Questions: Find the median of the frequency distribution. f x 42 1 43 3 46 3 51 1 52 2 54 2 55 2 56

Solution Steps

From the given frequency distribution:

\[ \begin{array}{|c|c|} \hline f & x \\ \hline 1 & 42 \\ 3 & 43 \\ 3 & 46 \\ 1 & 51 \\ 2 & 52 \\ 2 & 54 \\ 2 & 55 \\ 2 & 56 \\ \hline \end{array} \]

We construct the data points by repeating each value \( x \) according to its frequency \( f \):

\[ \text{Data points} = [42, 43, 43, 43, 46, 46, 46, 51, 52, 52, 54, 54, 55, 55, 56, 56] \]

The sorted data points are:

\[ \text{Sorted data} = [42, 43, 43, 43, 46, 46, 46, 51, 52, 52, 54, 54, 55, 55, 56, 56] \]

To find the median, we use the formula for the rank of the median:

\[ \text{Rank} = Q \times (N + 1) = 0.5 \times (16 + 1) = 8.5 \]

Since the rank is \( 8.5 \), we take the average of the values at positions \( 8 \) and \( 9 \) in the sorted list:

\[ Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{51 + 52}{2} = 51.5 \]

Final Answer