Transcript Ch 10: Normal Distribution & Standard Scores

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:
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–
–
–
–
–
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)