The-Normal-Distribution, Statics and Pro

statistics and probability pptx

1. The Normal Distribution RICO d. pACALA

2. NORMAL DISTRIBUTION To illustrate continuous random variables, we usually use a curve rather than a histogram for its distribution. The normal distribution is often referred to as Gaussian distribution in honor of Carl Friedrich Gauss , a German mathematician who first used the context of the normal curve to analyze astronomical data.

3. NORMAL DISTRIBUTION The normal distribution is a continuous probability distribution that describes data that clusters around the mean. The graph of the associated probability density function is bell–shaped, with a peak at the mean known as the bell curve.

4. NORMAL DISTRIBUTION Gaussian distribution is characterized by two parameters: mean (𝜇) and standard deviation (𝜎). Its main functions are: to determine the ordinates (y-values) of the graph that represents the distribution, and to determine the probabilities.

5. NORMAL DISTRIBUTION The normal distribution can be used to describe, at least approximately, any variable that tends to cluster around the mean. For example, the weights of adult females in Region V are roughly normally distributed. Most women have weight close to the mean, though a small number of outliers have a weight significantly above or below the mean. A histogram of female weights will appear similar to a bell curve if more data is used.

6. Histogram for the distribution of weights of Adult Female in Region V

7. Histogram for the distribution of weights of Adult Female in Region V Form the illustration , if the researcher selects a random sample of 100 adult female, measures their weight, and constructs a histogram, the researcher will get similar histogram in Figure (a). If the researcher increases the sample size and decreases the width of the classes, the histogram will look like in Figure (b) and (c). Lastly, if we possibly measure all the weights of all adult in Region V, the histogram will become a normal distribution.

8. NORMAL DISTRIBUTION If a random variable has a probability distribution whose graph is continuous, bell-shaped, and symmetric, it is called a normal distribution. The graph is called a normal distribution curve.

9. NORMAL DISTRIBUTION

10. NORMAL DISTRIBUTION The shape and position of a normal distribution curve depend on two parameters, the mean and the standard deviation. Each normally distributed variable has its own normal distribution curve, which depends on the values of the variable’s mean and standard deviation.

11. NORMAL DISTRIBUTION This shows two normal curves with same mean values but with different standard deviations .

12. NORMAL DISTRIBUTION above shows that the two normal curves have the same standard deviation with different mean values. They are identical in form but are centered in different positions along horizontal axis.

13. Remember this!! When the standard deviation is large , the normal curve is short and wide , while a small value for the standard deviation yields a taller and skinnier graph .

14. Summary of the Properties of the Theoretical Normal Distribution 1. A normal distribution curve is bell-shaped. 2. The mean, median, and mode are equal and are located at the center of the distribution. 3. A normal distribution curve is unimodal (i.e., it has only one mode). 4. The curve is symmetric about the mean, which is equivalent to saying that its shape is the same on both sides of a vertical line passing through the center.

15. Summary of the Properties of the Theoretical Normal Distribution 5. The curve is continuous; that is, there are no gaps or holes. For each value of X, there is a corresponding value of Y. 6. The curve never touches the x axis. Theoretically, no matter how far in either direction the curve extends, it never meets the x axis—but it gets increasingly close. 7. The total area under a normal distribution curve is equal to 1.00, or 100%. This fact may seem unusual, since the curve never touches the x axis, but one can prove it mathematically by using calculus.

16. Summary of the Properties of the Theoretical Normal Distribution 8. The area under the part of a normal curve that lies within 1 standard deviation of the mean is approximately 0.68, or 68%; within 2 standard deviations, about 0.95, or 95%; and within 3 standard deviations, about 0.997, or 99.7%.

17. Summary of the Properties of the Theoretical Normal Distribution Empirical Rule

18. Standard Normal Distribution it is important to note that a normal distribution can be converted into a standard normal distribution, the mean will become zero (𝝁 = 𝟎) and the standard deviation will become one (𝝈 = 𝟏). The corresponding distribution is called standard normal distribution and is commonly denoted by Z.

19. Finding Areas Under the Standard Normal Distribution Curve RICO D. PACALA

20. Areas under the Normal Curve In a normal distribution, the probability of two given values is equal to the area under the curve between these values. To manually compute the probability of any problem relative to normal distribution, we will use z-table to transform the value of random variable 𝑥 to z-score or standard score.

21. Areas under the Normal Curve To solve the probability using the areas under the normal curve, we will consider the following cases: CASE Illustration How to find I. Between O and any z-value Refer on the Standard Normal Table (Z table)

22. Areas under the Normal Curve CASE Illustration How to find II. To the left of Positive z-value Add 0.5 to the area between 0 and the z-value. III. To the left of Negative z-value Subtract the area between 0 and the z-value from 0.5. CASE Illustration How to find II. To the left of Positive z-value III. To the left of Negative z-value

23. Areas under the Normal Curve CASE Illustration How to find IV. To the Right of Positive z-value Subtract the area between 0 and the z-value from 0.5. V. To the Right of Negative z-value Add 0.5 to the area between 0 and the z-value. CASE Illustration How to find IV. To the Right of Positive z-value V. To the Right of Negative z-value

24. Areas under the Normal Curve CASE Illustration How to find VI. Between two Positive z-value To find the area between two z-scores with same signs, subtract the smaller area to the bigger area. VII. Between two Negative z-value CASE Illustration How to find VI. Between two Positive z-value VII. Between two Negative z-value

25. Areas under the Normal Curve CASE Illustration How to find VIII. Between negative and Positive z-value to find the area between two z-scores with opposite signs, we have to add the area of the two z – scores. CASE Illustration How to find VIII. Between negative and Positive z-value

26. SW04-022824 Find the area under the standard normal curve. 1. to the left of z=-1.83 2. to the left of z=2.23 3. to the right of z=3.01 4. to the right of z=-1.09 5. between z=-0.25 and z=1.86 6. between z=-1.37 and z=-0.03

27. “the new power is not money in the hand of the few, but the information in the hand of many” -John Naisbitt

28. STANDARD SCORE RICO D. PACALA

29. STANDARD SCORE In statistics, the standard score is the number of standard deviations by which the value of a raw score is above or below the mean value of what is being observed or measured. Raw scores above the mean have positive standard scores, while those below the mean have negative standard scores.

30. STANDARD SCORE A 𝒛-score can be placed on a normal distribution curve where the scores range from −3 standard deviations (which would fall to the left most part of the normal distribution curve) up to +3 standard deviations (which would fall to the far right of the normal distribution curve). In order to use 𝒛- score, one needs to know the mean (𝝁) and also the standard deviation (𝝈)

31. STANDARD SCORE The z-score is found by using the following equations: A. For Sample Where: Z=standard score x=raw score or observed value = sample mean S=sample standard deviation  

32. STANDARD SCORE The z-score is found by using the following equations: B. For Population Where: Z=standard score x=raw score or observed value µ= population mean =population standard deviation  

33. EXAMPLE 1: On final examination in Biology , the mean was 75 and the standard deviation was 12. Determine the standard score of the student who received a score of 60 assuming that the scores are normally distributed.

34. EXAMPLE 2: On the first periodic exam in statistics, the population mean was 70 and the population standard deviation was 9. Determine the standard score of a student who got a score of 88 assuming that the scores are normally distributed.

35. EXAMPLE 3: Luz scored 90 in an English test and 70 in a Physics test. Scores in the English test have a mean of 80 and a standard deviation of 10. Score in Physics test have a mean of 60 and a standard deviation of 8. In which subject was her standing better assuming that their scores in her English and Physics class are normally distributed?

36. EXAMPLE 4: In Science test, the mean score is 42 and the standard deviation is 5. Assuming the scores are normally distributed, what percent of the score is a. greater that 48? b. less than 50 c. between 30 and 48.

37. EXAMPLE 5: The mean height of grade 9 students at a certain high school is 164 centimeters and the standard deviation is 10 centimeters. Assuming the heights are normally distributed, what percent of the heights is greater than 168 centimeters?

38. EXAMPLE 6: In a math test, the mean score is 45 and the standard deviation is 4. Assuming normality, what is the probability that a scored picked at random will lie A. above score 50? B. below score 38?

39. EXAMPLE 7: Assuming that the scores of Grade 11 students in General Mathematics in their 2 nd quarter test are normally distributed with a mean of 48 and standard deviation of 6. If the z-score of student is 2, find his raw score.

40. EXAMPLE 8: The mean height of 1000 students at a certain elementary school is 140 cm and the standard deviation is 10 cm. Assuming that the height are normally distributed, how many students stand : a. between 120 and 145 cm? b. more than 150 cm?

41. EXAMPLE 9: Given a normal distribution with population and population variance , find the value of x that leaves 12.3% of the area to the left of the z-score of x.  

42. EXAMPLE 10: Given a normal distribution with population and population variance , find the value of x that leaves 30.5% of the area to the left of the z-score of x.  

43. EXAMPLE 11: Consider a normal distribution with a mean value of 120 and standard deviation of 6. Find the value of x: a. if the area from the mean to the z-scores is 19.5% and the z-score is negative. b. if the area represents the top 15% of the distribution.

44. SW05-030124 Answer the following by showing your complete solution: 1. The scores of students in the Final exam in Pre-calculus has a mean of 32 and a standard deviation of 5. Find the 𝒛-scores corresponding to each of the following: a. 37 b. 22 2. The scores of a group of students in a qualifying exam are normally distributed with a mean of 60 and standard deviation of 8. a. How many percent of the students got below 72? b. If there were 250 students who took the test, about how many students scored higher than 64?

45. SW05-030124 Answer the following by showing your complete solution: 3. An international university only admits top 5% of the total examinees in their entrance exam. The results of this year’s entrance exam follow a normal distribution with a mean of 285 and standard deviation of 12. What is the least score of an examinee who can be admitted to the university?