Psych 524, 10/12/05
p. 1/2
In-Class Exercise: Converting Raw Scores to Area
Step 1: convert raw score to z-score.... z = (score –mean)/(SD)
Step 2: draw a diagram
* make sure to shade in the area of interest
* make sure to label the x-axis with both raw scores and z-scores
* you may later note how much area falls in the shaded areas
Step 3: look up the z-score(s) in column 1 of Table D.2 (p 702) and find the area(s)
corresponding to the area(s) in your diagram
* if your area lies between the mean and your z-score, use column 2
* if your area lies beyond your z-score (between the z-score and the distribution’s tail),
use column 3
* if you need to find an area that does not touch the mean (e.g., between z = 1 and z =
1.5), you will find the areas between the mean and each of your 2 z-scores (you will use
them both in step 4)
Step 4: using the diagram as a guide, come up with your final answer
* if above a positive z: just use area beyond this positive z
* if above a negative z: add .5 to area between mean and negative z
* if below a positive z: add .5 to area between mean and positive z
* if below a negative z: just use area beyond this negative z
* if between a positive and negative z: add area between negative z and mean and area
between positive z and mean
* if between two positive z’s: subtract area between smallest z and mean from area
between largest z and mean
* if between two negative z’s subtract area between least negative z and mean from
area between most negative z and mean
Step 5: Make sure your answer makes sense! For example:
* answer is always between 0 and 1
* if diagram shows area taking up more than half of the distribution, answer should be
> .5
* if diagram shows area taking up less than half of the distribution, answer should be <
.5
Examples: Use a normal distribution with mean of 100 and SD of 15
1) area above a score of 110 = .251
2) area above 80 = .5 + .4082 = .908
3) area below 110 = .5 + .2486 = .749
4) area below 80 = .092
5) area between 80 and 110 = .4082+ .2486 = .657
6) area between 110 and 150 = .4996 - .2486 = .251
7) area between 70 and 80 = .4772- .4082 = .069
Psych 524, 10/12/05
p. 2/2
In-Class Exercise: Converting Area to Raw Scores
Step 1: draw a diagram
* remember that each half of the curve constitutes 50% of the area
* make sure to shade in the area of interest and label these areas with proportions
* you will later label the x-axis with both raw scores and z-scores
Step 2: look up the proportion of area in either column 2 or column 3 (see below) of Table D.2
(p 702) and find the z-score corresponding to the area in your diagram
* if your area lies between the mean and your cut-off point, use column 2
* if your area lies beyond your z-score (between the z-score and the distribution’s tail),
use column 3
* note that if you label your diagram such that you indicate the proportions both
between the mean and the cutoff and beyond the cutoff, you can use either proportion
(between or beyond) to look up your z-score, as long as you use the
correct/corresponding column
* in most cases, your exact proportion will lie between the two listed in the table; for
this course, simply use the tabled proportion that is closest to the given proportion
Step 3: convert the z-score you found in the table into a raw score: score = mean + z (SD)
Step 4: Make sure your answer makes sense! For example:
* if you’re asked what score separates the top 25% from the bottom 75%, your answer
should be higher than the mean
* remember the percentages associated with area between standard deviation units
- 68.3% is between +/- 1 SD
- 95.5% is between +/- 2 SD
- 99.7% is between +/- 3 SD
so, for example, if you are asked to find scores that mark off the middle 80% of the
distribution, the z-scores should be between 1 and 2 (80% is between 68.3% and
95.5%)…remember to convert the z’s to raw scores
Examples: Use a normal distribution with mean of 100 and SD of 15
1) score corresponding to top 5% X = 124.675
2) score corresponding to bottom 95% (how does this compare to #1?) X = 124.675
3) score corresponding to bottom 1% X = 65.00
4) score corresponding to top 80% X = 87.40
5) scores corresponding to middle 10% X = 98.05, 101.95
6) scores corresponding to middle 95% X = 70.6., 129.4