1. Topic 3 Z-Scores Unit 5 Topic 3

2. Explore Lindsay’s class wrote three diploma examinations. The results are shown in the table below. Relative to the other students in the class, on which diploma examination did Laura do best? How did you decide on which exam Laura did best on? Diploma ExamMean Standard Deviation Laura’s Mark English 30-1641285 Pure Math 3060876 Social Studies 30651689

3. Explore It is hard to tell! Laura had the highest mark in Social Studies, but the mean was highest in this course as well. Her score was 24% higher than the mean. Laura’s Math mark was 13% lower than her Social Mark, but the class average was also much lower. Her score in Math is only 16% higher than the mean, but the standard deviation is also lower! In order to make an accurate analysis, we need some way to compare her mark to the mean (while also taking the standard deviation into consideration).

4. Information Sometimes we can’t make direct comparisons between two or more categories because the means and standard deviations are different. To make a comparison, we need to convert the data to a standard normal distribution.

5. Information Consider a student who scores 18 out of 30 on a science test and 25 out of 40 on a math quiz. To determine which score is higher, we convert each mark to a common scale – percentages. In statistics, we convert data to a z-score, a standardized value that indicates the number of standard deviations a value is from the mean.

6. Information A standard normal distribution has the following properties: ▫ a mean of zero and a standard deviation of one ▫ z-scores below (to the left of) the mean are negative ▫ z-scores above (to the right of) the mean are positive

7. Information To calculate the z-score: where z is the z-score, x is a value, is the mean, and is the standard deviation.

8. Example 1 Calculating Z-Scores For each scenario given, calculate the z-score. If needed, round the z-scores to the nearest tenth. a) Edmonton’s average daily temperature in June is 21  C with a standard deviation of 3  C. On a particular day in June, the temperature is 15  C. b) The average on a math exam is 72% with a standard deviation of 7%. A student achieved a mark of 84%. c) The mean time for a sprint is 25.57 seconds with a standard deviation of 0.62 seconds. An athlete has a run time of 24.77 seconds. Try this on your own first!!!!

9. a) Edmonton’s average daily temperature in June is 21  C with a standard deviation of 3  C. On a particular day in June, the temperature is 15  C. Example 1: Solution a) Edmonton’s average daily temperature in June is 21  C with a standard deviation of 3  C. On a particular day in June, the temperature is 15  C. Make sure to calculate 15-21 before dividing by 3. In order to calculate in one step on your calculator, place brackets around 15-21 as shown below:

10. b) The average on a math exam is 72% with a standard deviation of 7%. A student achieved a mark of 84%. c) The mean time for a sprint is 25.57 seconds with a standard deviation of 0.62 seconds. An athlete has a run time of 24.77 seconds. Example 1: Solution

11. IQ tests are sometimes used to measure a person’s intellect. IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. a)Label the normal curve with the mean and the values for standard deviations. b)An IQ score of 125 is not easily determined with this graph. For an IQ score of 125, determine the z-score, to the nearest tenth. Example 2 Sketching and calculating the z-score Try this on your own first!!!!

12. Example 2: Solution a) Label the normal curve with the mean and the values for standard deviations. 100 +0.8 kg -0.8 kg 115 130 857055145 mean 100 standard deviation 15

13. Example 2: Solution 100115 130 857055145 b)An IQ score of 125 is not easily determined with this graph. For an IQ score of 125, determine the z-score, to the nearest tenth. The IQ score 125 falls between 1 and 2 standard deviation above the mean.

14. Example 3 Calculating the mean On a university entrance exam, the standard deviation is 6 and a score of 51 has a z-score of -1.5. Calculate the mean. Try this on your own first!!!!

15. Example 3: Solution On a university entrance exam, the standard deviation is 6 and a score of 51 has a z-score of -1.5. Calculate the mean. Cross multiply.

16. Need to Know: To compare data values that do not have the same means or standard deviations, we convert the data to a standard normal distribution. A standard normal distribution has a mean of zero and a standard deviation of 1.

17. Need to Know: To convert given data values to a standard normal distribution, we calculate z-scores: standardized values that indicate the number of standard deviations above or below the mean a data value falls. Z-scores are negative when they exist below the mean and positive when they exist above the mean. To calculate the z-score, use the formula You’re ready! Try the homework from this section.