How do compare two different sets of data. Suppose you are comparing gas mileage on two separate kinds of automobiles - say light trucks and compact cars. Assume the mean miles per gallon for the light trucks is 23.6 miles per gallon with a standard deviation of 3.6 miles per gallon and if the mean miles per gallon for compact cars is 28.7 miles per gallon with a standard deviation of 5.7 miles per gallon. If you are trying to compare a light truck with a miles per gallon rating of 27.5 and a compact car with a miles per gallon rating 31.2. Which one is more "unusual"? To solve this problem we need some way to standardize these scores - this way we would not have to know what scale was being used. The way to get a standard score is the z score:

The standard score or z-score, is the number of standard deviations that a given value x is above or below the mean. You calculate the z score using

So for the light truck described above: the z score is z=(28.7-23.6)/3.6=1.42 standard deviations above the mean. The z score for the compact car described above is z=(31.2-27.5)/5.7=0.65 standard deviations above the mean.

One big use of z-scores is differentiating between normal values and unusual values. Do you recall Chebyshev's Theorem? It stated that at least 75% of the values are within 2 standard deviations from the mean for any data set. So if you are more than 2 standard deviations away from the mean - this means you are a unusual value.

Example: According to the American Freshman that number of hours per week that college freshman spend studying has a mean of 7.06 hours with a standard deviation of 2.32 hours. Suppose Sally Simplestudent spends 2 hours per week studying. Does Sally spend an unusually small amount of time studying?

According to the z score: z = (2-7.06)/2.32 = -2.18, Sally is more than 2 standard deviations away from the mean, so her low amount of study time is unusual

Z scores provide a useful measure for making comparisons between different sets of data. Quartiles, Deciles and Percentiles are measures of position useful for comparing scores within one set of data.

You probably all took some type of college placement exam at some point (suppose the ACT). If your composite math score was say 28, it might have been reported that this score was in the 94^th percentile. What does this mean? This does not mean you received a 94% on the test. It does mean that of all the students who took that exam, 94% of them scored lower than you did (and 6% higher). For a set of data you can divide the data into three quartiles (Q1, Q2, Q3), nine deciles (D1,D2, ..., D9) and 99 percentiles (P1, P2, ...P99). The quartile Q1 separates the bottom 25% from the top 75%, Q2 is the median and Q3 separates the top 25% from the bottom 75%. To work with percentiles, deciles and quartiles - you need to learn to do two different tasks. First you should learn how to find the percentile that corresponds to a particular score and then how to find the score in a set of data that corresponds to a given percentile. Here is an example:

Suppose the following table represents the outcomes of a college basketball team (Iowa) in one particular season.

To find quartiles, deciles and percentiles, we want to sort the data - so here is the sorted data on points scored by Iowa:

Example1: Find the percentile for the score of 67. To find the percentile you should

1. Find the number of scores less than the given score (in this case 67)
2. Divide this by the total number of scores and then multiply by 100

So for the score 67 this is: (5/18)(100)=27.78, rounding up this gives about the 28^th percentile

There are many ways to do the other problem - finding the score corresponding to a given percentile. Here are some examples using the same set of data

Example 2: Find Q1 - the 25^th percentile (or first quartile)

To find a score for a given percentile - use the following technique:

1. Rank the data from lowest to highest
2. Compute L = (k/100)(n), where n is the number of scores and k is the percentile in question
3. If L is a whole number then the value of the kth percentile is midway between the Lth score and the next higher score
4. If L is not a whole number change it to a whole number by rounding UP to the next larger whole number and the value of the percentile is the Lth score counting from the lowest

So for the 25^th percentile, L = (25/100)(18) = 4.5. Since L is NOT a whole number we round L up to 5 and find the 5^th score counting from the bottom - this is the score 66

Example 3: Find the 50^th percentile of this data (this is Q2 or D5). Here it is:

L = (50/100)(18) = 9. Since L is a whole number the 50^th percentile is halfway between the 9^th score and the 10^th score. The ninth score is 71 and the tenth score is 72. To find the score halfway in between these two scores - add them and divide by 2. So Q2 = (71+72)/2 = 71.5

The flowchart on page 97 is quite useful