Standard Deviation — Definition & Calculation


Definition: Standard deviation is a measure of how far apart the data are from the average of the data. If all the observations are close to their average then the standard deviation will be small.

How to calculate standard deviation:
Suppose that an investor has $600 to invest and is considering investing all of it in the shares of one firm, currently trading at $30. The investor assesses a 0.75 probability that the shares will increase in market value to $33 over the coming period and a 0.25 probability that the share will decrease in its market value to $26. Assume that the firm will pay $1 dividend per share at the end of the year.

The payoffs from the proposed investment are as follows:

If shares increase: $33 x 20 shares + $20dividend = $680
If shares decrease: $26 x 20 shares + $20dividend = $540

PAYOFF

  ($)

RATE OF RETURN

PROBABILITY

EXPECTED RATE OF RETURN

VARIANCE

(1)

(2)

(3)

(4) = (2) x (3)

(5)

         

680

(680 – 600)/600 = 0.13

0.75

0.0975

(0.13 – 0.0725)^2 x 0.75 = 0.0025

         

540

(540 – 600)/600 = – 0.10

0.25

-0.025

(-0.1 – 0.0725)^2 x 0.25 = 0.0074

         
   

sum (x)

0.0725

Add the above two results to get σ² = 0.0099

The standard deviation is the square root of the variance. In the above example, the standard deviation is square root of 0.0099 i.e. 0.0995 or 9.95%

How to calculate the standard deviation using an ordinary calculator?
Key in 0.0099 and then press the √ key to get 0.0095 or 9.95%

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