Chapter 4 & 5The Normal Curve & z Scores

Normal CurveWhat 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?1.Most of the scores cluster around the middle of the distribution. The curve then drops down & levels out on both sides.2.It is symmetrical.3.The mean, median & the mode fall at the exact same point.4.It has a constant relationship with the standard deviation5.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.

Animation

Z-ScoresWhat 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 • To change a raw score to a z-score: subtract the mean from the raw

score & divide by the standard deviation

Gauss: Father of the Normal Curve

1775-1855

• 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?

Z-Scores Revisited• To find a raw score from a z-score: X= zSD +M -EX: z= 2 SD=17 M=150 **X=(2)(17) + 150 X= 34 +150 X= 184

• 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• 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

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