~Z-Scores: (Standard Scores)

~Used primarily for comparing scores from different data sets relative to their means.
~gives the number of standard deviations a given score is from its mean.

~Ex: if the mean of a data set is 80 & the S=5, then a Z-score of +1 would be 85

~Ex: Student A got 85 on a test whose mean=79 & s = 8. Student B got 74 on a test whose mean=70 s = 5. Who did better?  Answer: compute both Z-scores & the HIGHER would be the better score relative to its test.

~Formula:   Z = (x - m ) / S, where x is a score, m is the sample mean, S is the sample standard deviation

        Z = (x-m)/s, where x is a score, m is the population mean, s is the population standard deviation

Z = (85-79)/8 = 0.75,  Z = (74-70)/5 = 0.80, so, student B did better.


~Unusual values: more than 2 standard deviations away from the mean (Z>2 or Z<-2)

~Ex: All women in pro basketball have a mean height of 63.6" (m) & a standard deviation of 2.5" (s), Rebecca Lobo is 76" tall. Is she considered unusually tall for her sport?

Compute her Z-score:   Z = (76-63.6)/2.5 = 4.96 (very much so!)

~Quartiles: dividing a data set into 4 quarters.               _____._____._____._____

~the data points, L, Q1, Q2, Q3, H  are known as the 5 number summary & are important to known for the construction of Box-plots.  Q2 is also known as the median or P50 (later)

~How to find them:
1) Enter the data in a list. (the TI-83 will give you these)
2) Find Q2 (median) (50% of the data is below Q2 & 50% is above Q2)
3) For Q1, find the median for all scores L to Q2 (separates the bottom 25% from the top 75%)
4) For Q3, find the median for all scores Q2 to H (separates the bottom 75% from the top 25%)

~Note: when finding Q1 & Q3, use the scores in the data set.

~Ex: take the data set 2,5,3,4,7,0,11,2,3,8.  
       Sort: 0,2,2,3,3,4,5,7,8,11, n=10,
       x = 3.5 = Q2 = median
       Q1 = median of all scores from 0 to 3.5 = 2 = Q1
       Q3 = median of all scores from 3.5 to 11 = 7

~Percentiles: (used extensively in educational testing. Usually for large amounts of data)

~dividing or separating the data into 100 sections.
  The symbol for the k th percentile is Pk.

~Note:  Q1 = P25,  Q2 = P50, Q3 = P75.  Finding these scores is a bit tricky, so be careful.

~How to find them:
1) Sort the data from low to high. The TI-83 will do this for you after you enter them in a list.
2) To find the k th percentile, find its Location.
Take k% of n (# of scores) & look at the result. If the result is a whole  #, average the score in this position with the next one. If the result is a decimal, round up & take the score in that position.

~let’s check Q1 & Q3 from our last example. We have n=10, so find P25 & P75.

~For P25, take 25% of 10 = 2.5. Since the result is a decimal, round up to 3 & take the 3rd score (2).

~For P75, take 75% of 10 = 7.5. Since the result is a decimal, round up to 8 & take the 8th score (7).

~Ex: For the data set, 34,39,63,64,67,70,75,76,81,82,84,85,86,88,89,89,90,96,96,100
       Find P25, P33, P50, P75, P80

~For P25, take 25% of 20 = 5, since the result is a whole #, average the scores in the 5Th & 6Th place. We get, (67+70)/2 = 68.5 = Q1

~For P33, take 33% of 20 = 6.6, since the result is a decimal, round up to 7 & take the 7th score (75)

~For P50, take 50% of 20 = 10, since the result is a whole #, average the scores in the 10th & 11th place. We get, (82+84)/2 = 83 = Q2

~For P75, take 75% of 20 = 15, since the result is a whole #, average the scores in the 15th & 16th place. We get, (89+89)/2 = 89 = Q3

~For P80, take 80% of 20 = 16, since the result is a whole #, average the scores in the 16th & 17th place. We get, (89+90)/2 = 89.5

~Finding a percentile given a score: (This is going in reverse direction)

~Just find the number of scores BELOW the given score & convert it to a percent.

~Ex: In the above data set, the score of 86 is in what percentile?

~find the number of scores below 86, which is 12. So, 12/20 = .60 = 60%, so, 86 is in the 60th percentile (it’s also in the 61th percentile).
So, this might account for a few discrepancies you might encounter.

~Note: When your data set is relatively small (less than 100), a given score could be in more than one percentile. The ideal situation would be 100 scores, then each score would represent a different percentile. For very large data sets (usually the case), many scores are in a given
percentile.