July, 2000Guang Jin Statistics in Applied Science and Technology Chapter 6 The Normal Distribution

July, 2000Guang Jin 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

July, 2000Guang Jin 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.

July, 2000Guang Jin 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.

July, 2000Guang Jin 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.

July, 2000Guang Jin 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.

July, 2000Guang Jin 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

July, 2000Guang Jin 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)  - mean  - standard deviation

July, 2000Guang Jin 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.