Measures of Dispersion
Instructional objectives for the lesson
1. The student will be able to define dispersion.
2. The student will be able to define range.
3. The student will be able to define standard deviation.
4. The student will be able to calculate a range in a data set.
5. The student will be able to calculate a standard deviation of a data set.
Dispersion
The dispersion of a distribution refers to the variability of the distribution. It essentially answers questions on how stretched or how squeezed a distribution is. While there are several measures of dispersion, we will focus on two – the range and the standard deviation.
Range
The range in a set of data is the difference between the highest and lowest values in the set. When you are looking at a data set, you should sort the data from the lowest to the highest. Then you simply subtract the lowest from the highest value and that will give you your range. See the example below.
Johnny took four exams this semester in his statistics class and his scores were
94, 79, 89, 82.
The first thing you would do is sort them in order:
79, 82, 89, 94
Then you take the smallest, which would be 79 and subtract it from the largest, which would be 92.
94 – 79= 15
The range would be 13.
Standard Deviation
The standard deviation (SD) is a measure of the amount of dispersion or variation from the average. If the standard deviation is low, then the data points tend to be close to the average. If there is a high standard deviation, your data points tend to be spread out and have a larger range of values.
To find the standard deviation calculate the square root of the average of the differences from the average squared.
Using our data set from above, let’s calculate the standard deviation
79, 82, 89, 94
First, let’s calculate the average
(79 + 82 + 89 + 94) / 4 = 86
Second, let’s calculate the difference from the mean each data point is and then square each
79 – 86 = -9 (-9)2 = 81
82 – 86 = -4 (-4)2 = 16
89 – 86 = 3 (3)2 = 9
94 – 86 = 8 (8)2 = 64
Third, lets get the mean of these values
(81 + 16 + 9 + 64)/ 4 = 42.5
Lastly, the square root of 42.5 is 6.5. So your standard deviation is 6.5.
Additional Links
Below are some links that will provide additional information on dispersion
http://www.socialresearchmethods.net/kb/statdesc.php
http://www.dummies.com/how-to/content/how-to-interpret-standard-deviation-in-a-statistic.html
http://statistics.about.com/od/Descriptive-Statistics/a/What-Is-The-Range-In-Statistics.htm
Discussion Question
You and your classmates will be sent an email for access to a Google Doc to discuss this question.
Remember your responses should be respectful of other classmates feedback in this discussion.
When do you think a standard deviation might be useful in a real world application? Be specific and detailed. Please respond to two of your classmates ideas on the Google Doc.
Dispersion Assignment
By completing and submitting this assignment you are certifying it is your work. Please refer to the student's code of conduct as it discusses the ethics and responsibilities of earning an education at Youngstown State University.
Copy and paste the assignments into a Word document and complete the assignments.
Once you have completed both assignments, please submit to Nick Cascarelli at nvcascarelli@ysu.edu.
Matching. Place the letter for the description of each word to the definition that you feel best matches.
Calculate the measures of dispersion.
Calculate the range and the standard deviation for the data set below
5, 6, 6, 7, 10, 4, 6, 2, 12
Range __________
Standard Deviation __________