Question

show that for any two numbers a and b standard deviation is given by |a-b| /2​

Answer 1

Answer:Show that for any 2 numbers a and b, standard deviation is given by |a-b|÷2

Note that standard deviation is always positive or zero.

1. Take the mean of the numbers: (a+b)/2

2. Subtract the mean from each of the original numbers and square the differences:

a - (a+b)/2 = (2a - a - b)/2 = (a-b)/2

[(a-b)/2]2 = (a2-2ab+b2)/4

b - (a+b)/2 = (2b-a-b)/2 = (b-a)/2

[(b-a)/2]2 = (b2-2ab + a2)/4

3. Take the mean of the squares above

[(a2-2ab+b2)/4 + (b2-2ab+a2)/4]/2

= (2a2 - 4ab + 2b2)/8

= 2(a2 - 2ab + b2)/8

= (a2- 2ab + b2)/4

= (a-b)2/4

4. Take the square root of the above

√[(a-b)2/4] = ±(√(a-b)2)/2 = ±(a-b)/2

Since sd is always positive or zero

±(a-b)/2 = |a-b|/2

Step-by-step explanation: Hope it helped u