Questions: Find the median for the data items in the given frequency distribution. Score, x: 1, 2, 3, 4, 5, 6, 7, 8 Frequency, f: 1, 5, 6, 4, 3, 6, 3, 2 The median is (Type an integer or a decimal.)

Solution Steps

Given the frequency distribution of scores, we expand the data list based on the frequencies:

\[ \text{Scores: } [1, 2, 3, 4, 5, 6, 7, 8] \] \[ \text{Frequencies: } [1, 5, 6, 4, 3, 6, 3, 2] \]

The expanded data list is:

\[ \text{Data: } [1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 8, 8] \]

To find the median, we first determine the rank using the formula:

\[ \text{Rank} = Q \times (N + 1) = 0.5 \times (30 + 1) = 15.5 \]

Where \( N \) is the total number of data points, which is 30.

Since the rank is 15.5, we need to average the values at ranks 15 and 16. From the sorted data:

\[ \text{Sorted Data: } [1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 8, 8] \]

The values at these ranks are:

\[ X_{\text{lower}} = 4 \quad \text{(rank 15)} \] \[ X_{\text{upper}} = 4 \quad \text{(rank 16)} \]

Using the averaging formula:

\[ Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{4 + 4}{2} = 4.0 \]

Final Answer