Rules for Finding the Median in Any Data Set

📚 What is the Median?

The median is the middle value in a data set when the data is ordered from least to greatest. It's a measure of central tendency that is less affected by outliers than the mean (average). Finding the median depends slightly on whether you have an odd or even number of data points.

Let's explore how to find the median in different scenarios.

🔢 Median for Ungrouped Data

Ungrouped data is simply a list of individual data points. Here's how to find the median:

• 📍Step 1: Order the Data: Arrange the data in ascending order (from smallest to largest).
• 🔑Step 2: Identify the Middle Value:
  • If there is an odd number of data points, the median is the middle value. For example, in the dataset 2, 4, 6, 8, 10, the median is 6.
  • If there is an even number of data points, the median is the average of the two middle values. For example, in the dataset 2, 4, 6, 8, the median is (4+6)/2 = 5.

📊 Median for Frequency Tables

Frequency tables show how often each value occurs in a dataset. To find the median:

• 🧮Step 1: Calculate Cumulative Frequencies: Add up the frequencies as you go down the table. The last cumulative frequency should equal the total number of data points ($n$).
• 🔍Step 2: Find the Median Position: Calculate the median position using the formula: $(\frac{n+1}{2})$.
• 📈Step 3: Identify the Median Value: Find the value in the table where the cumulative frequency is first greater than or equal to the median position. That value is the median.

Example:

Total frequency ($n$) = 31. Median position = $(\frac{31+1}{2}) = 16$. The cumulative frequency first greater than or equal to 16 is 25, which corresponds to a value of 3. Therefore, the median is 3.

➗ Median for Grouped Data

Grouped data is presented in intervals or classes. The median is estimated using the following formula:

Median = $L + [(\frac{\frac{n}{2} - CF}{f}) * w]$

• 📍 $L$ = Lower boundary of the median class (the class containing the median)
• 🔑 $n$ = Total frequency
• 🧮 $CF$ = Cumulative frequency of the class before the median class
• 📈 $f$ = Frequency of the median class
• 📐 $w$ = Class width

Example:

Total frequency ($n$) = 28. Median position = $\frac{28}{2} = 14$. The median class is 30-40 (since the cumulative frequency reaches 22, exceeding 14). Therefore: $L = 30$, $CF = 12$, $f = 10$, $w = 10$. Median = $30 + [(\frac{14 - 12}{10}) * 10] = 30 + 2 = 32$.

💡Tips and Tricks

• ✔️ Double-check that your data is ordered correctly, especially for ungrouped data.
• ➕ For frequency tables, be careful when calculating cumulative frequencies.
• 📝 For grouped data, accurately identify the median class and its boundaries.