What is Standard Deviation?

Standard Deviation is a number that defines how close or far the examples of a set of data are from the mean (or average). It takes into account all the numbers and takes an average of their distance from the mean of the set. In simplest terms, it is how far the values are from the mean. In short, it describes the magnitude of the "spread" of the data.



When is Standard Deviation Useful?

Standard deviation is frequently used when there is a normal distribution (Diagram 1), or nearly normal distribution of data, meaning that the distribution follows the familiar "bell curve," with few outliers. Using standard deviation to measure the range of a graph that does not have a normal distribution, such as skewed right or skewed left histogram (show in Diagram 2), would not appropriately represent the spread of the graph. In this case, the IQR may be a better choice. A normal distribution of data would have a bell curve graph. The data is shaped as a bell because most of the data clusters around the mean, and the few outliers fall around the other parts of the graph. The narrower the bell curve, the smaller the standard deviation is. Thus, a wide, flat bell curve would mean there is a large standard deviation. Bell curves are useful to show the normal distribution of data and are a good representation of where the majority of data lies on a plot.


Standard deviation can be utilized in a probability distribution to find the probability of an event will occur. If the standard deviation is small, then the probability of having a data value far from the mean is small. On the other hand, a large standard deviation can lead to a larger probability.

What Affects Standard Deviation?

Standard Deviation is a resistant measure to addition and subtraction because standard deviation is just the average of distances from the mean in a data set. For example, if you add 3 to each value in a data set, the standard deviation would not change because the values are still the same distance from each other. However, standard deviation is not resistant to multiplication or division.

How Can I Derive the Standard Deviation?

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Diagram 1: The Normal Curve is a bell-shaped curve
Diagram 1: The Normal Curve is a bell-shaped curve


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Diagram 2: Skewed Left and Right Histograms


SAT2 Level Standard Deviation Problems

1.

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2. The largest value in a distribution of scores is doubled while other values remain the same.
Compared to the original standard deviation, the new stand deviation will be:

Larger
Smaller
Unchanged

3. The mean and standard deviation of a data set are 37 and 5 respectively. The new data set is created by
Doubling all of the values in the original data set
What is the standard deviation of the new set?


Explanation for Standard Deviation Problems
1. The reason why standard deviation stays the same is because its mean is merely increasing by 5. But, the actual scores are still the same distance in value from each other. So, the mean will increase by five points, making it 70, and the median will also increase by five, making it 76. The standard deviation though, is an average of the distance from the mean, so it would not be affected by a change in mean.

2. Answers is larger. The standard deviation will be larger because if one point is farther out that it was in the original distribution, this would make the average of the distances larger.

3. Answer is 10. The standard deviation would 10 because if all the values are multiplied by two, they will be 2x farther apart. That would mean that you would need to do 5 * 2 to find that the standard deviation of the new data set is 10.


Real Life Application
In stocks, one often wants a 'lock', which is a stock that will (theoretically) continue to steadily go up in small increments and is low risk. It is a consistent stock without much deviation. The consistency of the stock can be measured with standard deviation. If the standard deviation of the values of the stock is low from the last few years, it is very consistent - it's annual returns don't vary much. But if the standard deviation is high over time, it is a higher risk stock that sometimes has high returns and sometimes has low (or negative!) returns that one might wish to stay away from.


Bibliography:
Diagram 1-http://www.robertniles.com/stats/stdev.shtml
Diagram 2- http://rchsbowman.files.wordpress.com/2008/09/shape-skewed.jpg
Information about Standard Deviation-Mr. Postman's Single Variable Notes
Derivation of Standard Deviation-http://en.wikipedia.org/wiki/Standard_deviation


Problem 1 in SAT2 Level S.D. Problems-http://sat.collegeboard.com/practice/practice-question-next?pageId=practiceSubjectTestMathLevel2&conversationId=ConversationStateUID_1&followupAction=comeBack&practiceTestSectionIDKey=Subject.MATH_LEVEL_2&header=Mathematics%20Level%202&subHeader=SAT%20Subject%20Test%20in%20Mathematics%20Level%202%20Practi
Problems #2-3 in SAT2 Level S.D. Problems-Mr. Postman's Single Variable Assessment #3