1. Normal Distribution Farrokh Alemi Ph.D.
This lecture was organized by Dr. Alemi.
Farrokh Alemi Ph.D.

2. Normal Curve
Among all the distributions we see in practice, the most common is a symmetric, unimodal, bell curve distribution. It is so common, that it is known as the normal curve or normal distribution. Here we see a normal curve with mean of 52 and standard deviation of 6.

3. Normal Curve
Normal curves are always symmetric around their mean. The portion of the curve to the left of the mean is the mirror image of the curve to the right of the mean.

4. Normal Curve
Always unimodel, meaning that it has one high peak.

5. Normal Curve
But different variability. Here both curves have a mean of 52 but one has standard deviation of 6 and another 12.

6. Normal Curve
And different means. Here we see two normal curves with same variability around their means but different means, one at 76 and another at 52.

7. Distribution Parameters
Mean
Standard deviation
Because the mean and standard deviation describe a normal distribution exactly, they are called the distribution’s parameters.

8. Near Normal
Many variables are nearly normal, but none are exactly normal. Thus the normal distribution, while not perfect for any single problem, is very useful for a variety of problems.

9. Near Normal
Many processes are near normal. For example, as sample size increases the average of the sample becomes increasing like normal. Here we see 4 distributions. First one is normal, the second one is uniform, the third one is exponential and the fourth is parabolic. No matter what is the initial distribution, as the size of the sample increases, the average of the sample is distributed like a normal distribution.

10. Near Normal
Here we see the average of two observations. Already the distribution is becoming more like normal distribution.

11. Near Normal
Here we see the average of 5 observations. The symmetric uni-modal shape of normal distribution can be seen clearly for the first two distributions. The average of 5 observations from exponential distributions looks not completely symmetric but it is uni-modal. Even the average of 5 observations from parabolic distribution , seems to wiggle its way near to a normal uni-modal and symmetric distribution.

12. Near Normal
If taking the average of 30 observations, the distribution of average is near normal. As the number of observations averaged increases, the average increasingly has a normal distribution. This makes normal distribution relevant for analysis of average of observations, even when these observations are not normal.

13. Near Normal
A binomial distribution can be approximated by normal distribution when there are large number of observations. Here we see the histogram for binomial distribution of 16 patients having MRSA infection versus the same distribution curve drawn using normal curve. Normal distribution is pervasive and applies to many situations.

14. Standard Normal
A standard normal distribution has a mean of 0 and a standard deviation of 1. The most common value is the mean.

15. Standard Normal
68.27% of data fall within 1 standard deviation of standard normal curve. The area marked in yellow.

16. Standard Normal
95.45% of the data fall within 2 standard deviations. This is the white and the yellow area under the standard normal curve shown here.

17. Standard Normal
99.73% of data fall within 1 standard deviation of standard normal curve.

18. Standard Normal
Knowing how many standard deviations an observation is an observation away from the mean of a distribution tells us the probability of how likely it is to observe it. This provides an easy method of understanding how unusual an observation is.

19. Standard Normal
An observation becomes less frequent and more unusual depending on number of standard deviations it is away from the mean. At 1 standard deviation, 84% of observations fall below it.
84%

20. Standard Normal
At 2 standard deviations, 98% of observations fall below it.
98%

21. Standard Normal
At 3 standard deviations, 99% of the observations fall below it. The chances of observing a value more than 3 standard deviation away from the mean falls below 1 in 1000.
99%

22. Standard Normal Tables
Standard normal tables exists that tell us how common a value is based on how many standard deviation it is away from the mean. To count the number of standard deviation an observation is away from a mean, we need the Z score.

23. Z Score
If the observation is one standard deviation above the mean, its Z score is 1. If it is 1.5 standard deviations below the mean, then its Z score is This equation shows the Z score for an observation x that follows a distribution with mean µ and standard deviation σ. While the original distribution is centered around mean µ and has standard deviation of σ, the Z scores transform the distribution to a standard normal distribution.

24. Example
Assuming that patient ratings are normally distributed and we have 3 ratings of 3, 4, and 5. What is the mean and standard deviation?
Let us take a very simple example to demonstrate the power of normal distribution. Assume that patient ratings are normally distributed and we have 3 ratings of 3, 4, and 5. What is the mean and standard deviation?

25. Example
The mean is 4 and the standard deviation is 1. Given this mean and standard deviations we can answer other questions.

26. Example
If mean rating is 4 and the standard deviation is 1, what is the probability of observing a rating of 1?
In this question we are asking how unusual it is to observe a rating of 1. To answer this question we need to calculate the Z value.

27. Example
The Z value is -3. We can now look up in standard normal tables to see what percent of values are more than 3 standard deviation below the mean.

28. Standard Normal Table
This shows a standard normal table. The columns show the last two digit of the Z score. The rows show the Z score to the first digit. The cell values give the percent of values falling below the Z score.

29. Standard Normal Table
Zooming in, we can select for the second digit the column corresponding with 0.00 and the row corresponding with -3.0.

30. Standard Normal Table
We read that the probability of observing a Z of less than is Therefore observing a rating of 1 is quite rare.

31. Take Home Lesson
Normal distribution allows us to examine how rare an observed value is.

32. Do One
Assume that the boarding time in an emergency room is normal with average of 6 hours and standard deviation of 2 hours. What percent of patients stay after the day shift of 8 hours.
See if you can answer this question