3.6
Standard Deviation
Standard deviation: the measure of how closely all of the data in the data set surrounds the mean.
A normal distribution of data is represented when the majority of the data is found close to the
average of the set.
Variance
To calculate the standard deviation, you will first need to find the variance.
Variance of a data set is the average of each distance from the mean squared.
FOR EXAMPLE:
Annual fishing competition,
10 competitors who caught fish.
The total weights of their fish were 23 lbs., 37 lbs., 82 lbs., 49 lbs., 56 lbs., 70 lbs., 63 lbs., 72 lbs., 63 lbs. and
45 lbs.
The first step to calculate the variance is to find the mean of the data set.
Add 23 + 37 + 82+ 49 + 56 + 70 + 63 + 72 + 63 + 45, which equals 560. Then, divide 560 ÷ 10 = 56. The average
weight of fish caught was 56 lbs.
The next step to calculate the variance is to subtract the mean from each value.
The best way to set this up is in a table. Looking at the table, you can add a column for the mean to make the
subtracting easier. To get these totals, we will now subtract:
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
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





23 - 56 = -33
37 - 56 = -19
82 - 56 = 26
49 - 56 = -7
56 - 56 = 0
70 - 56 = 14
63 - 56 = 7
72 - 56 = 16
63 - 56 = 7
45 - 56 = -11
Next, we will take each of these differences and square them. So we will calculate:
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-33^2 = 1,089
-19^2 = 361
26^2 = 676
-7^2 = 49
0^2 = 0
14^2 = 196
7^2 = 49
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3.6


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16^2 = 256
7^2 = 49
-11^2 = 121
Finally, to calculate the variance, we will average each of these squared totals. To do so, add 1,089 + 361 +
676 + 49 + 0 + 196 + 49 + 256 + 49 + 121 = 2,846. Next, take the total, 2,846, and divide by the 10 data points.
2,846 ÷ 10 = 284.6, so the variance of this data set is 284.6.
How to Calculate the Standard Deviation
Calculating the standard deviation is simple once we've found the variance. To find the standard deviation,
we will simply take the square root of the variance. In the previous example, our variance was 284.6. The
square root of 284.6 is 16.9 when rounded to the tenths place. The standard deviation for the total weights
of fish caught was 16.9 lbs.
Steps to Finding the Standard Deviation
As you can see, finding the standard deviation is not too difficult. There is a specific series of steps that must
be carried out in order:
1.
2.
3.
4.
5.
Find the mean of your data set.
Subtract the mean from each of the data points.
Take each of the differences and square them.
Find the variance, which is the average of the squared differences.
Calculate the square root of the variance, which is the standard deviation.
What Can Standard Deviation Tell Me?
The standard deviation can help us determine if our data is a normal distribution. In a normal distribution,
most of your data will fall within one standard deviation of your mean. To calculate this range, you will add
and subtract the standard deviation to the mean.
In the example:
MEAN (the average weight of the fish caught) was 56 lbs.
The standard deviation was 16.9.
So to find the range of where most of the information will be, we will add and subtract the standard
deviation to the mean: 56 + 16.9 = 72.9 and 56 - 16.9 = 39.1.
This tells us that the majority of data for this set will be between 72.9 and 39.1, which represents one
standard deviation of the mean.
Lesson Summary
Even though standard deviation can seem difficult, you should now have the tools to calculate the standard
deviation for any set of data. Remember, standard deviation is the measure of how closely all of the data in
the data set surrounds the mean. By following these five steps, you can easily calculate the standard
deviation of any set of data:
1. Find the mean of your data set.
2. Subtract the mean from each of the data points.
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3.6
3. Take each of the differences and square them.
4. Find the variance, which is the average of the squared differences.
5. Calculate the square root of the variance, which is the standard deviation.
To check to see if a number is within one standard deviation of your mean, you will need to add and subtract
the standard deviation to the mean. This will give you a specific range.
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3.6
QUIZ AND ANSWERS
Find the variance for the following data set:
101, 106, 125, 142, 78, and 109
Your Answer
31
Watch Correct Answer Read Answer Explanation
Explanation
First find the mean:
(101 + 106 + 125 + 142 + 78 + 109) / 6 = 110
Now subtract the mean from each value:
101 - 110 = -9
106 - 110 = -4
125 - 110 = 15
142 - 110 = 32
78 - 110 = -32
109 - 110 = -1
Square each of these values and find the mean of the sum of these numbers:
(81 + 16 + 225 + 1024 + 1024 + 1) / 6 = 395
Find the standard deviation for the following data set:
101, 106, 125, 142, 78, and 109
Your Answer
19.9
Watch Correct Answer Read Answer Explanation
Explanation
First find the mean:
(101 + 106 + 125 + 142 + 78 + 109) / 6 = 110
Now subtract the mean from each value:
101 - 110 = -9
106 - 110 = -4
125 - 110 = 15
142 - 110 = 32
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3.6
78 - 110 = -32
109 - 110 = -1
Square each of these values and find the mean of the sum of these numbers:
(81 + 16 + 225 + 1024 + 1024 + 1) / 6 = 395
To find the standard deviation take the square root of the variance:
√395 = 19.9
Find the variance for the following data set:
16, 20, 14, 24, 16, 19, 30, and 8
Your Answer
2.64
Watch Correct Answer Read Answer Explanation
Explanation
Following the steps, find the variance by subtracting the mean from each number, then squaring each of
those numbers. Once that is finished, determine the average by adding them together and dividing by the
total number of numbers. For this number set the answer is 38.48
Find the standard deviation for the following data set:
16, 20, 14, 24, 16, 19, 30, and 8
Your Answer
6.2
Watch Correct Answer Read Answer Explanation
Explanation
Following the steps, find the variance by subtracting the mean from each number, then squaring each of
those numbers. Once that is finished, determine the average by adding them together and dividing by the
total number of numbers. To get the standard deviation take that average and find the square root. For this
number set the answer is 6.2
Find the range for one standard deviation in the following data set:
16, 20, 14, 24, 16, 19, 30, and 8
Your Answer
8 to 30
Watch Correct Answer Read Answer Explanation
Explanation
Following the steps, find the variance by subtracting the mean from each number, then squaring each of
those numbers. Once that is finished, determine the average by adding them together and dividing by the
total number of numbers. To get the standard deviation take that average and find the square root. To
calculate this range, you will add and subtract the standard deviation to the mean. For this number set you
will get 12.175 to 24.575
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