Normal Distribution, Biostatistic in health

Normal distribution in biostatistic

1. Normal Distribution By: Fuldisia Butt Acknowledgment: Muhammad Ishtiaq

2. What is data? Types of Variables? What is Probability?

3. Spread more to left All jumble up Spread more to right

4. Bell Shape Curve It looks like a bell

5. Types of Distribution

6. Normal (Gaussian) Distribution The normal distribution is a descriptive model that describes real world situations. It is defined as a continuous frequency distribution of infinite range (can take any values not just integers as in the case of binomial and Poisson distribution). This is the most important probability distribution in statistics and important tool in analysis of epidemiological data and management science.

7. Normal Distribution The mode represents the “high point” of the graph of the distribution Median represents the point where 50% of the area under the distribution is to the left and 50% of the area under the distribution is to the right. Mean represents the balancing point of the graph of the distribution

8. Normal Distribution For symmetric distributions with a single peak, such as the normal distribution, the Mean = Median = Mode. The inflection points are the points on the curve where the curvature of the graph changes. To the left of x = µ- σ and to the right of x = µ+ σ the curve is drawn upward.

9. Effect of µ and σ on Curve One density curve has µ = 0 and σ = 1 and the other has µ = 3 and σ = 1 Increasing the mean from 0 to 3 caused the graph to shift three units to the right but maintained its shape.

10. Description of effect of µ and σ on Curve One density curve has µ = 0 and σ = 1 and the other has µ = 0 and σ = 2 increasing the standard deviation from 1 to 2 causes the graph to become flatter and more spread out but maintained its location of center.

11. Properties of Normal Distribution

12. 1. It is symmetric about its mean, µ ( mirror images)

13. 2. Because there is Mean = Median = Mode. a single peak and the highest point occurs at x = µ

14. 3. It has inflection point at µ- σ and µ + σ

15. 4. The area under the curve to the right of µ equals the area under the curve to the left of µ which equals 1/2

16. 5. The area under the curve is 1.

17. 6. The Empirical Rule Approximately 68% of the area under the normal curve is between µ- σ and µ + σ Approximately 95% of the area under the normal curve is between µ-2 σ and µ +2 σ Approximately 99.7% of the area under the normal curve is between µ-3 σ and µ +3 σ See Figures on next slides

18. Another way of looking at 1 SD 68.26% of Cases

19. Another way of looking at 2 SD 95.44% of Cases

20. Another way of looking at 3 SD 99.72% of Cases

21. Interpreting the area under a Normal Curve Question The serum total cholesterol for males 20 to 29 years old is approximately normally distributed with mean µ = 180 and σ = 36.2 and based on data obtained from the National Health and Nutrition Examination Survey.

22. Step-1 & 2 Draw the normal curve with the mean µ = 180 labeled at the high point and the inflection points at µ- σ = 180-36.2 = 143.8 and µ+ σ = 180+36.2 = 216.2 Shade the region under the normal curve to the right of x = 200

23. Step-3 The two interpretations for the area of this shaded region are; the proportion of 20- to 29-year-old males that have high cholesterol is 0.2903 the probability that a randomly selected 20- to 29-year-old male has high cholesterol is 0.2903.

24. Exercise The relative frequency distribution given in Table 1 represents the heights of a pediatrician’s 200 three-year-old female patients. The raw data indicate that the mean height of the patients is µ = 38.72 i nches with standard deviation σ = 3.17 i nches.

25. Exercise (a ) Draw a relative frequency histogram of the data. Comment on the shape of the distribution. (b) Draw a normal curve with µ = 38.72 inches and σ = 3.17 inches on the relative frequency histogram. Compare the area of the rectangle for heights between 40 and 40.9 inches to the area under the normal curve for heights between 40 and 40.9 inches.

26. Solution of exercise Graph shows the relative frequency distribution. The relative frequency histogram is symmetric and bell-shaped.

27. Cont… This graph, the normal curve with µ = 38.72 and σ = 3.17 is superimposed on the relative frequency histogram. The figure demonstrates that the normal curve describes the heights of 3-year-old females fairly well. We conclude that the heights of 3-year-old females are approximately normal with µ = 38.72 and σ = 3.17

28. Cont… This graph also shows the rectangle corresponding to heights between 40 and 40.9 inches. The area of this rectangle represents the proportion of 3-year-old females between 40 and 40.9 inches. Notice that the area of this shaded region is very close to the area under the normal curve for the same region, so we can use the area under the normal curve to approximate the proportion of 3-year-old females with heights between 40 and 40.9 inches!

29. Cont…

30. Probability density function (or pdf ). Don’t feel threatened by this equation, because we will not be using it at this level. Instead, we will use the normal distribution in graphical form by drawing the normal curve,

31. Relation between a Normal Random Variable and a Standard Normal Random Variable z-score allows us to transform a random variable X with mean µ and standard deviation σ into a random variable Z with mean 0 and standard deviation 1. Z = X - μ Z indicates how many standard deviations away from the mean the point x lies. Z score is calculated to 2 decimal places. σ

32. Exercise The heights of a pediatrician’s 200 three-year-old female patients mean µ = 38.72 i nches with σ = 3.17 i nches. We wish to demonstrate that the area under the normal curve between 35 and 38 inches is equal to the area under the standard normal curve between the z-scores corresponding to heights of 35 and 38 inches.

33. Solution of Exercise figure shows the normal curve with mean µ = 38.72 with σ = 3.17. region between x = 35 and x = 38 is shaded. With mean µ = 38.72 with σ = 3.17, the standardized version of x = 35 is; 35-38.72/3.17 = -1.17 the standardized version of x = 38 is 38-38.72/3.17 = -0.23

34. Standard normal curve with the region between z = -1.17 and z = -0.23 shaded.

35. Standard Normal Distribution The standard normal distribution has a mean of 0 and a standard deviation of 1 The first method uses a table of areas that has been constructed for various values of Z. The second method involves the use of statistical software or a graphing calculator with advanced statistical features.

36. Standard Normal Curve to the Left of a z-Score Find the area under the standard normal curve that lies to the left of z = 1.68? Enclosed the row that represents 1.6 and the column that represents 0.08.The value located where the row and column intersect is the area we are seeking. The area to the left of z = 1.68 is 0.9535.

37. Standard Normal Curve to the Right of a z-Score Find the area under the standard normal curve to the right of z = -0.46? Find the row that represents -0.4 and the column that represents 0.06 in Table. Identify where the row and column intersect. This value is the area to the left of z = -0.46 is 0.3228

38. The area under the standard normal curve to the right of z = -0.46 is 1 minus the area to the left of z = -0.46 1-0.3228 = 0.6772

39. Standard Normal Curve between Two z-Scores Find the area under the standard normal curve between z = -1.35 and z = 2.01? = 0.9778-0.085 =0.8893

40. Application/Uses of Normal Distribution It’s application goes beyond describing distributions It is used by researchers and modelers. The major use of normal distribution is the role it plays in statistical inference. The z score along with the t –score, chi-square and F-statistics is important in hypothesis testing. It helps managers/management make decisions.