Data is often available in raw form and is organized using frequency distribution tables. For this, data is organized into classes and such data is called grouped data. The median of grouped data is found with the help of a cumulative frequency table. It is easy to find the median of ungrouped data by simply finding the middle term in the data set after arranging it in ascending or descending order. It can further be divided into continuous data and discontinuous data depending on the limits of the classes. If the upper limit of the first interval is equal to the lower limit of the second interval then it is called continuous data.

Example:

If the upper limit of the first interval is not equal to the lower limit of the second interval, then it is called discontinuous data.

Example:

Median is said to be the value of central tendency along with mean and mode for a given data set. It gives the middlemost value of the frequency distribution. Finding the median of ungrouped data is an easier task and can be done by arranging the values in ascending order and finding the middle term. If the number of terms is odd then the median is the (n+1)/2 th observation.

What is Median of Grouped Data?

We cannot find the median of grouped data just by checking the cumulative frequencies. The median is the middle value, and in grouped data, it lies inside one of the class intervals—not directly on a number. So, we must find the class interval that splits the total number of values into two equal parts. This special class is called the median class. Once we identify the median class, we use a formula to calculate the exact median value. This helps us get a more accurate answer, as the actual middle value is estimated from the data range.

Median of grouped data can be found using the formula:

Median = \( l\ + \left(\frac{\frac{N}{2}-C}{f}\right) \times h \)

where

l = Lower limit of the median class

f = Frequency of the median class

h = Size of the median class

C = Cumulative frequency of the class just before the median class

Median of Grouped Data Formula

The median is a measure of central tendency that represents the value separating the lower and upper halves of a dataset. When the data is in grouped form, i.e., divided into intervals, the calculation of the median is slightly different from that of ungrouped data.

The formula to calculate the median of grouped data is based on the assumption that the data is continuous and divided into intervals of equal width. The formula uses the cumulative frequency and frequency of the median class, as well as the width of the class interval, to determine the exact value of the median.

The median of grouped data can be calculated using the following formula:

\(Median = l+\left[\frac{(\frac{n}{2}-c)}{f}\right]\times h\).

Here, 

• l is the lower boundary of the median class.
• n is the total number of observations in the data set.
• c is the cumulative frequency up to the lower boundary of the median class.
• f is the frequency of the median class.
• h is the width of the median class interval.

How to Find the Median of Grouped Data

We can find the median of grouped data by following the below steps:

Step 1: Prepare the cumulative frequency column and obtain N = Σf and find N\2
Step 2: Look for the cumulative frequency just greater than N/2 and determine the corresponding class. This is the median class.
Step 3: Use the following formula to calculate the median:

Median = \( l\ + \left (\frac{\frac{N}{2}-C}{f}\right) \times h \)

Mean, Median, and Mode of Grouped Data

When we have grouped data (data arranged in intervals), we can use formulas to find the mean, median, and mode — three common measures of central tendency.

1. Mean (Average)

The mean is the average of all values in the dataset.

Formula:
Mean = ∑(fi × xi) / ∑fi

Where:

xi = mid-point of each class (average of lower and upper limits)

fi = frequency of the class

∑ = sum across all classes

2. Median (Middle Value)

The median is the middle value when data is arranged in order. For grouped data, we estimate it using this formula:

Formula:
Median = l + [(N/2 – cf) / f] × h

Where:

l = lower limit of the median class

N = total number of observations (N = ∑fi)

cf = cumulative frequency before the median class

f = frequency of the median class

h = class width (size of the class interval)

3. Mode (Most Frequent Value)

The mode is the value that appears most frequently. For grouped data, we use this formula:

Formula:
Mode = xk + h × [(fk – fk–1) / (2fk – fk–1 – fk+1)]

Where:

xk = lower limit of the modal class (the class with the highest frequency)

fk = frequency of the modal class

fk–1 = frequency of the class before modal class

fk+1 = frequency of the class after modal class

h = class width

Example 1: Calculate the median of the following distribution:

Solution:

We have, N = 49

Therefore, \( \frac{N}{2} = \frac{49}{2} = 24.5 \)

The cumulative frequency just greater than \( \frac{N}{2} \) is 26 and the corresponding class is 15 – 20.

Thus, 15 – 20 is the median class such that:

l = 15, f = 15, C = 11 and h = 5

Using the formula:

Median = \( l\ + \left (\frac{\frac{N}{2}-C}{f}\right) \times h \)

Median = \( 15\ + \left (\frac{24.5 – 11}{15}\right) \times 5 \)

Median = \( 15\ + \left(\frac{13.5}{3}\right) = 19.5 \)

Therefore, the median is 19.5

Example 2: Calculate the median of the following distribution:

Solution:

We have, N = 80

Therefore, \( \frac{N}{2} = \frac{80}{2} = 40 \)

The cumulative frequency just greater than \( \frac{N}{2} \) is 63 and the corresponding class is 40 – 60.

Thus, 40 – 60 is the median class such that:

l = 40, f = 37, C = 26 and h = 20

Using the formula:

Median = \( l\ + \left (\frac{\frac{N}{2}-C}{f}\right) \times h \)

Median = \( 40\ + \left (\frac{40 – 26}{37}\right) \times 20 \)

Median = \( 40\ + \left(\frac{280}{37}\right) = 47.56\)

Therefore, the median is 47.56.

Example 3: Anaya wrote down how many sandwiches she made each day for a week. The numbers were: 2, 5, 4, 3, 2, 6, 1

Solution: To find the median, we first put the numbers in order from smallest to biggest.

Original list:
2, 5, 4, 3, 2, 6, 1

Ordered list:
1, 2, 2, 3, 4, 5, 6

There are 7 numbers in total (which is an odd number), so the middle number is the 4th one.

So, the median = 3

The median number of sandwiches Anaya made is 3.

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