Chapter 4 & 5 The Normal Curve & z Scores

Normal Curve What is it? -It’s a unimodal frequency distribution curve *scores on X-axis & frequency on Y-axis -distributions observed in nature usually match it -it is a critical component for understanding inferential statistics What sets it apart from other frequency distributions? Most of the scores cluster around the middle of the distribution. The curve then drops down & levels out on both sides. It is symmetrical. The mean, median & the mode fall at the exact same point. It has a constant relationship with the standard deviation No matter how far out the tails are extended, they will never touch the X-axis because it is based on an infinite population. In other words, the normal curve is asymptotic to the abscissa.

Normal Curve EX: The following is a normal distribution of IQ scores M=100 SD=15

The Standard Normal Curve The “standard normal curve” is a normal curve that has been plotted using standard deviation units. It has a mean of 0 and a standard deviation of 1.00 The standard deviation units have been marked off in unit lengths of 1.00 on the abscissa. The area under the curve above these units always remains the same. IMPORTANT! Explain pic on slide: The mean is equal to an SD of 0. 34.13% of the cases always fall under the curve between the mean & 1 SD -if you add 1 SD below the mean & 1 SD above, you will always find 68% of the cases 13.59% of cases always fall between 1 & 2 SD -if you add 2 SD below & above the mean, you will find 95% of cases 2.15% of cases always fall between 3 & 4 SD -IF you add 3 SD below & above the mean, you will find 99% of cases *Go back to normal curve example slide & show where percentage of cases fall Animation

Z-Scores What is it? -the number of standard deviations a raw score is above or below the mean Why use it? -to figure out how a raw score compares to a group of scores In the previous slide we were ALSO describing z-scores Z-score table: used to obtain the precise percentage of cases falling between any z-score and the mean. -only shows positive z-scores because the curve is symmetrical so percentages would be the same for negative z-scores *p. 621, Table A Helpful hint: always draw the curve when working with z-scores -positive z-scores go to the right of the mean, negative go to left -the higher the z-scores the further it goes to the right, the lower the z-score the further to the left of the mean -shade in percentage of cases you’re referring to

Z-Scores: Calculation To find the percentage of cases between a z-score and the mean: -look it up in the z-score table! To find the percentage of cases above a z-score: -if positve z-score, look up z-score in table and subtract it from 50 -if negative z-score, look up z-score in table and add it to 50 To find the percentage of cases below a z-score: -if positive z-score, look up z-score in table and add it to 50 -if negative z-score, look up z-score in table and subtract it from 50 **this is the same way to calculate percentile --be sure to round to the nearest percentile --a z-score of 0 would be the 50th percentile To find percentage of cases between z-scores: -between a negative & postive z-score, look up both scores in table & add -between 2 positive or 2 negative z-scores, look up both scores in table & subtract Use Normal Curve handout (back 1-5) & draw curves on board/give examples To change a raw score to a z-score: subtract the mean from the raw score & divide by the standard deviation Z-Score Practice Worksheet Z-Score Homework due Next Class

Gauss: Father of the Normal Curve German Mathematician Often referred to as the greatest mathematician of all time “Perfect Pitch” Mathematical Prodigy People say he did math before he could talk! Developer of the Normal Curve or the “Gaussian Curve” Pair Share Topic: What does a Z-score tell you about a raw score? 1775-1855 Fun with Your Calculator Worksheet

Z-Scores Revisited To find a raw score from a z-score: X= zSD +M To find a raw score from a percentile -Use Table B to find z-score -Calculate the raw score: X= zSD +M -EX: 72nd percentile SD=17 M=150 **Table B: 72nd percentile is a z=0.58 X=(0.58)(17) + 150 X=9.86 + 150 X=159.86 To find a standard deviation from a z-score: -EX: X=184 M=150 z=2 **SD= 184-150/2 SD= 34/2 SD= 17 To find a mean from a z-score: M = X – zSD -EX: z= 2 SD=17 X=184 **M = 184 – (2)(17) M = 184 – 34 M = 150

T-Scores Z-Scores Revisited Worksheet What is a T-score? -a converted z-score with the mean always set at 50 & standard deviation at 10 -basically, it’s just another measure of how far a raw score is from the mean -always a positive number Calculating T-scores: T = z(10) + 50 -remember, the mean is always 50 & the standard deviation is always 10 -EX: z = 2 **T = (2)(10) +50 T = 20 + 50 T = 70 From T to z to raw scores -First, convert the T-score to a z-score: -Second, convert the z-score to a raw score: X= zSD +M -EX: M = 70 SD = 8.50 T = 65 **z = 65-50/10 z = 15/10 = 1.50 **X= (1.50)(8.50) + 70 X= 12.75 + 70 = 82.75 WHY use a T-score instead of a z-score? -We use T-scores when we have an N less than 30 because we need a more precise measure with small amounts of data like when comparing 2 raw scores **EX: Person takes 2 different IQ Tests & you want to know which one they did better on (calculate z for both then T for both & compare T scores) -Test 1: X= 110 M=100 SD=15 -Test 2: X =112 M=100 SD=18 -Answer T=56.70 on both tests so the performance was identical Z-Scores Revisited Worksheet

Other Normal Curve Transformations Normal Curve Equivalents -another popular standardized score (like the z-score or T-score) -calculated by setting the mean at 50 & the SD at 21 -larger Range than T-scores because of the larger SD Stanines -standardized score used mainly in educational psychology -divides normal curve into nine units where the Z-score divides into six units Grade Equivalent Scores -standardized scores popular in the field of education -based on the average score found for students in a particular grade at both the same age & time of year -A GE=6.9 indicates a score that a 6th grader in the 9 month of the school year receives on average -EX: a 3rd grader in the 1st month of school could take a test & get a GE=4.5 **That would mean they tested at the level of a 4th grader in the 5th month More on stanines -the 1st stanine is equal to a z-score of -1.75 & below -the scale then increases by .50 until the 9th stanine -the 9th stanine is equal to a z-score of +1.75 Z-Scores Revisited Homework due Next Class