Locating the Median of Data, Whether Grouped or Ungrouped.

The science of statistics involves procuring, scrutinizing, interpreting, and presenting massive quantities of numerical data. Captain John Graunt, hailing from London, is recognized as the pioneer of essential statistics due to his extensive research on the statistics of births and deaths. Among all the significant topics in statistics, "median" stands out. It serves as a measure of central tendency by referring to the middle-most value of the observations in a particular set of data. In this write-up, we discuss the process of determining the median for both grouped and ungrouped data.

What is Median?

Median denotes the value that stands as the middle-most observation in a given set of data.

How to calculate Median step by step?

To calculate the median, one must follow the steps below.

For ungrouped data:

Step 1. Arrange the values of the set in ascending order.

Step 2. Determine the total number of observations present. It is designated as "n."

Step 3. If "n" is an odd number, the median is equivalent to the [(n 1)/2]th observation.

Step 4. If "n" is an even number, the median equals the average of the (n/2)th observation and [(n/2) 1]th observation.

For grouped data:

Step 1. Create a table with three columns. The first column is for the class interval, the second for the frequency "f," and the third for the cumulative frequency "cf."

Step 2. Note down the frequency and corresponding class interval in their respective columns.

Step 3. On the cf column, plot the cumulative frequency. This is done by adding the frequency in each step.

Step 4. Find the sum of frequencies (∑f); it equals the final number on the cumulative frequency column.

Step 5. Calculate n/2 and determine the class whose cumulative frequency is bigger than, and nearest to n/2. This identifies the median class.

Step 6. Finally, utilize the formula:

• \(\begin{array}{l}\text{Median} = l \left ( \frac{\frac{n}-cf}{f} \right )\times h\end{array} \)
• Where l stands for the lower limit of the median class, n refers to the number of observations, cf denotes the cumulative frequency of the class preceding the median class, f indicates the frequency of the median class, and h represents the class size (assuming classes are of equal size).

• Formula

• For ungrouped data:
• Median = [(n 1)/2]th observation, if "n" is odd.
• Median = the mean of (n/2)th observation and [(n/2) 1]th observation, if "n" is even.
• For grouped data:
• \(\begin{array}{l}\text{Median} = l \left ( \frac{\frac{n}-cf}{f} \right )\times h\end{array} \)

• Solved Example

Example: A set of 9 unique observations has a median value of 20.5. If the four most massive observations in the dataset undergo an increase of 2 each, what will be the median value of the new set?

(A) Increased by 2

(B) Decreased by 2

(C) Twice the original median

(D) Remains the same as the original set

Solution:

Given n = 9; Median = 20.5; Median term = [(n 1)/2]th term = [(9 1)/2]th term

The 5th term in a set of data remains unchanged even if the largest 4 observations are increased by 2. This is because the median of the set is the 5th term. Therefore, the answer to the problem is option (D).

Additional Resources:

Properties Of Median

What is a Mean and Median?

Statistics Problems

Comparing Mean, Median, and Mode

Common Questions

How Do You Define the Median in Statistics?

The median is the value in the middle of a set of data.

How Can You Find the Median of a Set of Ungrouped Data?

To find the median of ungrouped data, arrange the data from smallest to largest. If the set is odd, the median is the middle number. If it is even, the median is the average of the two middle numbers.

What is the Formula for the Median of Grouped Data?

The formula for the median of grouped data is median = [l + ((n/2) – cf)/f)h], where cf is the cumulative frequency, l is the lower limit of median class, n is the number of observations, f is the frequency of median class, h is the class size (assuming equal size classes).

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