Objective
Here you will learn to calculate the standard deviation of sample sets and populations.
Concept
What is standard deviation? How is the standard deviation of a set related to variance? Is the standard deviation of a sample different from that of a population, the way it is with variation?
This lesson details the process of calculating standard deviation, and introduces a few examples of its use. After the lesson we’ll review the questions above, using the knowledge we have gained.
Watch This
http://youtu.be/HvDqbzu0i0E Khan Academy – StatisticsStandard Deviation
Guidance
Standard deviation is a very common term in statistics and it is not particularly difficult to calculate, particularly if you have already identified the variance of a set. The standard deviation is sort of a “reference difference from the mean” that you can use to evaluate the spread of the data in a set.
For instance, assume the mean of a particular set is 6 and the standard deviation is 4. If you are considering a value of 21, it is probably a very rare occurrence in that set since 21 is nearly 4 standard deviations away from the mean ( and 16 more than the mean would be 22). However, a value of 8 is much more likely, given that it is only of a standard deviation (SD) away from the mean.
Recall from the lesson Calculating Variance that calculating the variance of a set involves finding the arithmetic mean, subtracting each data point value from the mean and squaring the result, then finding the sum of the squared results and dividing by either the number of members of the set (population) or the number of members 1 (sample). See the first part of Example B for a review of finding the variance.
 Once you have the variance of a population, you are practically done finding the SD.
 To find the SD, simply take the square root of the variance . That’s it!
 One important difference between the variance and the standard deviation is that the units associated with variance are the square of the units of the original values, but the units associated with the standard deviation are the same as the units in the original set.
We will return to SD in our chapter on “Normal Distribution”, when we will further discuss the uses of the SD of both samples and populations.
Example A
What is the standard deviation of a set with of 14.6?
Solution: The standard deviation is simply the square root of the variance . As a formula: .
In this case we have:
Example B
What is the of set ?
Solution: First find the variation of the set:

Deviations and squared deviations:
Example C
Katrina wants to use the average scores of the top long jumpers at the 5 schools in her district to predict the average long jumps for top competitors at all schools in her state. Data for her district is below. Find the appropriate variance and standard deviation of the jumps.
Solution: Since Katrina intends to generalize from her sample data back to the population of jumpers in her state; we need to find the sample variance and corresponding sample standard deviation.

Start by finding the mean distance:
 As a decimal:

Deviations and squared deviations of each value:
 (Remember to divide by , since this is a sample)
Concept Problem Revisited
What is standard deviation? How is the standard deviation of a set related to variance? Is the standard deviation of a sample different from that of a population, the way it is with variation?
By now you should know that standard deviation is a measure of the spread of data, and is calculated as the square root of the variance. Since variance is calculated slightly differently for a sample than for a population, the deviation will differ similarly.
Vocabulary
Standard deviation is calculated by finding the square root of the variance. The standard deviation acts as a reference unit of difference from the mean in a set of data.
The variance is calculated as the sum of the squared differences from the mean, divided by either the number of values (for populations) or the number of values minus one (for samples).
Guided Practice
1. Find the mean , variance , and standard deviance of set .
2. Find the mean , variance , and standard deviance of set .
3. Which set has the greater standard deviation, or ?
4. Kevin takes a random sample of ages of students in his class, and gets the following values, what is the sample variance and standard deviation of the set?
Solutions:
1. Let’s start by finding the mean, since we will need it to calculate the others:
Now we calculate the deviation of each value from the mean and square it:
Now we sum the squared deviations: , and divide the total by the number of values: to get the variance.
Finally, to get the standard deviation , just take the square root of .
2. Start by finding
Next, find the squared variation from the mean for each value:
Sum the squared deviations and divide by the number of values to get the variance:
Finally, take the square root of the variance to get the standard deviation:
3. Follow the same series of steps to find the standard deviation of each set.
Set has the greater standard deviation
4. There are 13 values, with
 The sum of the squared deviations is: 15.2308, divide by 12 (since this is a sample!), to get the sample variance :
 The square root of the sample variance is the sample standard deviation:
Practice
Find and :
1. 265, 280.7, 293, 279, 314.2, 300, 289
2. 7200, 7020, 7165.9, 7100, 7196, 7112, 7116.1
3. 27, 20.3, 30.7, 40, 46, 36, 40, 33
4. 3607, 3600, 3600, 3631, 3600.6
5. 700, 700, 712, 736, 741, 716, 782
6. 3370, 3300.5, 3366, 3306.6, 3310, 3336, 3301.3
Calculate the sample standard deviation:
7. 34.4, 34, 34.7, 34.6, 34, 34.1, 31, 31.3
8. 989.22, 990.6, 992, 996.9, 981.1, 986, 975
9. 10, 16, 10.33, 10.63, 18, 17, 16.36, 10.46
10. 3240, 3260, 3250, 3280, 3280, 3300, 3310, 3270