# What is the standard deviation of {6, 2, 5, 5, 7, 3, 4, 8, 1, 2, 2}?

Feb 23, 2016

Mean =4.09 Standard Deviation = 2.195

#### Explanation:

Step-I
Given values of random variable are converted to a discrete frequency distribuion table.
Step-II
Then convert the frequencies to respective probabilities by dividing each freq. with the total freq.
Step-III
Now multiply each value of random variable with its respective prob. Xf(x). This column will be labeled as E(X).
Step-IV
Next find X^2
f(x)
Step-V
Calculate Values.
E(x) = X* f(x) =45/11=4.09
Var.(x) = E(x^2) ----- (E(x))^2
= 237/11 ----- (4.09)^2
= 2.195

Feb 23, 2016

Standard Deviation $\sigma = 2.193$

#### Explanation:

Note this just an alternate methodology to the one provided by Humayan Hassan and Kyx. Their method is completely valid, but I personally find it less simple.

Refer to the image below for actual data values.

Calculate the sum of the data values and from that the Mean (in this case $\mu \approx 4.09$).

For each data value calculate the difference or "Deviation" from this Mean.

For each data value find the Square of the "Deviation" (previous step).

Find the sum of the Squared Deviations.

Dividing the Sum of the Squared Deviations by the number of data values gives the Variance $\left({\sigma}^{2}\right)$ (n.b. "population variance"; for sample variance divide by the number of data values minus 1).

Taking the Square Root of the Variance gives the Standard Deviation. ...of course if you were doing this in a spreadsheet like Excel, you would simply enter the data and use the $S T D E V P \left(\ldots\right)$ function.