Q2 Cheat Sheet Z-scores

A. Below (PR): Determining the probability of finding a score below a certain score is the same as finding that score’s “percentile rank.”

B. Above (PA): Determining the probability of finding a score above a raw score is the opposite of percentile rank. The percentage of persons who “outscored” the raw score. the population.

C. Between(PB): Determining the probability of randomly drawing a score between two raw scores: convert the two raw scores to Z-scores and calculate the percentage between by adding or subtracting.

D. 95% Confidence Interval(95%CI): In every population, there is a score above the mean that includes 47.5% of the distribution, and a score below the mean that includes 47.5% of the distribution. We call it the “95% confidence interval” around the mean. We want to know the two scores between which 95% of all the scores in the population could be randomly drawn. At the same time, We isolate the 5% of scores that are unlikely to be randomly drawn. This problem is similar to a “find the percentage between two scores” problem, one score falls 47.5% below the mean, and the other falls 47.5% above the mean. (95%CI)

E. Finding a Raw Score when given a Percentage (% -> Y.): A percentile rank, or percentage above is given, and we find the raw score associated with that percentage. We do this by locating the percentage between the mean and Z associated with that percentage, look up Z score, and convert it to a raw score.

Master Cheat Sheet

(1)  Facts:  The sample mean:  = , the raw score: Y1 = (or two raw scores: Y1 and Y2), the standard deviation: s = . Determine which of the 5 problems are being requested: PR, PA, PB, 95%CI, or % -> Y.

(2)  Draw a normal curve, place the mean in the center, locate two standard deviations (raw scores) on either side of the mean.

(3)  a)  Place the Z-score for the mean (0) in the center below the mean. Locate two standard deviations (Z- scores) on either side of the mean.

   b)  Fill in the approximate percentile ranks for each whole standard deviation (Z score).

 (4) Find Z-obtained, or find Y1 (sometimes this step is delayed): Use the formula to convert the raw score to a Z-Score -- Zobt = (Y - )/s. Or, convert the Z-score to a raw score Y =  + (Z * s)

(5)  a)  Draw a dotted line through the normal curve at the mean, reminding us that 50% of the scores fall above the mean, and 50% fall below. Draw two arrows, one arrow extending to the left and one to the right of the dotted line at the mean. Insert the proportion .5000 above each half of the normal curve.

     b) Draw a dotted line through the curve approximately where Zobt will fall.

  c)  Draw an arrow between the two dotted lines extending between the mean and Zobt.

(6) Lightly shade the area of the normal curve that contains the scores of interest.

(7)  a. Locate the percentage or locate the Z-score.  Consult “The Normal Distribution” chart that contains the proportions in the normal curve between the mean and “Zobt.”

A quick lesson on how to read the chart: the example is an excerpt from the “The Normal Distribution” table in the appendix. The first decimal place of the Z-score (Zobt) is shown in the left hand column (1.3, 1.4, 1.5 etc.). The second decimal place of the Z-Score is arrayed across the top. Therefore the percentage between the mean and “Z” when Zobt is 1.30 is in the 1st column from the left: “.4032.” The percentage between the mean and “Z” for a Z-score of 1.31 is in the 2nd column from the left: “.4049.” The percentage between the mean and “Z” when Zobt is 1.32 is in the 3rd column from the left: “.4066,” and etc. (these percentages are underlined).

(8) Do the Math (sometimes this means returning to the previously delayed step 4) -- use these guidelines to find the percentage:

a.  If the raw score is below the mean (like it is here) subtract the percentage between the mean and Zobt from .5000, the requested percentage is the result (.5000 - .4332 = .0668)

b.  If the raw score is above the mean, add the percentage between the mean and Zobt to .5000, the requested percentage is the result.

 c.  Always determine how the shaded area relates to the percentage between the mean and the raw score (Zobt). If necessary, add to or subtract from 50% according to which area is shaded.