Interactive animation

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.

 The "Book of Animations" on your computer

 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.Can you explain that ?   Would the same phenomenon be observed with any other reasonable measure of group dispersion ?

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