MATH Resources for Academic Success
Measures of Position (3.4)
A measure of position describes the relative position of a data value in relation to the rest of the data in the set. There are several ways to describe this location. z-Score
One useful measure of position is the z-score. The z-score of a data point is how many standard deviations away from the mean the data point is. Since there are two different standard deviations (population and sample), there exist two different kinds of these:
population z-score, which represents how many population standard deviations away a data point is from the population mean, sample z-score, which represents how many sample standard deviations away a data point is from the sample mean.
For example, a data point with a z-score of 2.3 is 2.3 (slightly more than two) standard deviations above the mean. A z-score of -0.8 describes a data point that's 0.8 (almost one) standard deviation below the mean.
We can use the z-score to compare things between two different populations that otherwise couldn’t be compared. For example, men's basketball star Michael Jordan is 78 inches tall, and women's basketball star Rebecca Lobo is 76 inches tall. Obviously Jordan is taller than Lobo, but which player is relatively taller compared to other people of their gender? In other words, does Jordan's height compared to other men exceed Lobo's height compared to other women?
The mean height of adult American men, according to the National Health Survey, is 69.0 inches with a standard deviation of 2.8 inches. The mean height of adult American women is 63.6 inches, with a standard deviation of 2.5 inches.
Using the formula z-score = (score - mean)/standard deviation, Jordan's z-score is (78 - 69.0)/2.8 = 3.2, so his height is 3.2 standard deviations above the mean. Lobo's z-score is (76 - 63.6)/2.5 = 4.96, so her height is 4.96 standard deviations above the mean. Since Lobo's z-score is higher, that means that she's relatively taller (compared to other women) than Jordan is (compared to other men).
We will use z-scores a lot in this course, so make sure you understand what they represent and how to find them. See the links at the end of this document for additional information.
Percentile
Another measure of position is the percentile. The kth percentile of a data set is the value for which k% of the data is less than that value. For example, a SAT score in the 83rd percentile means that you scored better than 83% of the people who took the SAT.
A quick way to use percentiles is with quartiles. The quartiles break a data set into four pieces, each of which contains 1/4 of the data.
MATH Resources for Academic Success
Q2, the median, is the 50th percentile. Q1, the lower quartile, is the 25th percentile (i.e. the median of the first half of the data) and Q3, the upper quartile, is the 75th percentile (i.e. the median of the second half of the data). We can use quartiles to describe where a data point is in relation to the whole data set.
One thing we can do with quartiles is check for outliers. An outlier is a data point that is unusual compared to the rest. For example, I do a poll of 100 random people on campus, and ask them their ages. Most are between 18 and 20-something, but suppose that there happened to be one person who was 100 years old. This person was in my sample, but obviously was not representative of the population as a whole. We call data points like this outliers.
In order to find outliers, we need to compute the interquartile range, or IQR. The IQR is the range of the middle 50% of the data set, in other words, it's Q3 - Q1. If a data point is more than 1.5 IQRs below Q1 or above Q3, it's an outlier. It is a good idea to test for outliers when analyzing data, because those outliers can significantly change less resistant statistics (for example, the mean).
http://www.sophia.org/tutorials/range-and-interquartile-range-iqr--2
5-Number Summary & Boxplots (3.5)
The 5-number summary of a data set is a list of the five numbers we found when looking at the quartiles:
Minimum Q1 Q2 (Median) Q3 Maximum
This five number summary is part of the data the TI-84 outputs when you use 1-VarStat.
We can draw a picture of this summary using a boxplot.
Follow the directions in the book and videos to make a boxplot. You can also draw boxplots on the TI-84, directions on p.171 and my video for this week.
Resources
Prof Bryce's how-to to find the 5-number summary and draw boxplots on the TI-84:
http://www.youtube.com/watch?v=W8zRChviVvI
Khan Academy explanation of boxplots:
http://www.khanacademy.org/math/statistics/v/box-and-whisker-plots
http://www.khanacademy.org/math/statistics/v/reading-box-and-whisker-plots
More info about z-scores:
http://stattrek.com/statistics/dictionary.aspx?definition=z_score
http://statistics-help-for-students.com/What_are_Z_scores.htm
https://statistics.laerd.com/statistical-guides/standard-score.php