Standard Deviation Definition:
Standard deviation is a statistical
measure of spread or variability.The
standard deviation is the root mean square
(RMS) deviation of the values from their
arithmetic mean.
Variance
Definition: The square of the
standard deviation. A measure of the degree
of spread among a set of values; a measure
of the tendency of individual values to vary
from the mean value.
Formula:
Standard Deviation
 |
Population Standard Deviation
|
where Σ = Sum of
X = Individual score
M = Mean of all scores
N = Sample size (Number of scores)
Variance :
Variance = s
2
Standard Deviation Method1 Example: To
find the Standard deviation of 1,2,3,4,5.
Step 1: Calculate the mean and deviation.
|
X |
M |
(X-M) |
(X-M)2 |
| 1 |
3 |
-2 |
4 |
| 2 |
3 |
-1 |
1 |
| 3 |
3 |
0 |
0 |
| 4 |
3 |
1 |
1 |
| 5 |
3 |
2 |
4 |
Step 2:Find the sum of (X-M)
2
4+1+0+1+4 = 10
Step 3:N = 5, the total number of
values.Find N-1.
5-1 = 4
Step 4:Now find
Standard Deviation using the formula.
√10/√4 = 1.58113
Standard Deviation Method2 Example: To
find the Standard deviation of 1,2,3,4,5.
Step 1:First, square each of the scores.
| X |
X2 |
| 1 |
1 |
| 2 |
4 |
| 3 |
9 |
| 4 |
16 |
| 5 |
25 |
Step2: Use the formula
s = square root of[(sum of Xsquared -((sum
of X)*(sum of X)/N))/(N-1)]
= square root of[(55-((15)*(15)/5))/(5-1)]
= square root of[(55-(225/5))/4]
= square root of[(55-45)/4]
= square root of[10/4]
= square root of[2.5]
s = 1.58113
Population
Standard Deviation Example: To find the
Population Standard deviation of 1,2,3,4,5.
Perform the steps 1 and 2 as seen in above
example.
Step 3:Now find the
population standard deviation using the
formula.
√10/√5 = 1.414
Variance Example: To find the Variance of
1,2,3,4,5.
After finding the
standard deviation square the values.
(1.58113)
2 = 2.4999
Same for Population standard deviation.
(1.414)
2 = 2
