How to Find the Median of an Even Number of 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 an Even Number of Observations
1. What is the definition of a median in statistics?
In statistics, the median is the value that separates the higher half from the lower half of a data sample. To find it, the data must first be arranged in ascending or descending order. The median is considered a measure of central tendency and is particularly useful because it is not as affected by unusually large or small values (outliers) as the mean is.
2. How do you find the median for a dataset with an even number of observations?
To find the median when there is an even number of observations (n), you need to follow these steps:
- Step 1: Arrange all the observations in the dataset in ascending (smallest to largest) order.
- Step 2: Identify the two middle observations. These are the (n/2)th term and the ((n/2) + 1)th term.
- Step 3: Calculate the average (mean) of these two middle observations. This average is the median of the dataset.
3. What is the specific formula to calculate the median when 'n' (the number of observations) is even?
When the total number of observations, 'n', is an even number, the formula to calculate the median is:
Median = [ (n/2)th observation + ((n/2) + 1)th observation ] / 2
This formula essentially directs you to find the average of the two centermost values after the data has been sorted.
4. Can you provide an example of calculating the median for the even dataset: 4, 12, 8, 2, 10, 6?
Certainly. Let's calculate the median for the dataset: 4, 12, 8, 2, 10, 6.
- First, we arrange the data in ascending order: 2, 4, 6, 8, 10, 12.
- Next, we count the observations. Here, n = 6, which is an even number.
- Then, we find the two middle terms. These are the (6/2)rd = 3rd term and the ((6/2) + 1)th = 4th term.
- The 3rd term is 6 and the 4th term is 8.
- Finally, we find the average of these two terms: (6 + 8) / 2 = 14 / 2 = 7.
Therefore, the median of this dataset is 7.
5. Why is it necessary to take the average of two middle values for an even number of observations?
For a dataset with an even number of observations, there is no single middle value that can split the data into two equal halves. Instead, there is a pair of central values. Taking the average of these two central values provides a single point that best represents the center of the entire dataset. This calculated average acts as the true midpoint, ensuring that half of the data values lie below it and the other half lie above it.
6. What is the key difference between finding the median for an even versus an odd number of observations?
The key difference lies in how the middle value is identified after sorting the data:
- For an odd number of observations (n), the median is a single, distinct middle value, which is the ((n+1)/2)th observation in the sorted list.
- For an even number of observations (n), there are two middle values. The median is not a direct value from the dataset but is the average of the (n/2)th and ((n/2) + 1)th observations.
7. Does the median for an even number of observations always have to be one of the numbers from the dataset?
No, the median for a dataset with an even number of observations does not have to be a number from the original dataset. Since the median is the average of the two middle values, it can often be a decimal or a whole number that falls between those two values. For instance, in the dataset {2, 4, 6, 8, 10, 12}, the median is 7, which is not present in the original data.
8. How do extreme values (outliers) affect the median of an even dataset compared to the mean?
The median is highly resistant to outliers. In an even dataset, the median is calculated from the two central values only. An extreme high or low value (outlier) does not change which values are in the middle positions. The mean, however, is calculated using every value in the dataset. Therefore, a single outlier can significantly pull the mean up or down, making it a less reliable measure of center for skewed data. The median remains a more stable or 'robust' measure in such cases.
