## Tuesday, July 21, 2015

### Finding percentiles from the z-score list. Finding the proportion at an exact number on a list of a normally distributed set

Finding percentiles from the z-score list.
One of the things we can do with the z-score to proportion list is find a z-score that corresponds to the cut-off point for a particular percentile. Here are two examples at the 8th percentile and the 80th percentile.

The 8th percentile. because 8 < 50, we will look on the negative z-score side of the sheet. What we want to find is the two positions on the table where the values go from .08xx to .07xx. you will find then in the -1.4 row at the first two positions.

-1.40 -> 0.0808 (8 above 0.0800)
-1.41 -> 0.0793 (7 below 0.0800)

Because 7 < 8, -1.41 counts as the closest to the two, but we should check to see if the average -1.405 would be better. We look at far/close which is 8/7 ~= 1.142857... and if it's less than 3 (it is), we will use the average,so the z-score for the 8th percentile is -1.405.

The 80th percentile. because 80 > 50, we will look on the positive z-score side of the sheet. What we want to find is the two positions on the table where the values go from .79xx to .80xx. you will find then in the 0.8 row in the middle.

0.84 -> 0.7995 (5 below 0.8000)
0.85 -> 0.8023 (23 above 0.8000)

Because 5 < 23, 0.84 counts as the closest to the two, but we should check to see if the average 0.845 would be better. We look at far/close which is 23/5 = 4.6 and if it's more than 3 (it is), we will use the closest instead of the average,so the z-score for the 80th percentile is 0.84.

Finding the proportion at an exact number on a list of a normally distributed set. Consider the SAT values for average mu_x = 500 and sigma_x = 100. SAT scores are always round to the nearest 10, so if we want to find out what percentage of scores are at 600, we have to look at the proportion at 605 and 595.

Raw score 605 become the z-score (605-500)/100 = 105/100 = 1.05, which corresponds to 0.8531

Raw score 595 become the z-score (595-500)/100 = 95/100 = 0.95, which corresponds to 0.8289.

We subtract the small proportion from the big, .8531 - .8289 = .0242. This means about 2.42% of SAT takers will get exactly 600 on one section of the SAT.