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9/11/2013 NORMAL CURVE STANDARD SCORES AND THE NORMAL CURVE Prepared by: Jess Roel Q. Pesole CHARACTERISTICS OF THE NORMAL CURVE • 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 SIGNIFICANCE 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 • Some variables are assumed to 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) AREA UNDER THE NORMAL CURVE AREA UNDER THE NORMAL CURVE • PRINCIPLE # 1: The area under the normal curve is 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 9/11/2013 AREA UNDER THE NORMAL CURVE AREA UNDER THE NORMAL CURVE • PRINCIPLE # 3: Any given sigma distance above the mean contains the identical proportion of distance as the same sigma distance below the mean Important Proportions: • μ to 1σ = 34.13% of cases or p = 0.3413 • μ to 2σ = 47.72% of cases or p = 0.4772 • μ to 3σ = 49.87% of cases or p = 0.4987 AREA UNDER THE NORMAL CURVE Important Proportions: • -1σ to +1σ = 68.26% of cases or p = 0.6826 • -2σ to +2σ = 95.44% of cases or p = 0.9544 • -3σ to +3σ = 99.74% of cases or p = 0.9974 STANDARD SCORES (Z SCORES) STANDARD SCORES (Z SCORES) • 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) STANDARD SCORES (Z SCORES) • Characteristics: 1. The z-scores have the same shape as the set of raw scores. 2. The mean of the z-scores always equals zero. 3. Standard deviations of z-scores always equals 1. 2 9/11/2013 STANDARD SCORES (Z SCORES) • Example # 1 (Finding the z-score): You got a score of 80 in your midterm examination for history, with a mean of 60 and a 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 scores. STANDARD SCORES (Z SCORES) INTERPRETATION: Since the z-score of tiyr exam score in History is higher, this means that its relative position is higher than that of the z-score for the Economics exam. STANDARD SCORES (Z SCORES) = = − = − = = = = 1.82 = 1.15 STANDARD SCORES (Z SCORES) • Example # 2 (Finding the z-score): 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 Sd is 20. For the quantitative ability test, the mean for people in general is 0 and the Sd is 5. Which is this person’s stronger ability, verbal or quantitative? Explain and interpret your answer. 3 9/11/2013 STANDARD SCORES (Z SCORES) • Example # 3 (Finding the area given the raw score): 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? STANDARD SCORES (Z SCORES) Example # 3 (Finding the area given the raw score): STEP 1: Convert into zscores. Z = = = 0.60 STEP 2: Sketch your curve. STANDARD SCORES (Z SCORES) Example # 3 (Finding the area given the raw score): 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%. STANDARD SCORES (Z SCORES) Example # 3 (Finding the area given the raw score): ANSWER: The percentile rank of the child is 72.57 %. This means that 72.57 percent of those in the population who took the exam got a score lower than the child. STANDARD SCORES (Z SCORES) Example # 4 (Finding the area given the raw score): 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? STANDARD SCORES (Z SCORES) Example # 4 (Finding the area given the raw score): STEP 1: Convert into zscores. Z = = =1 STEP 2: Sketch your curve. 4 9/11/2013 STANDARD SCORES (Z SCORES) Example # 4 (Finding the area given the raw score): 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. ANSWER: 15.87% of the scores are found above 600. STANDARD SCORES (Z SCORES) Example # 5 (Finding the area given the raw score): STEP 1: Convert into zscores. Z = = = Z = = =1 0.5 STEP 2: Sketch your curve. 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 STANDARD SCORES (Z SCORES) Example # 5 (Finding the area given the raw score): 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? STANDARD SCORES (Z SCORES) Example # 5 (Finding the area given the raw score): 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 450. ANSWER: 53.28% of the scores are found between 450 and 600. • Pediatric data reveal that the average child is toilet trained at 26 months, but that there is a 2month 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