NORMAL CURVE STANDARD SCORES AND THE NORMAL CURVE AREA UNDER THE NORMAL CURVE AREA UNDER THE NORMAL CURVE 9/11/2013

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1 NORMAL CURVE AND THE NORMAL CURVE Prepared by: Jess Roel Q. Pesole Theoretical distribution of population scores represented by a bell-shaped curve obtained by a mathematical equation Used for: (1) Describing distributions of scores (2) Interpreting the standard deviation (3) Making statements of probability CHARACTERISTICS OF THE NORMAL CURVE 1. Symmetrical can be divided into the equal halves 2. Unimodal has only one peak of maximum frequency; it is also where the mean, median, and mode is found 3. Asymptotic theoretically, the tails never touch the base line but extend to infinity in either direction SIGNIFICANCE OF THE NORMAL CURVE Some variables are assumedto be normally distributed; as such, the sampling distributions of various statistics are known or assumed to be normal This assumption does not differ radically from the real world (ex. Height, IQ), but some variables may not conform to this assumption (ex. Distribution of wealth) PRINCIPLE # 1:The area under the normal curveis the area that lies between the curve and the base line containing 100% or all of the cases in any given normal distribution PRINCIPLE # 2:A constant proportion of the total area under the normal curve will lie between the mean and any given distance from the mean, as measured in sigma units (σ) 1

2 PRINCIPLE # 3:Any given sigma distance above the mean contains the identical proportionof distance as the same sigma distance below the mean Important Proportions: μto 1σ = 34.13% of cases or p = μto 2σ = 47.72% of cases or p = μto 3σ = 49.87% of cases or p = Important Proportions: -1σ to +1σ = 68.26% of cases or p = σ to +2σ = 95.44% of cases or p = σ to +3σ = 99.74% of cases or p = Transformed score that designates how many standard deviation units the corresponding raw score is above or below the mean Allows us to compare scores that are otherwise not directly comparable = (for population data) = (for sample data) Characteristics: 1. The z-scores have the same shape as the set of raw 2. The mean of the z-scores always equals zero. 3. Standard deviations of z-scores always equals 1. 2

3 Example # 1(Finding the z- You got a score of 80 in your midterm examinationforhistory,withameanof60anda standard deviation of 11. For economics, you got a score of 70, with a mean of 55 and a standard deviation of 13. Compare the relative position of the two = = = = 1.82 = = = = 1.15 INTERPRETATION: Since the z-score of tiyrexam score in History is higher, this means that its relative position is higher than that of the z-score for the Economics exam. Example # 2(Finding the z- A person scores 81 on a test of verbal ability and 6.4 on a test of quantitative ability. For verbal ability test, the mean for the people in general is 50 and the Sdis 20. For the quantitative ability test, the mean for people in general is 0 and the Sdis 5. Which is this person s stronger ability, verbal or quantitative? Explain and interpret your answer. 3

4 Example # 3(Finding the area given the raw A child scored 109 on the Wechsler IQ test. This test is scaled in the population to have a mean of 100 and a standard deviation of 15. What is its percentile rank? Example # 3(Finding the Z = = = 0.60 Example # 3(Finding the area given the raw STEP 3:Determine the percent of area at or below the z-score. NOTE: Use the table of the areas under the curve for this step. Based on the table, 22.57% of the distribution lie between the mean and z. Add 50% to cover the other end of the curve. This gives you 72.57%. Example # 3(Finding the The percentile rank of the child is %. This means that percent of those in the population who took the exam got a score lower than the child. Example # 4(Finding the area given the raw The Scholastic Achievement Test (SAT) is standardized to be normally distributed with a mean of 500 and a standard deviation of 100. What percentage of the SAT scores falls above 600? Example # 4(Finding the Z = = = 1 4

5 Example # 4(Finding the area given the raw STEP 3:Determine the percent of area at or below the z-score. Based on the table, 15.87% of the distribution lie beyond the z % of the scores are found above 600. Example # 5(Finding the area given the raw The Scholastic Achievement Test (SAT) is standardized to be normally distributed with a mean of 500 and a standard deviation of 100. What percentage of the SAT scores falls between 450 and 600? Example # 5(Finding the Z = = -0.5 Z = = = = 1 Example # 5(Finding the area given the raw STEP 3:Determine the percent of area at or below the z-score. Based on the table, 34.13% of the distribution lie between the mean and the z-score for 600. On the other hand, 19.15% of the distribution lie between the mean and the z-score for % of the scores are found between 450 and 600. EXERCISE IQ scores are normally distributed with a mean μ= 100 and a standard deviation σ= 15. Based on this distribution, determine: a. The percentage of scores between 88 and 120. b. The percentage of scores that are 110 or above c. The percentile rank corresponding to an IQ score of 125 Pediatric data reveal that the average child is toilet trained at 26 months, but that there is a 2- month standard deviation from this norm. A. What percentage of children are toilet trained by 23 months? B. A mother is concerned that her son was trained at 30 months. What is the percentile rank of her child? How common is it for toilet training to occur at or beyond 30 months? C. A mother is pleased that her son is trained at 18 months. What is the percentile rank of this prodigy? How likely is it that a child would be toilet trained by this age? 5