Understanding the z score
- 2. Steps in Finding the Areas Under the Normal Curve 1. Express the given z-value in 3 digit form. 2. Using the z-table, find the first two digits in the left/right column. 3. Match the 3rd digit with the appropriate column on the right. 4. Read the area (or probability) at the intersection of the row and column.
- 3. Find the area: From z=0 1.To z=1.36 2.To z=- 2.89 3.To z=0.05
- 5. Lesson for the Day
- 6. What is a Z-Score? z-score is the number of standard deviations from the mean a data point is. But more technically it’s a measure of how many standard deviations below or above the population mean a raw score is. A z-score is also known as a standard score and it can be placed on a normal distribution curve. Z- scores range from -3 standard deviations (which would fall to the far left of the normal distribution curve) up to +3 standard deviations (which would fall to the far right of the normal distribution curve). In order to use a z-score, you need to know the mean μ and also the population standard deviation σ.
- 7. z-score formula x z µ σ − = Where x represents an element of the data set, the mean is represented by and standard deviation by . µ σ
- 8. Analyzing the data Suppose SAT scores among college students are normally distributed with a mean of 500 and a standard deviation of 100. If a student scores a 700, what would be her z-score? Answer Now
- 9. Analyzing the data Suppose SAT scores among college students are normally distributed with a mean of 500 and a standard deviation of 100. If a student scores a 700, what would be her z-score? 700 500 2 100 z − = = Her z-score would be 2 which means her score is two standard deviations above the mean.
- 10. Analyzing the data • A set of math test scores has a mean of 70 and a standard deviation of 8. • A set of English test scores has a mean of 74 and a standard deviation of 16. For which test would a score of 78 have a higher standing? Answer Now
- 11. Analyzing the data 78-70 math -score = 1 8 z = 76-74 English -score= .25 16 z = To solve: Find the z-score for each test. A set of math test scores has a mean of 70 and a standard deviation of 8. A set of English test scores has a mean of 74 and a standard deviation of 16. For which test would a score of 78 have a higher standing? The math score would have the higher standing since it is 1 standard deviation above the mean while the English score is only .25 standard deviation above the mean.
- 12. Analyzing the data What will be the miles per gallon for a Toyota Vios when the average mpg is 23, it has a z value of 1.5 and a standard deviation of 5? Answer Now
- 13. Analyzing the data What will be the miles per gallon for a Toyota Vios when the average mpg is 23, it has a z value of 1.5 and a standard deviation of 2? 23 1.5 2 x − = 3 2 63 2x x == − The Toyota Vios would be expected to use 26 mpg of gasoline. x z − = µ σ Using the formula for z-scores:
- 14. Analyzing the data Suppose you have the population values 50 and 80 and their corresponding z-scores are -1 and 2, respectively. Is it possible to determine the population’s mean and standard deviation? If so, what are these values? If not, explain why it is impossible. Answer Now
- 16. Quote for the Day… There will always be one (or more) value that will exceed all others. — Emil J. Gumbel https://todayinsci.com/QuotationsCategories/S_Cat/Statistics- Quotations.htm
- 17. Analyzing the data There are three grades in a report card and you want to interpret in terms of performance: Mathematics (75), English (85) and Science (90). The means are 72, 83 and 88, respectively. The standard deviations are 3, 10, and 15, respectively. Is the information sufficient for you to compare the grades? If so, discuss your processes. If not, explain why it is impossible. Homework