Stats: Normal Distribution
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Definitions
Central Limit Theorem
Theorem which stats as the sample size increases, the sampling distribution of the sample means will become approximately normally distributed.
Correction for Continuity
A correction applied to convert a discrete distribution to a continuous distribution.
Finite Population Correction Factor
A correction applied to the standard error of the means when the sample size is more than 5% of the population size and the sampling is done without replacement.
Sampling Distribution of the Sample Means
Distribution obtained by using the means computed from random samples of a specific size.
Sampling Error
Difference which occurs between the sample statistic and the population parameter due to the fact that the sample isn't a perfect representation of the population.
Standard Error or the Mean
The standard deviation of the sampling distribution of the sample means. It is equal to the standard deviation of the population divided by the square root of the sample size.
Standard Normal Distribution
A normal distribution in which the mean is 0 and the standard deviation is 1. It is denoted by z.
Z-score
Also known as z-value. A standardized score in which the mean is zero and the standard deviation is 1. The Z score is used to represent the standard normal distribution.


Stats - Normal Distributions
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Any Normal Distribution
• Bell-shaped
• Symmetric about mean
• Continuous
• Never touches the x-axis
• Total area under curve is 1.00
• Approximately 68% lies within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations of the mean. This is the Empirical Rule mentioned earlier.
• Data values represented by x which has mean mu and standard deviation sigma.
• Probability Function given by
Standard Normal Distribution
Same as a normal distribution, but also ...
• Mean is zero
• Variance is one
• Standard Deviation is one
• Data values represented by z.
• Probability Function given by
Normal Probabilities
There is a table which must be used to look up standard normal probabilities. The z-score is broken into two parts, the whole number and tenth are looked up along the left side and the hundredth is looked up across the top. The value in the intersection of the row and column is the area under the curve between zero and the z-score looked up.
Because of the symmetry of the normal distribution, look up the absolute value of any z-score.
Computing Normal Probabilities
There are several different situations that can arise when asked to find normal probabilities.
Situation Instructions
Between zero and
any number Look up the area in the table
Between two positives, or
Between two negatives Look up both areas in the table and subtract the smaller from the larger.
Between a negative and
a positive Look up both areas in the table and add them together
Less than a negative, or
Greater than a positive Look up the area in the table and subtract from 0.5000
Greater than a negative, or
Less than a positive Look up the area in the table and add to 0.5000
This can be shortened into two rules.
1. If there is only one z-score given, use 0.5000 for the second area, otherwise look up both z-scores in the table
2. If the two numbers are the same sign, then subtract; if they are different signs, then add. If there is only one z-score, then use the inequality to determine the second sign (< is negative, and > is positive).
Finding z-scores from probabilities
This is more difficult, and requires you to use the table inversely. You must look up the area between zero and the value on the inside part of the table, and then read the z-score from the outside. Finally, decide if the z-score should be positive or negative, based on whether it was on the left side or the right side of the mean. Remember, z-scores can be negative, but areas or probabilities cannot be.
Situation Instructions
Area between 0 and a value Look up the area in the table
Make negative if on the left side
Area in one tail Subtract the area from 0.5000
Look up the difference in the table
Make negative if in the left tail
Area including one complete half
(Less than a positive or greater than a negative) Subtract 0.5000 from the area
Look up the difference in the table
Make negative if on the left side
Within z units of the mean Divide the area by 2
Look up the quotient in the table
Use both the positive and negative z-scores
Two tails with equal area
(More than z units from the mean) Subtract the area from 1.000
Divide the area by 2
Look up the quotient in the table
Use both the positive and negative z-scores
Using the table becomes proficient with practice, work lots of the normal probability problems!

Standard Normal Probabilities
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359
0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753
0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141
0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517
0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879
0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224
0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549
0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852
0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133
0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389
1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621
1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830
1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015
1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177
1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319
1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441
1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545
1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633
1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706
1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767
2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817
2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857
2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890
2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916
2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936
2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952
2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964
2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974
2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981
2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986
3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990
The values in the table are the areas between zero and the z-score. That is, P(0<Z<z-score)