About Standard Deviation

Standard deviation is a calculation of the distribution of different data points in a data set. Put more simply, standard deviation shows how far apart different numbers are from the mean (or average) of the entire group of data; it shows the observer how consistent the data is, and therefore can help indicate how reliably the data can be used to draw general conclusions. The term is used in statistics, probability, geometry and economics, and can be applied to a variety of real-world situations.

History

Sir Francis Galton first formulated standard deviation for his studies into hereditary human abilities. Published in 1869, Galton's book "Hereditary Genius: An Inquiry Into Its Laws And Consequences" explored the possibility that intelligence and intellectual interests might be inherited. Galton used the standard deviation to show that his data suggested the inherited nature of certain positive traits. Later, the data in Galton's book made it a foundational text for eugenics, the (often racist) scientific philosophy of attempting to improve the human genetic pool that was instrumental in the Nazi's Holocaust. The standard deviation, however, was an important contribution to both mathematics and science.

How To Calculate

Calculating the standard deviation of a data set is somewhat complicated, and most statisticians today use calculators or computer programs to make their calculations faster. Should you wish to find the standard deviation by hand, however, use the following method:
1. Find the mean of the data set.
2. Find the deviation of each piece of data from the mean by subtracting the mean from each quantity in the set.
3. Take the deviations from step 2 and square them.
4. Calculate the mean of all the squared deviations from step 3.
5. Find the positive square root of the mean you found it step 5. That is the standard deviation.

A Sample Calculation

Here's an example of how to calculate the standard deviation:
Say we have a data set as follows: 2, 4, 6, 8, 22, 24.
1. First we find the mean of the data:
2+4+6+8+22+24 = 66
66/6 = 11
2. Then we find the difference between the mean above and each of the numbers from the data set:
2-11= -9
4-11= -7
6-11= -5
8-11= -3
22-11= 10
24-11= 13
3. Now we square each of the numbers from step 2:
(-9) squared = 81
(-7) squared = 49
(-5) squared = 25
(-3) squared = 9
10 squared = 100
13 squared = 169
4. Next we find the mean of those deviations:
81+49+25+9+100+169 = 433
433/6 = 72 1/6
5. Last, we take the positive square root of the mean from step 4, which is approximately 8.495. Therefore, the standard deviation for this data set is about 8.495.

Special Cases

If all the numbers in a data set are equal, then the standard deviation is always 0. For example, if you took a survey of college students and asked how many classes they were taking that semester, and every single student answered "5", the data set would be entirely composed of 5s, and therefore none of the answers would deviate from any of the others. The mean would be 5, and therefore each number in the set would be exactly equal to the mean.

Applications

Standard deviation is used in a variety of fields and professions. Often, the standard deviation is used as a measurement of how accurate or reliable a set of data might be. For example, in a scientific study, the standard deviation of the data can indicate whether the data was consistent and therefore can help scientists consider how solid any scientific conclusion based on their data might be. In meteorology, the standard deviation of weather data can help an observer understand how reliable or predictable a certain weather forecast will be.

Standard deviation is also used in predicting the movement of the stock market and in determining how profitably or reliably an investment can be made to function. If there is a large amount of disparity between a stock's value or profit at different times, that could indicate that the stock is volatile; if there is a low standard deviation, that could indicate that a stock is solid and would make a safe investment.