Standard Deviation Formula

Introduction

In statistics, the standard deviation is a measure that reflects the amount of variation of the set of data values. It also stipulates how individual data differs from the average of a dataset. When the standard deviation is low, it means that the data points are closer to the mean, while a high standard deviation denotes that the data points fall within a wide range of values.

Formula for Standard Deviation

There are two general formulas for standard deviation: one for a population and one for a sample. We will go through each,

Population Standard Deviation Formula:

　　　 　 　　 　　 

Sample Standard Deviation Formula:　　 　 　　 　　  

                            

notations for Standard Deviation

σ = Standard Deviation

xi = Terms Given in the Data

x̄ = Mean

n = Total number of Terms

Standard Deviation Formula Based on discrete frequency Distribution

For discrete frequency distribution of the form:

x: x1, x2, x3, … xn and

f : f1, f2, f3, … fn

The formula for standard deviation becomes.

 

Here, N is given as:

N = n∑i=1 fi

Standard Deviation Formula for Grouped Data

The other standard deviation formula which is derived from the variance. The formula for the variance is,

Question Based on Standard Deviation Formula

Question: In a survey, 6 students were asked how many hours per day they study on an average? Their answers were given as : 2, 6, 5, 3, 2, 3. Calculate the standard deviation.

Solution:

 

Step A: Find the mean of data:

 

                    =3.5

 

Step 3: Now use the formula for finding the Standard Deviation

Sample Standard Deviation = 

 

 

 

      =√(13.5/[6-1])

 

      = √[2.7]

 

      = 1.643