Understanding the Z- score (Application on evaluating a Learner`s performance)
- 1. By: Lawrence P. Avillano, LPT
- 2. A z-score (aka, a standard score) indicates how many Standard Deviation an element or a score is from the mean is from the mean. A z-score can be calculated from the following formula
- 3. • It can be found using the formula z = (x – μ) / σ, : read z is equal to the quotient between the difference of the raw score and mean and the standard deviation. Where: • z - is the z - score • x - is the raw score • μ - is the (population) mean, but u can use x bar too • .σ - is the standard deviation
- 4. • A z-score less than zero (0) represents an element less than the mean. • A z-score greater than 0 represents an element greater than the mean. • A z-score equal to 0 represents an element equal to the mean.
- 5. • A z-score equal to 1 represents an element that is 1 standard deviation greater than the mean; a z-score equal to 2, 2 standard deviations greater than the mean; etc. • A z-score equal to -1 represents an element that is 1 standard deviation less than the mean; a z-score equal to -2, 2 standard deviations less than the mean; etc. • If the number of elements in the set is large, about 68% of the elements have a z-score between -1 and 1; about 95% have a z- score between -2 and 2; and about 99% have a z-score between -3 and 3.
- 6. In a 20 item test, the average score of the class is 16 and with 3.5 SD. Cherrie Pie scored 18. z = (x – μ) / σ z = (18– 16) /3.5 z = 2/3.5z = 0.57 That means that Cherrie Pie`s score is 0.57 (less than 1) standard deviations above the mean.
- 7. Other than said above, we can use Cherri Pies`s z-score of .57 to see how well she performed compared to other students or look at her class standing. • We may have to look through the z-table which can be found here http://www.statisticshowto.com/tables/z-table/ • Now take 0.57 • y - is 0.5 • x - is 0.07 We would get the z-value of 0.2157
- 9. • Multiply the z-value by 100: 0.2157*100 = 21.57 • It means that 21.57% of the class performed better than Cherrie Pie • Substract 21.57 from 100: 100-21.57 = 78.43 • It means that Cherrie Pie performed better than 78.43% of the test class. • That would help the teacher on making decisions with regards to Cherri Pie`s performance.