Standard+Deviation

Standard Deviation


 * __What is Standard Deviation?__**

In order to fully comprehend standard deviation, it is important to understand the normal distribution of data. Normal distribution of data usually forms a bell-curve graph, meaning that most points in the data set are close to the average, while very few points are at either extreme. For more information about Normal Distribution click here. Standard deviation is when a statistician examines a bell curve as shown above, however, normal distribution of the data is not needed to calculate the standard deviation. Click here to see another example of how data can be distributed. Standard deviation measures the spread of data, or how close points of data are to the Mean, or average. It is recognized by this symbol,. It is the average distance each point is away from the mean. Simply defined, standard deviation is the square root of the variance. So, what is the variance?

Variance, like standard deviation, is also a measure of the spread of data, but more specifically, measures the likelihood of a certain point in the set of data to vary from the mean.
 * __Variance__**

So how do we compute variance?

The formula for variance is as follows: Where represents standard deviation, ∑ is the “sum of”, x is a specific number in the set, µ is the mean, and N is the number of points.

where M represents the mean.
 * __Note:__** This is the variance of a population. When taking the variance of a sample, the same formula can be used (s^2 instead of [[image:1-1.gif]]), however this would make the result biased, meaning that the computed numbers would either overestimate or underestimate what is being determined. To give an unbiased estimate of [[image:1-1.gif]], use the formula:

If frequency is added into the mix, the formula must be altered: where f represents frequency and represents the mean. Remember, when you find the mean of set of data with frequency attached to each point, you must multiply that point by its frequency and then add all the products and divide by the addition of all the frequencies to achieve the correct mean. For example: The mean of this set of data is (3*10) + (5*2) + (6*1) divided by 13 = 3.538 Histograms are a good way to visualize the frequency of an occurrence.
 * Point in data || Frequency ||
 * 3 || 10 ||
 * 5 || 2 ||
 * 6 || 1 ||

Let’s see an example to fully understand Variance:

What is the variance of this set of numbers (assume population)? (2,6,7,8,9,13,25)


 * First, find the mean of this data set: (70/7) = 10
 * Next, subtract the mean from each individual number in the data set: (2-10 = -8) (6-10 = -4) (7-10 = -3) (8-10 = -2) (9-10 = -1) (13-10 = 3) (25-10 = 15)
 * Now square each answer and add them together: 64+16+9+4+1+9+225= 328
 * Divide that number by the amount of numbers in the set: 328/ 7 = 46.857

Try one on your own: Find the variance of (34, 23, 66, 29, 73, 36, 47)


 * __Determining Standard Deviation__**

Now that we know variance, standard deviation is simple. As stated before, standard deviation is the square root of the variance. Therefore, use the following formulas: (for a sample) (for a population)

(for frequency)

The standard deviation is small when all data points are close to the mean; the bell curve would appear steep. The standard deviation is large when all data points are far from the mean; the bell curve would be flattened out. To achieve a standard deviation of zero, every number in the set of data must be the same. Here you can see the difference between a small (numbers are closer to the mean rather than spread out) standard deviation (top) and one that is a bit larger (below).


 * Understand?**

Let’s take the example we worked out above and find the standard deviation. In the problem above, we discovered that the variance of the set of numbers (2,6,7,8,9,13,25) is about 46.857 To find the standard deviation just take the square root of this value (if you notice, the formulas for variance and standard deviation are the same except for the square root in the standard deviation formula). So, the standard deviation of that set is about 6.845

Can you figure out the standard deviation of example number two (34, 23, 66, 29, 73, 36, 47)?


 * Now that we understand standard deviation, let's go further:**

The z-score is another useful term for understanding standard deviation. It represents the amount of standard deviations a point is away from the mean. Z-score, or standard value, has the following formula:

How do we use this? Suppose you pick a point, 55, from a set of data that happens to have a standard deviation of 4 and a mean of 43. The standard value for that set of numbers is 3, meaning that the point 55 is three standard deviations away from 43, or the mean.

Okay, now that you understand the concept, let's try something a little more challenging (assume population for all):
 * Additional Practice**


 * 1) After giving the final exam, a teacher discovered that the average score was about a 74 and the standard deviation was about 7. Realizing the test was too hard, the teacher added 15 points to each score. What is the new mean and standard deviation?
 * 2) The standard deviation of a set of data is about 2.58 and the mean is 20. The data set consists of the numbers 16,17,18,19,20,21,22,23,24. What would happen to the standard deviation if the number 18 was added to the data set? What about 10?
 * 3) While doing your homework, you find that the standard deviation of a set of numbers is about 4.39 and the mean is about 21.857. However, you spilled sauce on one number of the set, so you only know 6 out of the 7 numbers in the set. These numbers are 20, 13, 28, 24, 22, 21. What would happen to the standard deviation if 31 was added to the set?


 * Answers to above problems:**
 * 1) The mean is 89 and the standard deviation is 7. The standard deviation remains the same because the same amount of points was added to every test, therefore the distance from the points to the mean remains the same.
 * 2) The standard deviation would decrease to about 2.53. If 10 was added, the standard deviation would increase to about 4.
 * 3) The number covered by the sauce is 25. The standard deviation would be about 5.23.

Bibliography: Niles, Robert. “Standard Deviation.” __RobertNiles.com__. 2008. 21 Dec. 2008 . “Standard Deviation.” __Managers-Net__. 2008. 21 Dec. 2008 . “Standard Deviation and Variance.” __Hyperstat Solutions Inc__. 2008. 21 Dec. 2008 . “Standard Deviation Tutorial.” __EasyCalculation.com__. 2008. 21 Dec. 2008 .