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Maths-

General

Easy

Question

If the mean deviation of number 1, 1 + d, 1 + 2d, …, 1 + 100d from their mean is 255, then d is equal to

10.0
20.0
10.1
20.2

The correct answer is: 10.1

Mean of a, b, 8, 5, 10 is 6. This implies that
$fraction numerator a plus b plus 8 plus 5 plus 10 over denominator 5 end fraction minus 6$
Therefore,
$a plus b equals 7$
Given that variance is 6.8. Therefore, variance is
$fraction numerator capital sigma open parentheses X subscript i end subscript minus A close parentheses to the power of 2 end exponent over denominator n end fraction equals fraction numerator left parenthesis a minus 6 right parenthesis to the power of 2 end exponent plus left parenthesis b minus 6 right parenthesis to the power of 2 end exponent plus 4 plus 1 plus 2 over denominator 5 end fraction equals 6.8$
Therefore,
$a to the power of 2 end exponent plus b to the power of 2 end exponent equals 25 rightwards double arrow a to the power of 2 end exponent plus left parenthesis 7 minus a right parenthesis to the power of 2 end exponent equals 25 rightwards double arrow a to the power of 2 end exponent minus 7 a plus 12 equals 0 text [using Eq. (1)] end text$
Therefore, a = 4, 3 and b = 3, 4.
Mean deviation is given by
$table row cell fraction numerator 1 over denominator n end fraction capital sigma open vertical bar x subscript i end subscript minus stack x with minus on top close vertical bar equals fraction numerator 1 over denominator n end fraction open square brackets table row cell vertical line 1 plus 50 d minus 1 vertical line plus vertical line 1 plus 50 d minus 1 minus d vertical line plus vertical line 1 plus 50 d minus 1 minus 2 d vertical line end cell row cell plus midline horizontal ellipsis plus vertical line 1 plus 50 d minus 1 minus 100 d vertical line end cell end table close square brackets end cell row cell rightwards double arrow 255 equals fraction numerator 1 over denominator 101 end fraction left square bracket 50 d plus 49 d plus 48 d plus midline horizontal ellipsis plus d plus 0 plus d plus midline horizontal ellipsis plus 50 d right square bracket end cell end table$
$equals fraction numerator 2 d over denominator 101 end fraction open square brackets fraction numerator 50 cross times 51 over denominator 2 end fraction close square brackets$
$rightwards double arrow d equals fraction numerator 255 cross times 101 over denominator 50 cross times 51 end fraction equals 10.1$

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