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
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Population Standard Deviation
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where Σ = Sum of
X = Individual score
M = Mean of all scores
N = Sample size (Number of scores)
Variance :
Variance = s2
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.
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
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