Section 3.4: Measures of Position
Objectives
By the end of this lesson, you will be able to...
 determine and interpret zscores
 determine and interpret percentiles
 determine and interpret quartiles
 check a set of data for outliers
In Sections 3.1 and 3.2, we discussed ways to describe a "typical" individual in a population or sample. In this next section, we'll talk about ways to describe an individual in relation to the population.
For a quick overview of this section, watch this short video summary:
zScores
It's fairly common for upperlevel statistics courses to have both undergraduate and graduate students. Given the exam scores listed below, can you determine which score is better relative to its peers, the undergraduate score of 83 or the graduate score of 88?


Actually, to answer this question, we need more information. In particular, we need a new way to describe relative position.
The zscore represents the number of standard deviations a data value is from the mean.
Population zScore  Sample zScore  


I can't overemphasize the importance of the meaning behind the zscore. Make a note of this now  you'll be seeing this again later on in the semester  it's very important!
Example 1
(continued)
So let's continue with our previous example. The sample mean of the undergraduate scores is 77.1, with a standard deviation of 10.73. That gives a zscore for the undergraduate 83 of:
z =  83  77.1  ≈0.55 
10.73 
With a sample mean of 85.75 and a standard deviation of 7.78, the graduate has a zscore of:
z =  88  85.75  ≈0.29 
7.76 
Since the undergraduate is more than 1/2 of a standard deviation above the mean (z = 0.55), that's a better relative score.
Note: You may have noticed that I went to the hundredths place for these zscores. That's standard practice.
Key: We use zscores when we want to compare two individuals from different populations, relative to their respective populations.
Percentiles
If you've ever taken a standardized exam like the PSAT, SAT, or ACT, you've seen in the report something about your percentile.
The kth percentile, denoted P_{k}, of a set of data divides the lower k% of a data set from the upper (100k)%.
Percentile ranks are used in a variety of fields:
 Special Education  students scoring below a certain percentile on specific tests qualify for services.
 Physicians  doctors usually track a child's weight and height and compare the growth to that of other children of the same age.
Unfortunately, there's no universally accepted way to calculate percentiles. Most software packages and calculators use a method similar to the one below (from your text), but you should be aware of the possibility of others.
Determining the kth percentile, denoted P_{k}
 Step 1: Arrange the data in ascending order.
 Step 2: Compute an index i using the formula
 Step 3:
 If i is an integer, the kth percentile, denoted P_{k}, is the ith value.
 If i is not an integer, the kth percentile is the mean of the observations on either side of i.
Example 2
Let's go back to the Jackson cousins we saw in Example 2 in Section 3.1. Suppose this time we add all the cousins, from little Zander at age 4 to Mae, who at age 18 is entering her first year at college.
Use the strategy above to find the 25th percentile by age.
Technology
Here's a quick overview of the formulas for finding percentiles in StatCrunch.

Note: Some software like Microsoft Excel interpolates instead of taking a simple average when calculating percentiles, so the results may differ slightly.
Determining the Percentile of a Data Value
The last thing we need to do with percentiles is to figure out the percentile of a particular individual. For example, if your Composite ACT score is a 28, what percentile does that leave you?
As before, there is no universally accepted way to calculate percentiles, but the following (from your text) is very common.
Finding the Percentile that Corresponds to a Data Value_{}
 Step 1: Arrange the data in ascending order.
 Step 2: Use the following formula to find the percentile of the value,
x.
percentile of x = number of data values less than x *100 n
Round this number to the nearest integer.
Example 3
Consider again the Jackson cousins we looked at in Example 2 above.
What is the percentile rank of James, the 14yearold?
Using the formula above, we calculate i as:
percentile of James =  9  *100 = 75 
12 
So James is the 75th percentile.
Quartiles
As the name implies, quartiles divide the data into four equal parts. Therefore the first quartile, Q_{1}, is the 25th percentile, the second quartile, Q_{2} is the 50th percentile (or the median), and the third quartile, Q_{3}, is the 75th percentile.
Example 4
Let's consider one of the sets of hypothetical exam scores we looked at in Section 3.2.
48 
57  58  65  68  69  71  73  73 
74  75  77  78  78  78  79  80  85 
87  88  89  89  89  95  96  97  99 
Find the quartiles.
Q_{1}:
So
the 1st quartile is x_{7}, or 71.
Q_{2}: Since there are 27 observations, the median is x_{14}, or 78.
Q_{3}:
So
the 3rd quartile is x_{21}, or 89.
Technology
Here's a quick overview of the formulas for finding quartiles in StatCrunch.

Note: Some software like Microsoft Excel interpolates instead of taking a simple average when calculating percentiles, so the results may differ slightly.
Checking for Outliers
One good use of quartiles is they give us a sense of what values might be extreme. In Statistics, we call these values outliers. There are various ways to check for outliers. Most depend on the distribution and often can only characterize observations as possible outliers. A common technique used is the following:
Checking for Outliers by Using Quartiles_{}
 Step 1: Determine the first and third quartiles
 Step 2: Compute the interquartile range: IQR = Q_{3}  Q_{1}
 Step 3: Determine the fences.
 Lower fence = Q_{1}  1.5(IQR)
 Upper fence = Q_{3} + 1.5(IQR)
 Step 4: If a value is less than the lower fence or greater than the upper fence it is considered an outlier.
Example 5
Let's look at those same exam scores we used in Example 4.
48 
57  58  65  68  69  71  73  73 
74  75  77  78  78  78  79  80  85 
87  88  89  89  89  95  96  97  99 
Use the above method to determine if there are any outliers.
IQR = Q_{3}  Q_{1} = 8971 = 18
Lower fence = Q_{1}  1.5(IQR) = 71  1.5(18) = 44
Upper fence = Q_{3} + 1.5(IQR) = 89 + 1.5(18) = 116
Since no values lie outside the fences, there are no outliers.