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Statistics in Applied Science and Technology

Chapter 6 The Normal Distribution

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Key Concepts in This Chapter

• The importance of normal distribution

• Properties of normal distribution

• Standard normal curve

• Z score

• Area under the normal curve

• Standard normal table

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The Importance of Normal Distribution

• Countless phenomena follow (or closely approximate) the normal distribution. For example: height, serum cholesterol, life span of light bulbs, etc.

• When a distribution of a variable given, inferences can be drawn as to how frequently certain observation are likely to occur.

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The Importance of Normal Distribution

• Mathematically speaking, normal distribution is easy to manipulate.

• Many statistical theory and methodology are developed based on the assumption that data are distributed approximately normally.

• Note: Certain non-parametric statistic methodology is required when distribution is not normal or unknown.

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Properties of the Normal Distribution

• Symmetrical bell-shaped curve.

• It is symmetrical about its mean, • Its standard deviation is expressed as • Values of the mean, median, and the mode

are always identical.

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Properties of the Normal Distribution (Cont’d)

• The total area under the normal curve represents the entire observations.

• The relative area between any two designated points is always the same (68.26% of the area is contained within , 95.45% within 2, and 99.74% within 3).

• The amount of area under the normal curve is directly proportional to the percentage of raw scores.

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• Standard normal curve is the one and only one normal curve with a mean of 0 and standard deviation of 1.

• Any normal distribution can be transformed into standard normal curve by creating a new variable z score.

Standard Normal Curve

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Area under the Normal Curve and Z score

• Z score can be calculated by:

• Area under the standard normal curve can be found in standard normal table (Table A)

x

Z - mean

- standard deviation

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Standard Normal Table

• Consists of columns of z cores coordinated with columns of proportions.

• Used to find proportion below a score, between two scores, beyond pairs of scores

• Proportion find in standard normal table equals to proportion of the entire observations.