Problems 15 and 16 use the following steps to approximate the median from grouped data.
Approximate the median of the frequency distribution in Problem 4.

Verified step by step guidance

1. Identify the median class by first calculating the total frequency \( N \) of the grouped data from Problem 4, then find \( \frac{N}{2} \). The median class is the class interval whose cumulative frequency just exceeds \( \frac{N}{2} \).
2. Determine the following values for the median class: the lower boundary \( L \), the frequency of the median class \( f_m \), and the cumulative frequency before the median class \( F \).
3. Find the class width \( h \) of the median class, which is the difference between the upper and lower boundaries of the class interval.
4. Use the median formula for grouped data: \[ \\text{Median} = L + \\left( \\frac{\\frac{N}{2} - F}{f_m} \\right) \\times h \]
5. Substitute the values of \( L \), \( N \), \( F \), \( f_m \), and \( h \) into the formula and simplify to approximate the median.

Key Concepts

Grouped Data and Frequency Distribution

Grouped data organizes raw data into intervals or classes with corresponding frequencies. A frequency distribution shows how many data points fall into each class, which helps summarize large datasets and is essential for calculating measures like the median.

Median of Grouped Data

The median is the middle value that divides the data into two equal halves. For grouped data, the median is approximated using the class interval containing the median position, since individual data points are not known exactly.

Median Approximation Formula

To approximate the median in grouped data, use the formula: Median = L + [(N/2 - F) / f] × w, where L is the lower boundary of the median class, N is total frequency, F is cumulative frequency before median class, f is frequency of median class, and w is class width.