Question

Properties of standard deviation and variance

Answer 1

Answer:

Property 1

If a constant c is added to each value of a population function, then the new variance is the same as that of the old variance. The new standard deviation is also the same as that of the old standard deviation.

Property 2

If each data item of a population function is multiplied by a constant c, the new variance is c2 times the old variance. The new standard deviation is |c| times the old standard deviation.

Step-by-step explanation:

Proof of Property 1

Consider the data items:

x1, x2, x3, . . . xn having mean.

Add c to each data item: x1 + c, x2 + c, x3 +c, . . . xn + c and the new mean = + c

New d2

= (x1 - )2 + (x2 - )2 + . . . + (xn - )2

= old d2

Note: If the values are equal, the square root of the variance will be equal. Therefore, the standard deviations are also equal.

Proof of Property 2

Consider the population function x1, x2, x3, . . . xn whose mean is and variance d2 .

New data items: cx1, cx2, cx3, . . . cxn having new mean = c.

New variance =

= =

= c2´ the old variance

New standard deviation =