Median – Definition, Formula, Properties & Examples

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The median, in statistics, is the middle value of the given list of data when arranged either in ascending or descending order. It is one of the representative figures along with the arithmetic mean and mode. These figures, i.e., median, arithmetic mean, and mode are also called measures of central tendencies. While the arithmetic mean is the ratio of the sum of all observations to the total number of observations, the median is the middlemost value in the list of data items.

Let’s understand how to find median class in statistics, what is median and how it is calculated along with its properties using examples.

What is Meant by Median in Statistics?

The median of a set of data is the middlemost number or centre value in the set. The median is also the number that is halfway into the set.

To find the median, the data should be arranged first in order of least to greatest or greatest to the least value. A median is a number that is separated by the higher half of a data sample, a population, or a probability distribution from the lower half. The median is different for different types of distribution.

Further, the calculation of the median depends on the number of data points. For an odd number of data, the median is the middlemost data, and for an even number of data, the median is the average of the two middle values.

Median for Ungrouped Data

To find the median of a given set of data, the first step is to arrange that set of data either in ascending or descending order. After that, count all the data items of that data set. 

• If there are odd numbers of data items in a data set, identify the middle value of that data set. That middle value or the data point in the middle will be the median. 
• If there are even numbers of data points in the given data set, identify the two data points in the middle, add those two and divide them by 2. This way we find the average of these two data items. This average value is the median.

Median for Odd Number of Observations

The steps followed to find the median of an odd number of observations in a data set are:

Step 1: Arrange the data either in ascending or descending order.

Step 2: Note down the count of data items $n$.

Step 3: Calculate $\frac{n + 1}{2}$.

Step 4: Locate $\left(\frac{n + 1}{2} \right)^{th}$ value in the data set.

Step 5: The value located in step 4 is the median.

Examples

Ex 1: Find the median of the data set $12$, $19$, $14$, $17$, $16$.

Data items after arranging them in ascending order: $12$, $14$, $16$, $17$, $19$

The number of observations in the data set is $5$ ($5$ is an odd number).

$n = 5$

$\frac{n + 1}{2} = \frac{5 + 1}{2} = \frac{6}{2} = 3$.

Median is $3^{rd}$ data item = $16$.

Alternatively,

Data items after arranging in descending order: $19$, $17$, $16$, $14$, $12$

The number of observations in the data set is $5$ ($5$ is an odd number).

$n = 5$

$\frac{n + 1}{2} = \frac{5 + 1}{2} = \frac{6}{2} = 3$.

Median is $3^{rd}$ data item = $16$.

Median for Even Number of Observations

The steps followed to find the median of an even number of observations in a data set are:

Step 1: Arrange the data either in ascending or descending order.

Step 2: Note down the count of data items $n$.

Step 3: Calculate $\frac{n}{2}$ and $\frac{n}{2} + 1$.

Step 4: Locate $\left(\frac{n}{2} \right)^{th}$ and $\left(\frac{n}{2} + 1\right)^{th}$ values in the data set. Let $\left(\frac{n}{2} \right)^{th}$ value be $m$ and $\left(\frac{n}{2} + 1\right)^{th}$ value be $n$.

Step 5: Compute $\frac{m + n}{2}$.

Step 6: The value computed in step 5 is the median.

Examples

Ex 1: Find the median of the data set $18$, $28$, $22$, $26$, $21$, $25$.

Data items after arranging in ascending order: $18$, $21$, $22$, $25$, $26$, $28$

The number of observations in the data set is $6$ ($6$ is an even number).

$n = 6$

$(\frac{n}{2} = \frac{6}{2} = 3$ and $\frac{n}{2} + 1 = \frac{6}{2} + 1 = 3 + 1 = 4$

$m = 3^{rd}$ value => $m = 22$

$n = 4^{th}$ value => $n = 25$

$\frac{m + n}{2} = \frac{22 + 25}{2} = \frac{47}{2} = 23.5$

Median = $23.5$.

Alternatively,

Data items after arranging in descending order: $28$, $26$, $25$, $22$, $21$, $18$

The number of observations in the data set is $6$ ($6$ is an even number).

$n = 6$

$(\frac{n}{2} = \frac{6}{2} = 3$ and $\frac{n}{2} + 1 = \frac{6}{2} + 1 = 3 + 1 = 4$

$m = 3^{rd}$ value => $m = 25$

$n = 4^{th}$ value => $n = 22$

$\frac{m + n}{2} = \frac{25 + 22}{2} = \frac{47}{2} = 23.5$

Median = $23.5$.

Median for Grouped Data

When data is present in the form of a frequency table, it is called grouped data. The process of calculating the median of the data involves finding the cumulative frequency and the median class. After that, the formula for computing the median is used. The formula used for calculating the median for grouped data is $\text{Median = } l + \left(\frac{\frac{n}{2} – cf}{f} \right) \times h$.

where, $l$ is the lower limit of the median class

$n$ is the total of frequencies

$cf$ is the cumulative frequency of the median class

$f$ is the frequency of a class preceding the median class

$h$ is the class width of the median class

How to Find Median Class in Statistics

The first step is to check whether the frequency distribution has an exclusive class interval or an inclusive class interval. 

Inclusive Class Interval

The inclusive class interval is used for discrete data. In an inclusive class interval, the lower limit of a class does not get repeated in the upper limit of the preceding class. The interval $\text{a} – \text{b}$ in inclusive class interval includes all values between $a$ and $b$ including both $a$ and $b$. In inequality form, it can also be represented as $a \le x \le b$.

For example, the inclusive class interval $5 – 10$ includes all the values from $5$ to $10$ including $5$ and $10$, i.e., the values included in the inclusive class interval $5 – 10$ include values $5$, $6$, $7$, $8$, $9$ and $10$. 

Note: Inclusive class interval includes only finite fixed values.

Exclusive Class Interval

The exclusive class interval is used for continuous data. In an exclusive class interval, the lower limit of a class is repeated in the upper limit of the preceding class. The interval $\text{a} – \text{b}$ in exclusive class interval includes all values between $a$ and $b$ including both $a$ but excluding $b$. In inequality form, it can also be represented as $a \le x \lt b$.

For example, the exclusive class interval $5 – 10$ includes all the values from $5$ to $10$ including $5$ but excluding $10$, i.e., the values included in the inclusive class interval $5 – 10$ include values $5$, $5.4$, $6$, $6.89$, $7$, $8$, $9$, $9.987$, $9.999$, and so on. 

Note: Exclusive class interval includes infinite values within the class interval.

Converting Inclusive Class Interval to Exclusive Class Interval

The steps followed to convert an inclusive class interval to an exclusive class interval are:

Step 1: Find the difference between the upper limit of a class interval and the lower limit of the next class interval.

Step 2: Divide the difference obtained in Step 1 by 2.

Step 3: Add the number obtained in Step 2 to the upper limit of each class interval.

Step 4: Subtract the number obtained in Step 2 from the lower limit of each class interval.

Step 5: The class intervals obtained are exclusive class intervals.

Examples

Ex 1: The following frequency distribution is in inclusive class interval form. Convert it to exclusive class interval form.

Select any class interval (Other than the first or last).

Selected class interval $30 – 39$.

Upper limit of $30 – 39$ is $39$.

Class interval next to $30 – 39$ is $40 – 49$.

Lower limit of $40 – 49$ is $40$.

Difference of $39$ and $40$ is $40 – 39 = 1$.

Dividing $1$ by $2$, we get $\frac{1}{2} = 0.5$.

Now, add $0.5$ to an upper limit of all class intervals and subtract $0.5$ from a lower limit of all class intervals.

$10 – 19$ becomes $10 – 0.5 – 19 + 0.5 = 9.5 – 19.5$

$20 – 29$ becomes $20 – 0.5 – 29 + 0.5 = 19.5 – 29.5$

$30 – 39$ becomes $30 – 0.5 – 39 + 0.5 = 29.5 – 39.5$

$40 – 49$ becomes $40 – 0.5 – 49 + 0.5 = 39.5 – 49.5$

$50 – 59$ becomes $50 – 0.5 – 59 + 0.5 = 49.5 – 59.5$

After setting the frequency distribution table in exclusive class interval form, find the cumulative frequency of each of the class intervals.

Computing Cumulative Frequencies

The steps followed to compute cumulative frequencies.

Step 1: Add one more column to the right of the Frequency column for Cumulative Frequency.

Step 2: Copy the frequency of the first interval to the cumulative frequency column.

Step 3: Move to the next class interval.

Step 4: Add the frequency of the current class interval to the cumulative frequency of the preceding class interval.

Step 5: The sum obtained in Step 4 is the cumulative frequency of the current class interval.

Step 6: Repeat Step 3 to Step 5, till the last class interval is reached.

Step 7: The table obtained is the Cumulative Frequency table.

Note: The cumulative frequency of the last class interval is always equal to the sum of all frequencies.

Now, the next step is to locate the median class.

To locate the median class follows the following steps.

Step 1: Find the total number of observations($n$), i.e., the sum of all the frequencies.

Step 2: In the cumulative frequency column identify the class interval in which the $\frac{n}{2}$ falls. 

Step 3: The class interval identified in Step 2 is the median class.

How to Find Median of Grouped Data?

After locating the median class, the following steps are used to find the median of the grouped frequency distribution.

Step 1: Find the lower limit $\left(l \right)$, and the frequency $\left(f \right)$ of the median class.

Step 2: Find the cumulative frequency $\left(c \right)$ of the class just preceding the median class. 

Step 3: Find the width $h$ of the median class using the formula $h = \text{Upper Limit} – \text{Lower Limit}$

Step 4: Apply the formula for median for grouped data: $\text{Median} = l + \left(\frac{\frac{n}{2} – c}{f} \right)\times h$.

Properties of Median

Following are the important properties of a median of a data set.

• The Median is the central value of data (Positional Average).
• Data has to be arranged in ascending/descending order to find the middle value or median.
• Not every value is considered while calculating the median.
• The Median doesn’t get affected by extreme points.

What is the Use of Median in Statistics

The median is less sensitive to skewed data and outliers than the mean. Extreme values pull the mean away from the center of the distribution, making it potentially misleading. It might not be near the most common values in the distribution.

For example, the mean is not a good statistic for summarizing annual income because that is a right-skewed distribution. A few highly affluent people can increase the mean dramatically, giving a misleading view of yearly incomes. For this type of data, the median is more accurate.

Points to Remember

Practice Problems

1. Find the median of the following set of data: $12$, $11$, $13$, $11$, $16$. 

2. Find the median of the following set of data: $12$, $18$, $24$, $18$, $11$, $20$, $29$, $41$, $20$. 

3. The time taken, in minutes, for a group of children to complete a puzzle is recorded.

Find the median time taken by the group of children to complete the puzzle. 

4. What is the median of the first $50$ natural numbers?

5. What is the median of the first $10$ prime numbers? 

6. The numbers $3$, $7$, $13$, $14$, $16$, $19$, $20$ and $x$ are arranged in ascending order. If the mean of the numbers is equal to the median, find the value of $x$.

7. The median of a set of eight numbers is $4.5$. Given that seven of the numbers are $7$, $2$, $13$, $4$, $8$, $2$, and $1$, find the eighth number. 

FAQs

Conclusion

The median of a set of data is the middlemost number or centre value in the set. In this article, we learned how to find median class in statistics and use it to find the median of a grouped frequency distribution.

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