[BINNY]

Calculating Standard Deviation _ Long Procedure


There is a little more math involved in calculating the standard deviation, but it is not advanced. The standard deviation is simply the square root of the average squared deviation of the data from the mean. Before you allow this definition to scare you off, let’s calculate the standard deviation for our dataset of child weights together:

13 22 26 38 36 42
49 50 77 81 98 110

• Step 1: First, we calculate the mean (or average) of the data. Fortunately, we’ve already done this. The mean is 53.5.
• Step 2: Now, subtract the mean from every item in the set. Often, a table is helpful in performing these calculations. I have performed these calculations below:

Child Weight	Subtract the Mean	Difference
13	53.5	-40.5
22	53.5	-31.5
26	53.5	-27.5
38	53.5	-15.5
36	53.5	-17.5
42	53.5	-11.5
49	53.5	-4.5
50	53.5	-3.5
77	53.5	23.5
81	53.5	27.5
98	53.5	44.5
110	53.5	56.5

Notice that if you were to sum all the numbers in the "Difference" column, you would get a sum of zero. This makes sense, of course, because by definition, the mean should be the exact middle value equidistant to each of the points in the dataset, so the positive and negative differences will always balance each other out. Adding these numbers will always result in zero, regardless of how condensed or dispersed the data values are about the mean.

• Step 3: Square the difference between each number and the mean(the third column in the table above).

  In order to prevent our sum of the Difference column from resulting in zero, we can square these numbers. Remember this means multiplying each number by itself (-40.5 times -40.5 equals 1640.25). The new squared differences are now as follows:
  Difference Squared:
  1640.25
  992.25
  756.25
  240.25
  306.25
  132.25
  20.25
  12.25
  552.25
  756.25
  1980.25
  3192.25
  

• Step 4: Sum the squared differences. We get a sum of 10,581.
• Step 5: Divide this sum by the number of items (for a sample, you would instead divide by n-1; remember n is the count).

  Since our dataset was a sample of child weights, we will divide by n-1 (12-1=11). The answer is 961.9 This number is called the variance.
  

• Step 6: Take the square root of the variance to find the standard deviation.

  Taking a square root converts the variance from squared units to the original units of measurement. Our standard deviation is 31.0.
  

Yeah! We did it! Phew...






Does anyone know the shortest method if the data set given is something like the above post.