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) implies x = 9.62 + 0.26x)
(x – 0.26x = 9.62 implies 0.74x = 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 fd^{2}}{sum f}})
| (x) | (f) | (fx) | (d = |x – bar{x}|) | (d^{2}) | (fd^{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 |
(Mean(bar{x}) = frac{sum fx}{sum f})
= (frac{200}{50}
= 4
S.D = (sqrt{frac{100}{50}})
= (sqrt{2})
= 1.4142