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This tutorial is intended to assist students in understanding the normal curve. Created by Del Siegle University of Connecticut 2131 Hillside Road Unit 3007 Storrs, CT 06269-3007 Modified by Dr. Finney, Foss High School May 10, 2011 Your next mouse click will display a new screen. c. 2000 Del Siegle [email protected] Suppose we measured the right foot length of 30 teachers and graphed the results. Number of People with that Shoe Size If our second subject had a 9 inch foot, we would add her to the graph. As we continued to plot foot lengths, a 8 pattern would begin to emerge. 7 6 5 4 Your next 3 mouse 2 click will 1 display a 4 5 6 7 8 9 10 11 12 13 14 Length of Right Foot new screen. c. 2000 Del Siegle Assume the first person had a 10 inch foot. We could create a bar graph and plot that person on the graph. 4 5 6 7 8 9 10 11 12 13 14 Length of Right Foot new screen. c. 2000 Del Siegle Number of People with that Shoe Size Notice how there are more people (n=6) with a 10 inch right foot than any other length. Notice also how as the length becomes larger or smaller, there are fewer and fewer people with that measurement. This is a characteristics of many variables that we measure. There is a tendency to have most measurements in the middle, and fewer as we approach the high and low extremes. If we were to connect the top of each bar, we 8 would create a frequency polygon. 7 6 5 4 Your next 3 mouse 2 click will 1 display a 8 7 6 5 4 3 2 1 4 5 6 7 8 9 10 11 12 13 14 Length of Right Foot c. 2000 Del Siegle Number of People with that Shoe Size You will notice that if we smooth the lines, our data almost creates a bell shaped curve. You will notice that if we smooth the lines, our data almost creates a bell shaped curve. 8 7 6 5 4 3 2 1 4 5 6 7 8 9 10 11 12 13 14 Length of Right Foot Your next mouse click will display a new screen. c. 2000 Del Siegle Number of People with that Shoe Size This bell shaped curve is known as the “Bell Curve” or the “Normal Curve.” 9 8 7 6 5 4 3 2 1 12 13 14 15 16 17 18 19 20 21 22 Points on a Quiz Your next mouse click will display a new screen. c. 2000 Del Siegle Number of Students Whenever you see a normal curve, you should imagine the bar graph within it. The Nowmean, lets look mode, at quiz andscores median forwill 51all students. fall on the same value in a normal distribution. 12 13 13 12+13+13+14+14+14+14+15+15+15+15+15+15+16+16+16+16+16+16+16+16+ 17+17+17+17+17+17+17+17+17+18+18+18+18+18+18+18+18+19+19+19+19+ 14 14 14 14 19+ 19+20+20+20+20+ 21+21+22 = 867 15 15 15 15 15 15 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 22 16 16 16 16 16 16 16 16 867 / 51 = 17 17 17 17 17 17 17 17 17 17 18 18 18 18 18 18 18 18 19 19 19 19 19 19 21 21 9 8 7 6 5 4 3 2 1 22 12 13 14 15 16 17 18 19 20 21 22 Points on a Quiz Your next mouse click will display a new screen. c. 2000 Del Siegle Number of Students 20 20 20 20 Your next mouse click will display a new screen. c. 2000 Del Siegle Normal distributions (bell shaped) are a family of distributions that have the same general shape. They are symmetric (the left side is an exact mirror of the right side) with scores more concentrated in the middle than in the tails. Examples of normal distributions are shown to the right. Notice that they differ in how spread out they are. The area under each curve is the same. Mathematical Formula for Height of a Normal Curve The height (ordinate) of a normal curve is defined as: where m is the mean and s is the standard deviation, p is the constant 3.14159, and e is the base of natural logarithms and is equal to 2.718282. x can take on any value from -infinity to +infinity. Your next mouse click will display a new screen. This information is not needed for EPSY341. It is provided for your information only. c. 2000 Del Siegle f(x) is very close to 0 if x is more than three standard deviations from the mean (less than -3 or greater than +3). If your data fits a normal distribution, approximately 68% of your subjects will fall within one standard deviation of the mean. Approximately 95% of your subjects will fall within two standard deviations of the mean. Your next mouse click will display a new screen. c. 2000 Del Siegle Over 99% of your subjects will fall within three standard deviations of the mean. The mean and standard deviation are useful ways to describe a set of scores. If the scores are grouped closely together, they will have a smaller standard deviation than if they are spread farther apart. Small Standard Deviation Large Standard Deviation Same Means Different Standard Deviations Different Means Same Standard Deviations Different Means Different Standard Deviations Your next mouse click will display a new screen. c. 2000 Del Siegle Click the mouse to view a variety of pairs of normal distributions below. z-score -3 -2 -1 0 1 2 3 T-score 20 30 40 50 60 70 80 IQ-score 65 70 85 100 115 130 145 200 300 400 500 600 700 800 SAT-score Your next mouse click will display a new screen. c. 2000 Del Siegle When you have a subject’s raw score, you can use the mean and standard deviation to calculate his or her standardized score if the distribution of scores is normal. Standardized scores are useful when comparing a student’s performance across different tests, or when comparing students with each other. Your assignment for this unit involves calculating and using standardized scores. The number of points that one standard deviations equals varies from distribution to distribution. On one math test, a standard deviation may be 7 points. If the mean were 45, then we would know that 68% of the students scored from 38 to 52. 31 38 On another test, a standard deviation may equal 5 points. If the mean were 45, then 68% of the students would score from 40 to 50 points. 45 52 59 Points on Math Test 30 35 63 40 45 50 55 Points on a Different Test 60 c. 2000 Del Siegle 24 Your next mouse click will display a new screen. Data do not always form a normal distribution. When most of the scores are high, the distributions is not normal, but negatively (left) skewed. Skew refers to the tail of the distribution. 8 7 6 5 4 3 2 1 4 5 6 7 8 9 10 11 12 13 14 Length of Right Foot Your next mouse click will display a new screen. c. 2000 Del Siegle Number of People with that Shoe Size Because the tail is on the negative (left) side of the graph, the distribution has a negative (left) skew. When most of the scores are low, the distributions is not normal, but positively (right) skewed. 8 7 6 5 4 3 2 1 4 5 6 7 8 9 10 11 12 13 14 Length of Right Foot Your next mouse click will display a new screen. c. 2000 Del Siegle Number of People with that Shoe Size Because the tail is on the positive (right) side of the graph, the distribution has a positive (right) skew. When data are skewed, they do not possess the characteristics of the normal curve (distribution). For example, 68% of the subjects do not fall within one standard deviation above or below the mean. The mean, mode, and median do not fall on the same score. The mode will still be represented by the highest point of the distribution, but the mean will be toward the side with the tail and the median will fall between the mode and mean. Your Negative or Left Skew Distribution Positive or Right Skew Distribution c. 2000 Del Siegle next mouse click will display a new screen. • One use of the normal curve is to explore Standard Scores. Standard Scores are expressed in standard deviation units, making it much easier to compare variables measured on different scales. • There are many kinds of Standard Scores. The most common standard score is the ‘z’ scores. • A ‘z’ score states the number of standard deviations by which the original score lies above or below the mean of a normal curve. 17 c. 2000 Del Siegle Standard Scores • The normal curve is not a single curve but a family of curves, each of which is determined by its mean and standard deviation. • In order to work with a variety of normal curves, we cannot have a table for every possible combination of means and standard deviations. 18 c. 2000 Del Siegle The Z Score The Z Score xi x Z s 19 c. 2000 Del Siegle • What we need is a standardized normal curve which can be used for any normally distributed variable. Such a curve is called the Standard Normal Curve. • The Standard Normal Curve (z distribution) is the distribution of normally distributed standard scores with mean equal to zero and a standard deviation of one. • A z score is nothing more than a figure, which represents how many standard deviation units a raw score is away from the mean. 20 c. 2000 Del Siegle The Standard Normal Curve • For scores above the mean, the z score has a positive sign. Example + 1.5z. • Below the mean, the z score has a minus sign. Example - 0.5z. • Calculate Z score for blood pressure of 140 if the sample mean is 110 and the standard deviation is 10 • Z = 140 – 110 / 10 = 3 21 c. 2000 Del Siegle Example Z Score • Interpreting a raw score requires additional information about the entire distribution. In most situations, we need some idea about the mean score and an indication of how much the scores vary. • For example, assume that an individual took two tests in reading and mathematics. The reading score was 32 and mathematics was 48. Is it correct to say that performance in mathematics was better than in reading? 22 c. 2000 Del Siegle Comparing Scores from Different Distributions Z Scores Help in Comparisons 23 c. 2000 Del Siegle • Not without additional information. One method to interpret the raw score is to transform it to a z score. • The advantage of the z score transformation is that it takes into account both the mean value and the variability in a set of raw scores. Created by Del Siegle 2131 Hillside Road Unit 3007 Storrs, CT 06269-3007 [email protected] c. 2000 Del Siegle University of Connecticut