Standard DeviationStandard deviation is the best measure of dispersion. Standard deviation is also known as root mean square deviation.
 
Standard deviation is the positive square root of the average of squared deviations taken from the arithmetic mean or more simply, the square root of the variance. WOW!! Let's see if we can simplify that!!
 
Standard deviation can be represented by the abbreviation S, sd, or sigma.
 
The formula is sigma =+sqrt{frac{1}{n} sum (x_i- overline{x}) ^2}, where x_i are the data values, overline{x} is the mean or average, and n is the number of pieces of data.
 
Properties
If all the observations assumed by a variable are constant then the standard deviation is zero. Standard deviation is unaffected due to change of origin but is affected due to change of scale.Let's look at examples illustrating the two methods for handling data.
 
Example 1: A large metropolitan elementary school has 8 kindergarten classes with enrollments of the following number of students: 16, 18, 19, 20, 22, 22, 23 and 25. Find the standard deviation.
 
Solution:
1. Find the mean: overline{x} =frac{16+18+19+20+22+22+23+25}{8} =frac{165}{8} =20.625
 
2. Let's look at the table we will need for the problem: List the data in the left column, the mean in the second column, their difference in the third column and the square of that difference in the last column.
 
 
3. To find the variance, find the sum of the values in the last column and then divide by the number of data values.
text{Var}=frac{sum (x_i-overline{x})^2}{n} =frac{21.3906+6.8906+2.6406+0.3906+1.8906+1.8906+5.6406+19.1406}{8}
=frac{59.8748}{8} =7.48435
4. To find the standard deviation, take the positive square root of the variance.
sigma =+sqrt{text{Var}} =+sqrt{7.48435} =+2.800
Sometimes we will have values in our data set that appear multiple times. Rather than constructing a table where each piece of data has its own line, we can condense the table taking into account the frequency with which certain data entries appear.
Example 2: Find the standard deviation for the data set: {14, 12, 20, 12, 20, 20, 14, 18, 20, 12, 18, 15, 12, 20, 14}
Solution: It might be easier to start by ordering the data: {12, 12, 12, 12, 14, 14, 14, 15, 18, 18, 20, 20, 20, 20, 20}1. Find the mean: overline{x} =frac{4(12)+3(14)+15+2(18)+5(20)}{15} =frac{241}{15} =16.0overline{666} approx 16.07
2. Let's look at the table we will need for the problem. Although there are 15 values, we notice that there are many duplicated.
Therefore, we will condense the table, taking into account the frequency of each different entry.
x_i f overline{x} x_i-overline{x} (x_i-overline{x} )^2 f(x_i-overline{x} )^212
4
16.07
-4.07
16.5649
66.2596
14
3
16.07
-2.07
4.2849
12.8547
15
1
16.07
-1.07
1.1449
1.1449
18
2
16.07
1.93
3.7249
7.4498
20
5
16.07
3.93
15.4449
77.2245
3. To find the variance, find the sum of the values in the last column and then divide by the number of data values. Remember to divide by 15, the number of pieces of data, not 5, the number of rows!
text{Var}=frac{sum (x_i-overline{x})^2}{n}
=frac{66.2596+12.8547+1.1449+7.4498+77.2245}{15}
=frac{164.9335}{15} =10.9955
4. To find the standard deviation, take the positive square root of the variance.
sigma =+sqrt{text{Var}} =+sqrt{10.9955} approx +3.3159