How to Calculate the Median When You Have Even Observations
In statistics, the median of a group of observations is the value in the middle of the given data, where half of the data lies above it, and the other half lies below it. The group of observations or the data is sometimes grouped or ungrouped. The number of observations is also of two types; an odd and an even number of observations. Today we will learn about the median for even numbers of both grouped and ungrouped data.
Median for a Given Data
The Formula of the Median for Even Numbers
The median of the even number of observations is $(\dfrac{{(\dfrac{n}{2})th + (\dfrac{{n + 1}}{2})th}}{2})$, where n stands for the number of observations. If the number of observations is even, for example, 8, 12, 14, etc., then you have to use the above formula to find the median.
This is how you can calculate the value of the median for even numbers.
Questions Related to Median for Even Numbers
As you have learnt about the median formula for even numbers of data, here are some problems that will help you understand the formula better.
1. Find the median of the given data; “24, 33, 22, 30, 21, 25, 34, and 27"
Solution:
The number of observations in the given question is 8. Hence, you must use the median formula for an even number of data points.
The next step is to arrange the given data in ascending order. Hence, the order of the numbers will be 21, 22, 24, 25, 27, 30, 33, and 34.
The value of n is 8.
Following the formula for the median for an even number of observations, you can follow the steps given below.
$\begin{array}{l}\dfrac{{{{(\dfrac{n}{2})}^{th}}observation + {{(\dfrac{{n + 1}}{2})}^{th}}observation}}{2}\\ = \dfrac{{{{(\dfrac{8}{2})}^{th}}observation + {{(\frac{{8 + 1}}{2})}^{th}}observation}}{2}\\ = \dfrac{{{4^{th}}observation + {{(4 + 1)}^{th}}observation}}{2}\end{array}$
Since the values of the 4th observation and the (4+1)th observation, that is, the 5th observation, are 25 and 27, respectively, we will replace these values in the above equation.
$\begin{array}{l} = \dfrac{{25 + 27}}{2}\\ = \dfrac{{52}}{2}\\ = 26\end{array}$
Hence, the median of the given data is 26.
2. Find the median of the given data; “2, 3, 4, 5, 6, and 7”
Solution:
The number of observations in the given question is 6, which is an even number.
We will first arrange the given data in ascending order.
As the data is already grouped, we can proceed.
We will use the formula of the median for even numbers.
The value of n=6
$\begin{array}{l}\dfrac{{{{(\dfrac{n}{2})}^{th}}observation + {{(\dfrac{{n + 1}}{2})}^{th}}observation}}{2}\\ = \dfrac{{{{(\dfrac{6}{2})}^{th}}observation + {{(\dfrac{{6 + 1}}{2})}^{th}}observation}}{2}\\ = \dfrac{{{3^{rd}}observation + {{(3 + 1)}^{th}}observation}}{2}\end{array}$
Since the values of the 3rd observation and (3+1)th observation, i.e., the 4th observation, are 4 and 5, respectively, you have to replace these values in the above equation.
$\begin{array}{l} = \dfrac{{4 + 5}}{2}\\ = \dfrac{9}{2}\\ = 4.5\end{array}$
3. Find the median of the given data; “100, 80, 90, and 40”
Solution:
The number of observations in the given question is 4, which is even.
Hence, we will arrange the given data in ascending order. Hence, the order of the given data will be 40, 80, 90, and 100.
We will use the median for even numbers formula for the question.
Since n = 4, we will replace its value in the formula.
$\begin{array}{l}\dfrac{{{{(\dfrac{n}{2})}^{th}}observation + {{(\dfrac{{n + 1}}{2})}^{th}}observation}}{2}\\ = \dfrac{{{{(\frac{4}{2})}^{th}}observation + {{(\dfrac{{4 + 1}}{2})}^{th}}observation}}{2}\\ = \dfrac{{{2^{nd}}observation + {{(2 + 1)}^{rd}}observation}}{2}\end{array}$
Since the values of the 2nd and 3rd observations are 80 and 90, respectively, we will replace these values in the above equation.
$\begin{array}{l} = \dfrac{{80 + 90}}{2}\\ = \dfrac{{170}}{2}\\ = 85\end{array}$
Conclusion
So, today you have learnt about the median, its definition, and the two types of observation, which are odd number of observations and even number of observations. You have also learnt the formula to find out the value of the median for an even number of observations. By following these few steps, you can easily get the value of the median.
FAQs on Median for Even Number of Observations: Simple Guide
1. What is the formula for the median of an even number of observations?
The formula for finding the median when there is an even number of observations is to take the average of the two middle numbers after arranging the data in order. The formula is given by:
$$\text{Median} = \frac{\left(\text{n/2}^{\text{th}}\ \text{term}\right) + \left((\text{n/2} + 1)^{\text{th}}\ \text{term}\right)}{2}$$
- n is the total number of observations
- Arrange data in ascending (or descending) order before applying the formula
2. How to find the median of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10?
To find the median of the set 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (which has 10 numbers), use the median formula for even observations:
- First, arrange the numbers in order (already arranged).
- Here, n = 10 (an even number).
- The two middle terms are the 5th and 6th numbers: 5 and 6.
- Calculate the average: $$\text{Median} = \frac{5 + 6}{2} = 5.5$$
3. How do you calculate the median of a data set with an even number of observations that have been ordered from least to greatest?
To calculate the median of an ordered data set with an even number of observations:
- Identify the two numbers in the center. If there are n values, these are the (n/2)th and ((n/2) + 1)th numbers.
- Find their average: $$\text{Median} = \frac{\text{Middle Value 1} + \text{Middle Value 2}}{2}$$
- This technique provides the most accurate measure of central tendency for even-numbered data sets in statistics.
4. What are the steps to find the median in statistics for even data sets?
The steps to find the median for an even number of observations are:
- Arrange the data in ascending or descending order.
- Count the total number of observations (n).
- Locate the two middle terms at positions n/2 and (n/2) + 1.
- Calculate the average of these two numbers using: $$\text{Median} = \frac{\text{(n/2)th value} + \text{((n/2)+1)th value}}{2}$$
5. Can the median value be a non-integer when dealing with even numbers of data points?
Yes, the median can absolutely be a non-integer (decimal or fraction) when dealing with an even number of observations. This is because the median is calculated as the average of the two central values, which may not be whole numbers. For example, if the middle terms are 8 and 9, the median would be $$\frac{8 + 9}{2} = 8.5$$. At Vedantu, students learn to work confidently with all types of number results in mathematical statistics.
6. How is the median for an even number of observations different from the mean?
The median and mean are both measures of central tendency but differ in calculation and sensitivity to extreme values:
- Median: For even sets, it is the average of the two central numbers. The median is not influenced by outliers.
- Mean: Calculated by dividing the sum of all values by the number of observations; it can be skewed by extremely high or low values.
7. What happens to the median if the data set includes repeated values in the center?
If a data set with an even number includes repeated values at the center, the median is still calculated as the mean of those two middle terms—even if they are equal. For example, in the set {2, 4, 6, 6, 8, 10}, the two centermost values are both 6, so: $$\text{Median} = \frac{6 + 6}{2} = 6$$. Vedantu’s teachers help students interpret repeated data points confidently in real exam scenarios.
8. Why is ordering the data essential before finding the median in even-sized sets?
Ordering the data is essential before finding the median—even for even sets—because the median reflects the middle of the distribution. Arranging the numbers ensures accuracy when identifying the two central values, and avoids mistakes in statistical calculations. Vedantu strongly emphasizes data organization during problem-solving practice.
9. How are real-life problems solved using the median for even number of observations?
In real-life applications, calculating the median for even data sets helps in:
- Finding the typical household income in economics
- Determining the median score in class assessments
- Evaluating central values in surveys and research reports
