The standard deviation measures by how much the data values differ from the mean. The mean x is given by the sum of the data values divided by the number of observations. The symbol σ, read as sigma, represents the standard deviation. Let's review the process for finding the standard deviation.
Following this process, let's find the standard deviation of an example data set. Keep in mind that there are many possible examples.
{4,4,4,4,4,4}
In this case, we have 6 values. The mean will be the sum of the values in the set divided by 6.
x=4+4+4+4+4+4/6 ⇒ x= 4
With this value, we can now calculate the deviation of each value — the difference between the data value and the mean. Then we will square each deviation. Let's do this in a table.
x
x
x-x
(x-x)^2
4
4
0
0
4
4
0
0
4
4
0
0
4
4
0
0
4
4
0
0
4
4
0
0
Now, we need to find the mean of the squared deviation. This is called the variance of the data set. It is represented as σ^2. Let's do it!
σ^2=0+0+0+0+0+0/6 ⇓ σ^2=0
Finally, to find the standard deviation, we have to calculate the square root of the variance. Recall that the square root of 0 is also 0.
σ=sqrt(0) ⇒ σ=0
Therefore, we have an example data set with a 0 standard deviation.
Negative Standard Deviation?
Let's recall the standard deviation formula.
σ=sqrt((x_1-x)^2+(x_2-x)^2+...+(x_n-x)^2/n)
Note that, since each difference is squared, the numerator of the radicand is a sum of positive values. Additionally, n is positive because it is the number of values in the data set. This means that we will have the square root of a nonnegative number. Furthermore, because the standard deviation measures the distance from the mean, it cannot be negative.
