An approximate median can be found for data that have been...

An approximate median can be found for data that have been grouped into a frequency distribution. First it is necessary to find the median class. This is the class that contains the median value. That is the n/2 data value. Then it is assumed that the data values are evenly distributed throughout the median class. The formula is $$\mathrm{MD}=\frac{n / 2-\mathrm{cf}}{f}(w)+L_{m}$$ $\begin{aligned} \text { where } & n=\text { sum of frequencies } \\ & \mathrm{cf}=\text { cumulative frequency of class immedi- } \\ & \text { ately preceding the median class } \end{aligned}$ $$\begin{aligned} w &=\text { width of median class } \\ f &=\text { frequency of median class } \\ L_{m} &=\text { lower boundary of median class } \end{aligned}$$ Using this formula, find the median for data in the frequency distribution of Exercise $16 .$

An approximate median can be found for data that have been grouped into a frequency distribution. First it is necessary to find the median class. This is the class that contains the median value. That is the n/2 data value. Then it is assumed that the data values are evenly distributed throughout the median class. The formula is
$$\mathrm{MD}=\frac{n / 2-\mathrm{cf}}{f}(w)+L_{m}$$
$\begin{aligned} \text { where } & n=\text { sum of frequencies } \\ & \mathrm{cf}=\text { cumulative frequency of class immedi- } \\ & \text { ately preceding the median class } \end{aligned}$
$$\begin{aligned} w &=\text { width of median class } \\ f &=\text { frequency of median class } \\ L_{m} &=\text { lower boundary of median class } \end{aligned}$$
Using this formula, find the median for data in the frequency distribution of Exercise $16 .$

Key Concepts

Linear Interpolation

Linear interpolation is a technique used to estimate a value within two known values by assuming a linear distribution between them. In the context of grouped data, it is used to approximate the median within the median class by assuming that data values are uniformly distributed across the class interval.

Class Interval and Width

A class interval represents a range of data values in a grouped frequency distribution, and the class width is the extent or size of these intervals, calculated as the difference between the upper and lower boundaries. Both are essential in applying formulas to estimate measures like the median.

Median in Grouped Data

The median is the value that separates the lower half of the data from the upper half. In a grouped frequency distribution, because the exact data values are not known, the median is estimated by first identifying the median class—where the cumulative frequency equals or exceeds half the total frequency—and then using an estimation formula.

Grouped Frequency Distribution

A grouped frequency distribution organizes raw data into intervals (or classes) and records the number of observations (frequencies) within each interval. This method simplifies large data sets and enables easy identification of patterns and central tendencies.

Cumulative Frequency

Cumulative frequency is the running total of frequencies up to each class in a distribution. It is used to determine the median class by showing how the frequencies accumulate across the classes, allowing one to identify the point at which half of all observations have been accounted for.

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