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    Normal distribution curve and the association to individually

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    This explains various basic theories and components and provides step-by-step
    explanation with examples, of the premise behind the normal distribution curve and how it relates to the definition, measurement, and computation of individually administered intelligence tests

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    Solution Preview

    A normal distribution has a bell-shaped density curve described by its mean and standard deviation. The density curve is symmetrical, centered about its mean, with its spread determined by its standard deviation. The normal distribution refers to a family of continuous probability distributions described by the equation:

    Normal equation. The value of the random variable Y is:

    Y = {1/ [ σ * sqrt(2π) ] } * e-(x - μ)2/2σ2

    where X is a normal random variable,
    μ is the mean, σ is the standard deviation,
    π is approximately 3.14159, and e is approximately 2.71828.

    The random variable X in the normal equation is called the normal random variable. The normal equation is the probability density function for the normal distribution. In terms of the normal curve, the graph of the normal distribution depends on - the mean and the standard deviation. The mean determines the location of the center of the graph, while the standard deviation determines the height and width of the graph. When the standard deviation is large, the curve is short and wide; when the standard deviation is small, the curve is tall and narrow. All normal distributions tend to look like symmetric, bell-shaped curves.

    In this example, the left curve is shorter and wider than the curve on the right, since it has a bigger standard deviation. The normal distribution is a continuous probability distribution, with several implications for probability:
    The total area under the normal curve is equal to 1.
    The probability that a normal random variable X equals any particular value is 0.
    The probability that X is greater than a equals the area under the normal curve bounded by a and plus infinity (as indicated by the non-shaded area seen below).
    The probability that X is less than a equals the area under the normal curve bounded by a and minus infinity (as indicated by the shaded area seen below).

    Additionally, every normal curve (regardless of its mean or standard deviation) will and should conform to the following "rule".
    About 68% of the area under the curve falls within 1 standard deviation of the mean.
    About 95% of the area under the curve falls within 2 standard deviations of the mean.
    About 99.7% of the area under the curve falls within 3 standard deviations of the mean.
    Such points are usually referred to as the empirical rule or the 68-95-99.7 rule. Given a normal distribution, most outcomes will be within 3 standard deviations of the mean.
    An Example

    An average light bulb manufactured by the Acme Corporation lasts 300 days with a standard deviation of 50 days. Assuming that bulb life is normally distributed, what is the probability that an Acme light bulb will last at most ...

    Solution Summary

    The random variable X in the normal equation is called the normal random variable. The normal equation is the probability density function for the normal distribution. In terms of the normal curve, the graph of the normal distribution depends on - the mean and the standard deviation. The mean determines the location of the center of the graph, while the standard deviation determines the height and width of the graph. When the standard deviation is large, the curve is short and wide; when the standard deviation is small, the curve is tall and narrow.

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