Since \(P(X < 54.1) = 0.5\), the mean \(\mu = 54.1\) because the median of a normal distribution is the mean.

For \(P(X > 50.9) = 0.8665\), we find the corresponding \(z\)-score. The probability \(P(X > 50.9) = 0.8665\) implies \(P(X < 50.9) = 1 - 0.8665 = 0.1335\).

Using the standard normal distribution table, \(P(Z < z) = 0.1335\) corresponds to \(z = -1.11\).

Standardizing, we have:

\(-1.11 = \frac{50.9 - 54.1}{\sigma}\)

Solving for \(\sigma\):

\(\sigma = \frac{50.9 - 54.1}{-1.11} = 2.88\)