Questions: Median of a data set The following list contains the number of movies that a group of 12 students saw in theaters in the past 12 months. 13, 8, 9, 5, 20, 12, 13, 14, 15, 16, 7, 4 Find the median number of movies that the students saw.

Solution Steps

The first step in finding the median is to sort the given data set. The number of movies that the students saw is:

\[ \{13, 8, 9, 5, 20, 12, 13, 14, 15, 16, 7, 4\} \]

After sorting, the data becomes:

\[ \{4, 5, 7, 8, 9, 12, 13, 13, 14, 15, 16, 20\} \]

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

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

where \( Q = 0.5 \) (for the median) and \( N = 12 \) (the number of data points). Thus,

\[ \text{Rank} = 0.5 \times (12 + 1) = 0.5 \times 13 = 6.5 \]

Since the rank is \( 6.5 \), we need to find the average of the values at ranks \( 6 \) and \( 7 \):

• \( X_{\text{lower}} = 12 \) (the 6th value in the sorted list)
• \( X_{\text{upper}} = 13 \) (the 7th value in the sorted list)

Using the averaging formula:

\[ Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{12 + 13}{2} = \frac{25}{2} = 12.5 \]

Final Answer