1. 2.5 Normal Distributions and z-scores

2. Comparing marks Stephanie and Tavia are both in the running for the Data Management award. Stephanie has 94% and Tavia has 92%, from different classes. Who deserves the award? What if I told you it was Tavia? Why?

3. Stephanie’s class: mean 78,  = 9.36 Tavia’s class: mean 73,  = 8.19 Distributions are different Fair comparison not possible …yet

4. Standard Normal Distribution Mean 0, standard deviation 1 Can translate each element of a normal distribution to standard normal distribution by finding number of  a given score is away from the mean –this process is called standardizing

5. z-scores z = The number of standard deviations a given score x is above or below the mean z = z-score –Positive: value lies _________ the mean –Negative: value lies _________ the mean above below

6. Example 1: Calculating z-scores Consider the distribution Find the number of standard deviations each piece of data lies above or below the mean. A)x = 11B) x = 21.5 Note: z-scores are always rounded to 2 decimal places

7. Example 2: Comparing data using z-scores Stephanie and Tavia are both in the running for the Data Management award. Stephanie has 94% and Tavia has 92%. If Stephanie’s class has a mean of 78 and  = 9.36, and Tavia’s class has a mean of 73 and  = 8.19. Who deserves the award?

8. Example 2 Use z-scores: Stephanie:Tavia Tavia’s z-score is higher, therefore her result is better.

9. z-Score Table appendix B, pp. 398-399 of text Determines percentage of data that has equal or lesser z-score than a given value Example: P(z < -2.34) = 0.0096 Only 0.96 % of the data has a lower z-score, and 1 – 0.0096 = 99.04% of the data has a higher z-score

10. Note Notice z-score table does not go above 2.99 or below –2.99 Any value with z-score above 3 or less than –3 is considered an outlier –If z > 2.99, P(z < 2.99) = 100% –If z < -2.99, P(z < -2.99) = 0% If z = 0, P(z < 0) = 50% –The data point is the mean

11. Percentiles The kth percentile is the data value that is greater than k % of the population Example z = 0.40 z = 1.67 65.54 % of the data are below this data point. It is in the 66 th percentile. 95.25 % of the data are below this data point. It is in the 96 th percentile.

12. Example 3: Finding Ranges Given, find the percent of data that lies in the following intervals: A) 3 8.6 For x = 3,For x = 6,

13. Example 3: Finding Ranges Given, find the percent of data that lies in the following intervals: A) 3 8.6 So 29.20% of the data fills this interval.

14. Example 3: Finding Ranges Given, find the percent of data that lies in the following intervals: A) 3 8.6 For x = 8.6, So 23.27% of the data lies above 8.6.