What is a standard deviation?
In statistics, standard deviation measures the data set dispersion relative to its mean. It is measured as a square root (sq. root) of the variance. The process standard deviation was given by Karl Pearson in 1893.
Standard deviation is calculated by the sq. root of the variance by summing each data’s point of deviation relative to the data set’s mean.
A standard deviation is a statistical tool primarily used in finance. It is used to calculate the annual rate of return of an investment. It shows the investment’s market index.
To learn this concept of standard deviation, you need to know about the standard deviation formula or the standard deviation calculator.
Standard deviation Formula
It would be best to use a standard deviation calculator to find the solution of any problem. There are two types of series, discrete and continuous. There are four methods for calculating standard deviation under discrete and continuous series:
- Actual Mean Method
- Assumed Mean Method
- Step-deviation method
Actual Mean Method
Here is the formula for the actual mean method for the discrete series:
σ = √∑d2/n
Example of the actual mean method:
Calculate the standard deviation of the following data:
5, 15, 25
X = 5+15+253 = 453 = 15
| X | d (x-x) | d2 |
| 5 | -10 | 100 |
| 15 | 0 | 0 |
| 25 | 10 | 100 |
| 0 | 200 |
σ = √∑d2/n
= 2003
= 66.666
= 8.165
Assumed Mean Method
Here is the standard deviation formula for assumed mean method for the discrete series:
σ =√∑d2/n- (√∑d/n)2x c
d = X- AX
You need to take an assumed mean from the X column. In this method, you need to calculate using the
Example of the assumed mean method:
Calculate the standard deviation of the following data:
5, 15, 25
| X | d (x-A X) | d2 |
| 5 | 0 | 0 |
| 15 | 10 | 100 |
| 25 | 20 | 400 |
| 30 | 500 |
Assumed mean = 5
σ = √∑d2/n- (√∑d/n)2x c
σ = 5003 – 3032
σ = 166.667 – 102
σ = 166.667 – 100
= 66.667
= 8.165
Step Deviation Method
Here is the standard deviation formula for step deviation method for the discrete series:
σ =√∑d12/n- (√∑d1/n)2x c
d = x- assumed mean
Example of the step deviation method:
Calculate the standard deviation of the following data:
5, 15, 25
| x | d ( x – 5) | d1 (d/10) | d12 |
| 5 | 20 | 2 | 4 |
| 15 | 10 | 1 | 1 |
| 25 | 0 | 0 | 0 |
| 3 | 5 |
Assumed Mean = 5
σ = √∑d12/n- (√∑d1/n)2x c
σ = 53 – 332 x 10
σ = 1.667 – 12 x 10
σ = 1.667 – 10
σ = -8.333
Actual Mean Method
Here is the formula of the actual mean method of the continuous series:
= √∑fd2/n
d= m – X
An example for the actual mean method:
In this method, you will need to first calculate the mean of the data and then calculate the deviations from it.
| CI | f | m | fm | d (m – X) | fd | fd2 |
| 10-20 | 4 | 15 | 60 | -20 | -80 | 6400 |
| 20-30 | 8 | 25 | 200 | -10 | -80 | 6400 |
| 30-40 | 12 | 35 | 420 | 0 | 0 | 0 |
| 40-50 | 16 | 45 | 720 | 10 | 160 | 25600 |
| 40 | 1400 | 38400 |
X = fmf= 140040 = 35
= √∑fd2/n
= 3840040= 960
Assumed Mean Method
Here is the standard deviation formula for assumed mean method for the continuous series:
= √∑fd2/n- (√∑fd/n)2
d = X- AX
| CI | f | m | d (m -A X) | fd | fd2 |
| 10-20 | 4 | 15 | 10 | 40 | 1600 |
| 20-30 | 8 | 25 | 0 | 0 | 0 |
| 30-40 | 12 | 35 | 10 | 120 | 14400 |
| 40-50 | 16 | 45 | 20 | 320 | 102400 |
| 40 | 480 | 118400 |
Assumed mean is 45
= 11840040 – 480402
= 2972- 122
= 2972-144
= 2828
= 53.179
Standard deviation example
Here is a standard deviation example of step deviation method for continuous series:
standard deviation formula:
=√∑fd12/n- (√∑fd1/n)2x c
d = x- assumed mean
| CI | f | m | d(m – AX) |
d1 ( d/ c) |
fd1 | fd12 |
| 0-10 | 2 | 5 | -10 | -2 | -4 | 8 |
| 10-20 | 4 | 15 | 0 | 0 | 0 | 0 |
| 20-30 | 6 | 25 | 10 | 2 | 12 | 144 |
| 30-40 | 8 | 35 | 20 | 4 | 36 | 1296 |
| 20 | 44 | 1448 |
Assumed Mean = 15
c = class interval = 5
σ = √∑fd12/n- (√∑fd1/n)2x c
σ = 144820 – 44202 x 5
σ = 72.4 – 2.22x 5
σ = 72.4 – 4.84 x 5
σ = 72.4 – 4.84 x 5
σ = 67.56 x 5
= 67.56 x 5
= 8.219x 5
= 41.097
Conclusion
Understanding the concept of standard deviation is simple and easy. You need to follow this process properly by using the standard deviation calculator. When you follow the process step by step, you will understand the concept of standard deviation.