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# Normal Distributions

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# Normal Distribution - Answer Key

## Distribution of SAT scores

### Topic

Distribution of SAT Scores

### Vocabulary

• Probability Distribution
• Normal Distribution
• Standard Deviation of Normal Distribution
• Mean
• Correlation Coefficient

### Student Exploration

#### How and why does the scoring of the SAT change to correspond with the population of test-takers?

The SAT is a standardized test for college admission

The SAT is the most common standardized exam that students in the United States take for admission to college. The test has been around since 1901 and is now administered across the world. It consists of three 800-point parts; Mathematics, Critical Reading (also referred to as “Verbal”) and Writing. Scores on each section range from 200 to 800 points, however there are not 800 questions on the test; the points are scaled scores. Students’ raw test scores are converted to the point scores based on a normalized curve. Every year the creators of the SAT test work to change the conversion table of the raw test scores to the point test scores in order to keep up with the changes in the population of test-takers. The histogram graphs below demonstrate this for the 1990 SAT and are from the College Board, the organization that creates the SAT test, http://professionals.collegeboard.com/profdownload/pdf/200211_20702.pdf.

The normal distribution curve of the Verbal SAT scores from 1990

The normal distribution curve of the Math SAT scores from 1990

1. Create a probability distribution table for the verbal section of the SAT in 1990.

2. Create a probability distribution table for the math section of the SAT in 1990.

Math Scores Percentage of Total
200-240 1.2
250-290 5.3
300-340 9.6
350-390 12.7
400-440 13.9
450-490 14.4
500-540 13.5
550-590 11.1
600-640 8.2
650-690 5.8
700-740 3
750-800 1.2

3. What was the mean and standard deviation of the scores on the verbal section of the SAT in 1990?

4. What was the mean and standard deviation of the scores on the math section of the SAT in 1990?

(Need help finding the mean? First the midpoint of each bar/bin in the histogram and then multiply that by the percentage of test takers that got that score, this will give you the weight each score based on how many students got the score. Then find the average score by adding up all the weighted scores and dividing the total by 100).
The mean score on the math section is 475 and the standard deviation is 1.358 (or a score of 100) for the math section of the SATs in 1990. Technically a normal distribution should have a standard deviation of 1 (or 100 as a score in this situation) but it is difficult and rare for this to occur naturally with a set of real data.
http://www.ltcconline.net/greenl/courses/201/descstat/meanSDGrouped.htm

5. Between what scores do the middle 68% of test takers fall? Answer for both the Math and Verbal SAT in 1990.

68% of the Math SAT test takers in 1990 fall between the scores of 350 and 590, but it is actually 65.6%.

6. The top 16% of test takers have what score or higher? Answer for both the Math and Verbal SAT in 1990.

The top 16% of test takers have a score of 600 or higher on the Math SAT, but it is actually 18.2%.

7. Why do the creators of the SAT make the test scores correspond with a normal distribution curve every year? Explain.

The creators of the SAT scale the raw test scores to the scaled test scores so that the scores are representative of the group of people that take the test. This makes it so that there is a universal meaning to the scores. Furthermore it allows universities to better be able to use the scores as an indication of students’ skills and ability compared to other students. In addition it makes it so that low and high scores are less frequent and that the majority of the test takers will fail in a certain range, for the SAT 65.5% score between a 350 and 590.

8. Do you think that it is fair that the creators of the SAT make the test scores correspond with a normal distribution curve? Why or why not? Explain.

Students’ answers will vary.

### Extension Investigation

For more information on the history of the scoring of the SAT exam read, http://professionals.collegeboard.com/profdownload/pdf/200211_20702.pdf !

Here are some follow-up activities to compare the SAT scores to the Advanced Placement (AP) test scores. The AP tests are the end of the year exams for AP courses. The AP exams are scored on a five-point scale, from one to five. Students that earn a score of three or higher can receive college credit to most universities, this varies depending on the policies of each university.

1. Look up the AP scores for one course/test.
2. Compare the AP and SAT scores. What is the mean and standard deviation for the AP scores?
3. What are the scores of the middle 68% of test takers for both tests? What are the scores of the top 16% of test takers for both tests?
4. Research for the AP and SAT scores of a group of students and then create a scatter plot graph of the students’ results on the two tests. What do you notice? What does the graph tell you about the data?
5. Find the linear regression equation for the scatter plot.Determine the correlation coefficient between the two exams. What do you notice? What does the equation and correlation coefficient tell you about the data?

AP & SAT scores in California: http://www.cde.ca.gov/ds/sp/ai/

### Connections to other CK-12 Subject Areas

• Histogram
• Discrete Random Variables
• Normal Distributions

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