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CLASS NOTES: The Normal Curve, z-scores & Other Normal Curve Transformations CONCEPT CALCULATION/EXAMPLES APPLICATION The normal curve: The “normal curve” is as theoretical distribution. But, is used for describing data as so many distributions of people-related measurements come so close to this ideal normal curve. Features of a normal curve: * Most of the scores cluster around the middle of the distribution. As distance from the middle increases in either direction, there are fewer & fewer scores. * It is symmetrical. Its two halves are mirror images of each other. * All three measures of central tendency (mean, median & mode) all fall precisely in the center or midpoint of the distribution. * There is a constant relationship w/ the standard deviation. * When you go out one full standard deviation from the mean the curve above the abscissa reaches its point where the point changes direction & begins going out more quickly than it goes down. * No matter how far out the tails are extended, they will never reach the x-axis. Probability & the normal distribution The percentage of a normal distribution b/t the µ of 0 & σ of 1 is 34.13% The space b/t 1 σ & 2 σ is 13.59% Beyond the 2nd σ is 2.28%. This is the same for both above & below the mean. So, 50% of scores fall above the mean & 50% of scores fall below the mean. There are larger examples of probability & the normal distribution at the end of these notes. Standardized Distribution: Composed of scores that have been transformed to create predetermined values for a mean and SD. Standardized distributions are used to make dissimilar * Data is often transformed into standard scores to obtain more information about a score or set of scores. * Two or more different sets of distributions comparable. z-score or standard score: The purpose is to identify & describe the exact location of every score in a distribution. A z-score specifies the precise location of each X value within a distribution. The sign of the z-score (+ or -) signifies whether the score is above the mean (positive) or below the mean ( - ). The numerical value of the zscore specifies the distance from the mean by counting the number of standard deviations b/t X and µ. The z-score always consists of 2 parts: a sign (+ or -) and a magnitude. z-score formula to be used for transforming back & forth from x values to z-scores scores are converted to have the same mean & SD so that information & data can be compared. * Standard scores make it possible to compare different scores or different individuals even though they may come from completely different distributions. z-score formula for a population: z=x-µ σ z-score formula for a sample: z=X-M s Example: x = 120 µ = 100 σ = 10 z = 120 – 100 10 20 = 2.00 10 The look of a standardized distribution. This formula is used when you want to convert a raw score to a standard score, or z-score. X - µ is a deviation score measuring the distance b/t a particular score & the standard mean. σ is used so that the z-score measures distance in terms of SD units. For a sample, X represents the score, while M represents the sample standard mean & s is the sample standard deviation score. In this example available, the raw score given is 120. You want to standardize that score to a mean of 100 & SD of 10. Plug the numbers in the formula & you get an answer of 2.00 indicating that the raw score of 120 would fall 2 SD’s above the mean when the mean is 100 & SD is 10. Standardized data are all represented by a bell curve. When converting to z-scores, the mean will always be 0 & the SD will always be + or – 1. Z-scores are not the only standard scores though. Others include SAT scores, IQ scores, etc.. A standard score is any score in which raw scores have been converted from its raw form to a standard that has the same mean & deviation score. Converting from standard scores to raw scores X = µ + zσ This formula is used when you have a standard score, or a z-score & you wish to convert back to a raw score. Example: In this example, you are given a standard mean of 50 & a standard deviation of 10. The z-score is – 3.333, which means that the actual score lies 3.333 standard deviations below the mean. Following through w/ the formula for converting standard scores back into raw, the raw score comes out to be 17. µ = 50 z = - 3.333 σ = 10 X = 50 + (- 3.333)(10) X = 16.67 X = 17 Converting z-scores to standard deviation Finding the range Converting z-score to the mean More about Z-Scores: If every X value is transformed into a z-score, then the distribution of zscores will have the following properties: The shape of the z-score distribution will be the same as the original distribution of raw scores The z-score distribution will always have a mean of 0. The z-score distribution will always have a SD of 1. SD = X – M Z This formula is used when you have a z-score & you wish to convert to a standard deviation, or the standard distance of a score from the mean. The X value minus the mean over the z-score The range is roughly six times the SD (6)(SD) M = X – (Z)(SD) This formula is used when you have a z-score & you wish to convert this score to obtain the mean, or average. The X value minus the z-score multiplied by the SD. The Unit Normal Table: This table lists proportions of the normal distribution for a full range of possible z-score values. The body always corresponds to the larger part of the distribution whereas the tail always represents the smaller section whether it is on the right or the left. B/c the normal distribution is symmetrical, proportions on the right side are exactly the same as proportions on the left. Although z-score values can change depending upon whether or not they are on the right or the left of the mean (+ or - ) the proportions will always be positive. The look of the unit normal table as represented w/in a normal distribution. (A) z (D) (C) (B) Between Above Below z mean and z z 0.00 0.5000 0.5000 0.0000 0.01 0.5040 0.4960 0.0040 0.02 0.5080 0.4920 0.0080 0.03 0.5120 0.4880 0.0120 0.04 0.5160 0.4840 0.0160 0.05 0.5199 0.4801 0.0199 0.06 0.5239 0.4761 0.0239 0.07 0.5279 0.4721 0.0279 0.08 0.5319 0.4681 0.0319 0.09 0.5359 0.4641 0.0359 To the left in an example or a portion of the z unit normal table. Column A lists z-scores. Column B lists the area that falls below that particular z-score. Column C represents the area that falls above the z-score, & column D represents the area that falls between the mean & the zscore. If a question asks you to determine the distance under the normal curve “greater” than the given score, you obtain the z-score, then you will mark that score on your normal distribution & “color in” the space to the right of that score. If it asks you to determine the distance under the normal curve “less” than the given score, you obtain the z-score, mark it on your normal distribution & “color in” the space to the left of the given score. Locate your zscore on the UNT. If more than 50% of your distribution is “colored in,” then you will use the proportion in the body. If less than 50% of your distribution is “colored in,” then you will use the proportion in the tail. This picture corresponds with column B (the area below the zscore) The area C corresponds with column C (the area that falls above the z-score) The area D corresponds with column D (the area that Other Standardized Distributions Based on Z-Scores: Before standardizing a set of raw scores into scores w/ a specific mean & SD, raw scores are transformed into z-scores & then into scores w/ a specific mean & SD SAT scores (Scholastic Aptitude Test) 200 300 400 IQ scores (Intelligence Tests) 65 70 T-scores 20 Calculating T-scores 85 30 40 500 600 700 800 SAT scores have a µ of 500 & σ of 100 100 115 130 145 IQ tests are typically converted to a µ of 100 & σ of 15 50 T-score values range from 20 to 80 & are all positive scores as are the SAT’s & IQ tests. 60 70 80 X = (z)(SD) + M Or T = (z)(10) + 50 Scroll down to view the percentages covered under the normal curve….. These tables below represent the distance covered under the normal curve. The first curve indicates the percentages between each half standard deviation (or z-score) above (+) & below (-) the normal curve. The second curve indicates the same, but shows the percentages covered under the normal curve between each whole standard deviation (or z-score) & their negative counterpart (ex: +1.00 to -1.00 = 68.27%). The total percentage covered under the normal curve is 100%. If none of the space under the normal curve is indicated, that would then be represented by 0%.