

    Q 6: If scores are normally distributed with a mean of 30 and a standard deviation of 5, about what percent of the scores is greater than 37?

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