Topics
- histogram and shapes
- percentiles and their meaning
- frequency curves: basically the connecting of the tops of the rectangles in a histogram
- shapes are frequency curves are interpreted similarly to those of histogram
- bell shaped or symmetric distributions and the Empirical Rule
- normal curve: special type of symmetric
- standardized scores
- standardized scores for a normal curve or normal distribution: the standard normal
- percentiles and proportions for standard normal scores using Table 8.1
- finding observed value from a percentile
A standardized score or z-score is simply the number of standard deviations an individual falls
above or below the mean for the whole group.
A pth-percentile score represents the percentage of the scores that fell at or below your score.
E.g. if you scored in the 80th percentile on an exam this would be interpreted as “80% of all
scores on this exam fell at or below your score.”
Examples
1. Notre Dames’ Department of Philosophy Graduate Program has no automatic cut-off based on
GRE scores, but average scores for students admitted recently are: 93rd percentile in verbal and
84th percentile in quantitative. These two sections of the GRE are scaled to a score range of 200
to 800. Assume the GRE scores are approximately normal with the following means and SD:
Verbal: mean of 465 and SD of 110
Quantitative: mean of 584 and SD of 70
If you were applying to ND what scores would you need to meet their average percentile
requirements?
1. find the z-score for each percentile: Verbal: 0.93 corresponds to a z-score of 1.48 and
Quantitative: 0.84 corresponds to a z-score of 1.00
2. Through algebra, z-score = (observed – mean)/SD converts to: Observed = Z*SD + Mean.
Plugging and chugging results in the following GRE scores for each section:
Observed Verbal = (1.48)*(110) + 465 = 627.8
Observed Quantitative = (1.00)*(70) + 584 = 654
2. What percentile is a score of 410 on Verbal?
z-score = (410 – 465)110 = – 55/110 = – 0.5 From Table 8.1 this z-score of – 0.5
corresponds to a proportion of 0.31 or the 31st percentile. This would mean that 31% of the
scores were at or below 410 on the verbal section.
3. Of you classes this semester you have one that is very difficult, calculus, and another that is
relatively easy, InArt. Your first exam scores are 60% and 80% respectively. You mention these
scores to your parents and they are visibly upset over the calc score. However, you recall from
your stat 100 class the importance of standardizing scores to place these results on a comparable
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scale. The mean and SD for physics was 50% and 5% and for InArt these statistics were 85% and
2.5%. Calculating the z-score for each we have:
Z-scorecalc = (0.6 – 0.5)/0.05 = 2 and Z-scoreInArt = (0.8 0.85)/0.025 = – 2
By comparison, your physics score was two standard deviations above the mean while your InArt
score was 2 standard deviations below the mean. If the scores of these two exams followed a
bell-shape distribution, then approximately 97.5% of the exam scores in physics were at or below
your grade (i.e. your score is in the 97.5 percentile) while your InArt score is in the 2.5 percentile
or 2.5% of the InArt scores were at or below your 80!
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