A biased \(5\)-sided spinner gives the scores \(1\), \(2\), \(3\), \(4\) and \(5\) with the probabilities shown in the table, where \(p\) and \(q\) are constants.
| Score | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) |
| Probability | \(0.2\) | \(0.3\) | \(p\) | \(p\) | \(q\) |
Given that the variance of the score is \(1.61\), calculate the mean score.[7]
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