The table shows the scores obtained when a fair die was thrown a number of times. If the probability of obtaining a 3 is 0.26, find the (a) median (b) standard deviation of the distribution.

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The table shows the scores obtained when a fair die was thrown a number of times. If ...

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The table shows the scores obtained when a fair die was thrown a number of times.

Score	1	2	3	4	5	6
Frequency	2	5	x	11	9	10

If the probability of obtaining a 3 is 0.26, find the (a) median
(b) standard deviation of the distribution.

Explanation

Score	1	2	3	4	5	6
Frequency	2	5	x	11	9	10

P(3) = 0.26

\frac{x}{37 + x} = 0.26

x = 0.26 (37 + x) ⟹ x = 9.62 + 0.26 x

x - 0.26 x = 9.62 ⟹ 0.74 x = 9.62

x = \frac{9.62}{0.74} = 13

Total toss frequency = 37 + x
= 37 + 13
= 50.
(a) Median =

\frac{n + 1}{2}

\frac{50 + 1}{2}

= 25.5
From the table, 25.5 occurs at 4 which is the median.
(b) Standard deviation =

\sqrt{\frac{\sum f d^{2}}{\sum f}}

$x$	$f$	$\fx$	$d = \| x - \bar{x} \|$	$d^{2}$	$f d^{2}$
1	2	2	-3	9	18
2	5	10	-2	4	20
3	13	39	-1	1	13
4	11	44	0	0	0
5	9	45	1	1	9
6	10	60	2	4	40
$\sum$	50	200			100

M e a n (\bar{x}) = \frac{\sum f x}{\sum f}

\frac{200}{50} = 4 S . D = \(\sqrt{\frac{100}{50}}

\sqrt{2}

= 1.4142

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