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.
Verified Correct Answer: .
Step-by-Step Mathematical Solution
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
Source: Myschool Past Questions Archive (Record #52774)