6-Understautyrteyyunding-the-Z-Scores.pptx

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1. UNDERSTANDING THE Z-SCORES DOMINIC DALTON L. CALING Statistics and Probability | Grade 11

2. Lesson Objectives At the end of this lesson, you are expected to: relate a random variable distribution to a normal variable distribution; understand the concept of the z-score; convert a random variable to a standard normal variable and vice-versa; and solve problems involving random and normal variables.

3. Pre-Assessment

4. Lesson Introduction z-score is stated to be a measure of relative standing. These scores represent distances from the center measured in standard deviation units. There are six z-scores at the base line of the normal curve: three z scores to the left of the mean and three z-scores to the right of the mean.

5. Discussion Points The z-score The areas under the normal curve are given in terms of z-values or scores. Either the z-score locates X within a sample or within a population. The formula for calculating z is: where: X = given measurement = population mean = population standard deviation = sample mean = sample standard deviation  

6. Discussion Points For any population, the mean and the standard deviation are fixed. Thus, the z formula matches the z-values one-to-one with the X values (raw scores). That is, for every X value there corresponds a z-value and for each z-value there is exactly one X value.

7. Discussion Points The z values are matched with specific areas under the normal curve in a normal distribution table. Therefore, to find the percentage associated with X, we must find its matched z-value using the z-formula. The z-value leads to the area under the curve found in the normal curve table, which is a probability , and that probability gives the desired percentage for X.

8. EXAMPLE 1 Given the mean, = 50 and the standard deviation, = 4 of a population of Reading scores. Find the z-value that corresponds to a score X = 58.  

9. EXAMPLE 1 This conversion from raw score to z-score is shown graphically From the diagram, we see that a score X = 58 corresponds to z = 2. It is above the mean. So we can say that, with respect to the mean, the score of 58 is above average.

10. EXAMPLE 2 Locate the z-value that corresponds to a PE score of 39 given that = 45 and = 6.   With respect to the mean, the score 39 is below the population mean. We can also say that the score 39 is below average.

11. EXERCISES

12. Summary The areas under the normal curve are given in terms of z -values or scores. Either the z -score locates X within a sample or within a population. The formula for calculating z is:

13. Summary Raw scores may be composed of large values, but large values cannot be accommodated at the base line of the normal curve. So, they have to be transformed into scores for convenience without sacrificing meanings associated with the raw scores.

14. THANK YOU!