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Ch 10: Normal Distribution & Standard Scores Mon. Feb 23rd, 2004 Normal Distribution (ND) Common theoretical distribution w/bellshaped, symmetrical curve – Mean, median, mode all occur at peak Properties of the ND: – – – – – – Area under it = 1.0 (100% scores) 50% scores fall above midpt, 50% below Betw Mean & + or - 1 SD = 34.13% of scores Betw 1 & 2 SD (pos or neg) = 13.59% of scores Beyond 2 SD (pos or neg) = 2.28% of scores …or 68-95-99.7 rule (68% between mean and +/1 SD; 95% betw mean and +/- 2 SD; 99.7% betw mean and +/- 3 SD. Z scores By remembering these %s, if we have a ND, can determine % scores falling betw any score & the mean Use of z scores (standardized scores) – The difference betw any score & the mean in standard deviation units Z = (y – ybar) / Sy – Notice sign & magnitude of z score… (cont.) – If pos z score, raw score was above the mean; if neg z score below the mean – Magnitude indicates how many standard deviations the score is away from the mean SAT mean = 500, Sy = 100, your score (y) = 620 Z = 620 – 500 / 100 = 1.2 You scored 1.2 standard deviations above the mean Could compare to your friend, y = 540 Z = 540 – 500 / 100 = .4 (scored 4/10’s of standard deviation above mean) Converting z to y May also need to transform z to raw score: Y = Ybar + Z(Sy) Someone tells you they scored 2 SD above the mean SAT; what raw score? Y = 500 + 2(100) = 700 SAT score Standard Normal Table (aka Unit Normal Table) Table showing proportion of scores corresponding to a certain z score Appendix B in book: 3 columns – Col A gives z scores (all pos since symmetrical – just find same z for neg) – Col B gives proportion of scores betw the z score & mean – Col C gives proportion of scores beyond the z score Using the table… Start by sketching distribution, label mean & Sy, then score you’re interested in – Shade relevant area of distribution – Translate y into a z score, use table Examples: IQ test has ybar=100, Sy = 15 What is probability of having IQ < 85? >130? What IQ score is needed to be in top 5%? What is probability of having IQ between 90 &120? Lab 11 Click button for “ND” demo, move flags & find % associated w/scores Use unit normal table (either link from lab or your book)…note how the tables differ slightly. SPSS – lab11.sav dataset…create z scores (follow directions in lab)