1. Math II UNIT QUESTION: Can real world data be modeled by algebraic functions? Standard: MM2D1, D2 Today’s Question: How is a normal distribution used to curve test scores? Standard: MM2D1d

2. The Normal Distribution Section 7.4

3. The normal distribution and standard deviations

4. Length of Right Foot Number of People with that Shoe Size 8765432187654321 4 5 6 7 8 9 10 11 12 13 14 Suppose we measured the right foot length of 30 teachers and graphed the results. Assume the first person had a 10 inch foot. We could create a bar graph and plot that person on the graph. If our second subject had a 9 inch foot, we would add her to the graph. As we continued to plot foot lengths, a pattern would begin to emerge.

5. Length of Right Foot 8765432187654321 4 5 6 7 8 9 10 11 12 13 14 If we were to connect the top of each bar, we would create a frequency polygon. Notice how there are more people (n=6) with a 10 inch right foot than any other length. Notice also how as the length becomes larger or smaller, there are fewer and fewer people with that measurement. This is a characteristics of many variables that we measure. There is a tendency to have most measurements in the middle, and fewer as we approach the high and low extremes. Number of People with that Shoe Size

6. Length of Right Foot 8765432187654321 4 5 6 7 8 9 10 11 12 13 14 Number of People with that Shoe Size You will notice that if we smooth the lines, our data almost creates a bell shaped curve.

7. 8765432187654321 4 5 6 7 8 9 10 11 12 13 14 Length of Right Foot Number of People with that Shoe Size You will notice that if we smooth the lines, our data almost creates a bell shaped curve. This bell shaped curve is known as the “Bell Curve” or the “Normal Curve.”

8. Whenever you see a normal curve, you should imagine the bar graph within it. 12 13 14 15 16 17 18 19 20 21 22 Points on a Quiz Number of Students 987654321987654321

9. The mean,mode,and median 12 13 14 15 16 17 18 19 20 21 22 Points on a Quiz Number of Students 987654321987654321 12+13+13+14+14+14+14+15+15+15+15+15+15+16+16+16+16+16+16+16+16+ 17+17+17+17+17+17+17+17+17+18+18+18+18+18+18+18+18+19+19+19+19+ 19+ 19+20+20+20+20+ 21+21+22 = 867 867 / 51 = 17 12 13 14 14 15 15 15 16 16 16 16 17 17 17 17 17 17 17 17 17 18 18 18 18 19 19 19 20 20 21 22 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 22 will all fall on the same value in a normal distribution. Now lets look at quiz scores for 51 students.

10. Normal distributions (bell shaped) are a family of distributions that have the same general shape. They are symmetric (the left side is an exact mirror of the right side) with scores more concentrated in the middle than in the tails. Examples of normal distributions are shown to the right. Notice that they differ in how spread out they are. The area under each curve is the same.

11. The normal distribution and standard deviations The total area under the curve is 1. In a normal distribution: 34% 13.5% 2.35%

12. The normal distribution and standard deviations Approximately 68% of scores will fall within one standard deviation of the mean In a normal distribution:

13. The normal distribution and standard deviations Approximately 95% of scores will fall within two standard deviations of the mean In a normal distribution:

14. The normal distribution and standard deviations Approximately 99.7% of scores will fall within three standard deviations of the mean In a normal distribution:

15. When you have a subject’s raw score, you can use the mean and standard deviation to calculate his or her standardized score if the distribution of scores is normal. Standardized scores are useful when comparing a student’s performance across different tests, or when comparing students with each other. z-score -3 -2 -1 0 1 2 3 T-score 20 30 40 50 60 70 80 IQ-score 65 70 85 100 115 130 145 SAT-score 200 300 400 500 600 700 800 2.35% 13.5% 34% 34% 13.5% 2.35%

16. The number of points that one standard deviations equals varies from distribution to distribution. On one math test, a standard deviation may be 7 points. If the mean were 45, then we would know that 68% of the students scored from 38 to 52. 24 31 38 45 52 59 66 Points on Math Test 30 35 40 45 50 55 60 Points on a Different Test On another test, a standard deviation may equal 5 points. If the mean were 45, then 68% of the students would score from 40 to 50 points. 2.35% 13.5% 34% 34% 13.5% 2.35%

18. Using standard deviation units to describe individual scores 1001101209080 -1 sd1 sd2 sd-2 sd What score is one sd below the mean?90 What score is two sd above the mean? 120 Here is a distribution with a mean of 100 and standard deviation of 10:

19. Using standard deviation units to describe individual scores Here is a distribution with a mean of 100 and standard deviation of 10: 1001101209080 -1 sd1 sd2 sd-2 sd How many standard deviations below the mean is a score of 90? 1 2 How many standard deviations above the mean is a score of 120?

20. Using standard deviation units to describe individual scores Here is a distribution with a mean of 100 and standard deviation of 10: 1001101209080 -1 sd1 sd2 sd-2 sd What percent of your data points are < 80? 2.50% 84% What percent of your data points are > 90?

21. Class Work In class: Workbook Page 277 #1-21

22. Homework Monday in class: Finish the Workbook page 277 #1-21 and then do page 266 #1-11