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Interactive animation |
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Standard deviation
By definition, the (positive) square root of the variance.
* We will denote "
" the Standard
Deviation of a distribution
(and its variance ²).
* We will denote "s" the Standard Deviation of a sample.
Although the variance is the favorite quantity to
study the dispersion of a group of numbers around its average
value, it is expressed in a unit that is the square of the unit
the quantity is measured in. For example, if you are interested in deposits
in bank accounts expressed in $, then the variance in expressed in ($)²,
hardly a popular currency unit. But the Standard Deviation is expressed
in the same unit as the quantity that is being considered and can therefore
be represented graphically.
The next animation illustrates the concept of Standard Deviation.
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The "Book of Animations" on your computer |
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On either side of the average are displayed segments that are 1 Standard Deviation long.
Drag red points with your mouse, and observe the influence on the Standard Deviation. * When you push a point towards
the group average, the Standard Deviation decreases. When you pull it away from
the group average, the Standard Deviation increases. * What is the largest possible
value of the Standard Deviation you can obtain (within the limits of the illustration) ?
How would you characterize the situation(s) conducive to this largest
value ? * You can easily obtain configurations where both S-D ticks are "inside" the group of red points. How would you characterize these configurations in words ? You can also obtain configurations where one S-D tick is inside, and the other one outside the group of points. How would you characterize such configurations in words ? Now try to have both ticks outside the group. You can't. Why ? The definition of the Standard Deviation is rather
arbitrary (just as that of the variance). Would the same phenomenon be
observed with any other reasonable measure of group dispersion ? * Find a configuration where
one red point is positioned exactly under the left S-D tick. Now concentrate
on the position of the right S-D tick. Move the red point to the right : the
right S-D tick moves to the right too. Get the point back to its original
position, and now move it to the left. The right S-D ticks moves again to the
right. |
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