The median is the middle value in a set of numbers. It is used to show the central point of the data. When the numbers are arranged in order from smallest to largest, the median splits the data into two equal halves. This means that 50% of the values are less than or equal to the median, and the other 50% are greater than or equal to it.

The median is especially helpful when we want to find the typical value in a set, especially if the data contains very high or low values (called outliers) that could affect the average (mean).

 There are different ways to calculate the median based on the type of data:

• For ungrouped data (a simple list of numbers), we sort the numbers and pick the middle one.
• For grouped data (data shown in intervals), we use a formula to find the median.
• For discrete data, we look at the frequency of each number.

How to Find Median

Before you can find the median, it's important to understand what it means. The median is the middle value in a group of numbers that are arranged from smallest to largest. It divides the data into two equal parts — half the values are less than or equal to the median, and half are greater than or equal to it.

The median helps us find the center of the data, especially when the data has very small or very large values that might make the average (mean) misleading. For example, in a set like 2, 3, 4, 5, and 100, the mean would be high because of 100, but the median (which is 4) gives a better idea of the typical value.

The median is mostly used with quantitative data (numbers), but it can also be used with qualitative or ranked data (like ratings such as good, better, best) when the values can be ordered.

In short, the median is a helpful tool to find the center of your data, especially when the data is not evenly spread out (skewed). It is a reliable measure to understand the middle point of your dataset.

Check out this article on Comparison of Quantity.

To determine the median value in a sequence of numbers, the numbers must first be arranged in ascending order.

• If there is an odd amount of numbers, the median value is the number that is in the middle, with the same amount of numbers below and above.
• If there is an even amount of numbers in the list, the median is the average of the two middle values.

Grouped data is data that has been bundled together in categories. The steps for finding the median differ depending on whether you have an odd or an even number of data points. If there are two numbers in the middle of a dataset, their mean is the median.

Learn more about the Sum of Harmonic Progression here.

When the Observation is Odd

Here are the steps to find the median of grouped data with odd numbered datasets.

Step 1: Order the values from low to high.

Step 2: Calculate the middle position. Use the formula \(\frac{(n+1)}{2}\) where n is the number of values in your dataset.

Step 3: Find the value in the middle position.

Here’s an example to help you understand better. Consider the weekly expense of an average college student.

Step 1: Order the values from low to high.

Ordered dataset

Step 2: Calculate the middle position. Use the formula \(\frac{(n+1)}{2}\) where n is the number of values in your dataset. Here n = 5, thus, the median is at \(\frac{(5+1)}{2}\) = 3 i.e. at 3rd position.

Step 3: Find the value in the middle position.

Here the middle position is 3rd as we found out in the previous step.

Thus, the median is 350.

When the Observation is Even

In an even-numbered dataset, there isn’t a single value in the middle of the dataset, so we have to follow a slightly different procedure. Here are the steps to find the median of grouped data with even numbered datasets.

Step 1: Order the values from low to high.

Step 2: Calculate the two middle positions. The middle positions are found using the formulas \(\frac{n}{2}\) and \((\frac{n}{2}) + 1\) where n is the number of values in your dataset.

Step 3: Find the two middle values.

Step 4: Find the mean of the two middle values. To find the median, calculate the mean by adding together the middle values and dividing them by two. The mean value of the two middle values is the median of the dataset.

Here’s an example to help you understand better. Consider the weekly expense of an average college student.

Step 1: Order the values from low to high.

Ordered dataset

Step 2: Calculate the two middle positions. The middle positions are found using the formulas \(\frac{n}{2}\) and \((\frac{n}{2}) + 1\) where n is the number of values in your dataset. Here n = 6, thus, the two middle positions are \(\frac{n}{2}\) = 3rd and \((\frac{n}{2}) + 1\) = 4th.

Step 3: Find the two middle values.

Middle values

The middle values are 350 and 500.

Step 4: Find the mean of the two middle values.

To find the median, calculate the mean by adding together the middle values and dividing them by two.

Calculating the median

Median: \(\frac{(350+500)}{2}=425\)

Learn about the various harmonic mean formulas here.

How to Find Median of Grouped Data

Ungrouped data is the data you first gather from an experiment or study. The data is raw — that is, it’s not sorted into categories, classified, or otherwise grouped. An ungrouped set of data is basically a list of numbers.

The median for a specific observation in grouped data cannot be determined by examining cumulative frequencies. The data will fall into a class interval around its middle value. Thus the value inside the class interval that splits the entire distribution in half must be found. To solve this problem, we must determine the median class.

The cumulative frequencies of each class and n/2 must be determined in order to determine the median class. Locate the class that has a cumulative frequency that is bigger than (or closest to) n/2 next. The group is referred to as the median class.

Use the formula below to determine the median value after determining the median class.

Median = \(l+ \left ( \frac{\frac{n}{2}-m}{f} \right )\times{c}\)

l is the lower limit of the median class

n is the number of observations

f is the frequency of median class

c is the class size

m is the cumulative frequency of class preceding the median class.

Let’s understand this with the help of an example.

Example: The following are the marks scored by the students in the Summative Assessment exam. Calculate the median.

Solution:

Median class=(N/2)th value

= (50/2)th value

= 25th value

Median class=20 – 30

l = 20, N/2 = 25, m = 9, f = 15 and c = 10

Median = \(l+ \left ( \frac{\frac{n}{2}-m}{f} \right )\times{c}\)

Substitute.

Median = \(20 + (\frac{25 – 9}{15})\times10\)

\(= 20 + \frac{16}{15}\times10\)

= 20 + 10.6

= 30.6

≈ 31

How to Find Median of Discrete Data

The median of a distribution with a discrete random variable depends on whether the number of terms in the distribution is even or odd. The median is the value of the term in the centre when the number of terms is odd. This value is chosen so that the number of terms with values higher than or equal to it and the number of terms with values lower than or equal to it are the same.

In the event that there are an even number of terms, the median is the average of the two terms in the middle, with the result that the number of terms with values more than or equal to it is the same as the number of terms with values less than or equal to it.

The median of a distribution with a continuous random variable is the value m such that the probability is at least 1/2 (50%) that a randomly chosen point on the function will be less than or equal to m, and the probability is at least 1/2 that a randomly chosen point on the function will be greater than or equal to m.

Check out the three types of mean arithmetic mean, geometric mean and harmonic mean.

How to Find Median From A Frequency Table

Median from a frequency table is when we find the median average from a data set which has been organised into a frequency table.

Follow these steps below to find the median from a frequency table.

Step 1: Put the results in numerical order (in a frequency table this will already be done).

Step 2: Count the total amount of results and add one

Step 3: Divide this by 2 to find the the position of the middle result

Step 4: Find the middle result in the numerically ordered list or frequency table

Step 5: You will then have the median of the set of results

How to Find Median Formula

Let’s summarize our learnings so far on how to find the median in terms of three simple formulae that you must apply whenever you need to find the median of any data.

Median of Ungrouped Odd Data

The median is the value of \(\frac{(n+1)}{2}\) term where n is the number of terms in the given data set.

Median of Ungrouped Even Data

Median is given by the following formula:

Median = \(\frac{[(\frac{n}{2})^{th}\text{ term } + [(\frac{n}{2})+1]^{th}\text{ term }]}{2}\)

Median of Grouped Data

Median = \(l+ \left ( \frac{\frac{n}{2}-m}{f} \right )\times{c}\)

Summary

Here are some key points for you to remember:

• The Mean, Median and Mode are measures of central tendency within a distribution of numerical values. The mean is more commonly known as the average. The median is a center value. The mode is the most frequent value.
• The middle value in a sorted list of numbers is called the median, and it can be used to describe a data collection more accurately than the average.
• The median is the same as the 50th percentile for the set of numbers. In other words, the median is the middle of a set of numbers with half of the values less than the median and half the values greater than the median.
• When there are outliers in the series that could affect the average of the numbers, the median is occasionally utilised instead of the mean.
• The number in the middle, with the same number below and above it, when there are an odd number of integers, is the median value.
• The middle pair must be identified, added together, and divided by two in order to obtain the median value if the list has an even number of integers.
• The median, mean, and mode are all the same in a normal distribution.

Properties of Median 

The median is a way to find the middle value of a set of numbers. It helps us understand where the center of the data lies. Here are some easy-to-understand points about the median:

​1. Middle Value: The median shows the middle number when all values are arranged in order.

2. Odd or Even Data:

• If there is an odd number of values, the middle number is the median.
• If there is an even number of values, the median is the average of the two middle numbers.

3. Not Affected by Outliers: The median doesn’t change much even if there are very big or very small values (called outliers) in the data.

4. No Need for Normal Order: Unlike the average (mean), the median works well even if the data is not evenly spread.

5. Can Be Used for Different Data Types: The median is useful for both numbers and categories (like ranks or labels), not just for numerical data.

Important Notes on Median 

Here are some simple and important points to help you understand the median better:

• The median is the middle value in a set of numbers. It is also called a positional average because it depends on the position of numbers, not their total.
• To find the median, you must first arrange the data in order — either from smallest to largest (ascending) or from largest to smallest (descending).
• While calculating the median, you don’t need to use every value in the same way as you do in the mean. You only focus on finding the value that sits in the center of the list.
• One big advantage of the median is that it is not affected by extreme values (very high or very low numbers). For example, in the data 5, 6, 7, 8, 100, the number 100 doesn’t change the median as much as it would change the mean.

Solved Examples on How to Find Median

Here are some solved examples on how to find the median.

Solved Example 1: Find the median for the following data set:102, 56, 34, 99, 89, 101, 10, 54.

Solution:

Here n = 8

Step 1: Order the values from low to high.

Ordered dataset

10, 34, 54, 56, 89, 99, 101, 102.

Step 2: Calculate the two middle positions. The middle positions are found using the formulas \(\frac{n}{2}\) and \((\frac{n}{2}) + 1\) where n is the number of values in your dataset. Here n = 8, thus, the two middle positions are \(\frac{n}{2}\) = 4th and \((\frac{n}{2}) + 1\) = 5th.

Step 3: Find the two middle values.

The middle values are 56 and 89.

Step 4: Find the mean of the two middle values.

To find the median, calculate the mean by adding together the middle values and dividing them by two.

Calculating the median

Median: \(\frac{(56+89)}{2}=72.5\)

Solved Example 2: The following table gives the weekly stipend of 200 students in a college. Find the median of the weekly stipend.

Solution:

Median class=(N/2)th value

= (200/2)th value

= 100th value

Median class=2000 – 3000

l = 2000, n/2 = 100, m = 74, f = 54 and c = 1000

Median = \(l+ \left ( \frac{\frac{n}{2}-m}{f} \right )\times{c}\)

Substitute.

Median = \(2000 + (\frac{100 – 74}{15})\times1000\)

\(= 2000 + \frac{26}{54}\times1000\)

= 2000 + 481.5

= 2481.5

Example 3: Let’s say the ages of a group of students are:
{21, 18, 25, 30, 22, 27, 19}
We will find the median of this data step by step.

Step 1: Arrange the numbers in ascending order (from smallest to largest):
{18, 19, 21, 22, 25, 27, 30}

Step 2: Count how many numbers there are.
There are 7 values, so n = 7.

Step 3: Use the formula for the median when the number of values (n) is odd:
Median = (n + 1) / 2
= (7 + 1) / 2 = 4

Step 4: The 4th value in the arranged list is 22.

The median of the set {21, 18, 25, 30, 22, 27, 19} is 22.

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