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    How to Find Median for Grouped and Ungrouped Data

    Understand how to find median for grouped and ungrouped data in Statistics. The step by step solutions help students to learn to find the median for grouped and ungrouped data.

    JEEIIT JEE Study MaterialHow to Find Median for Grouped and Ungrouped Data

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    How to Find Median for Grouped and Ungrouped Data

    Statistics refers to the collection, analysis, interpretation and presentation of masses of numerical data. Captain John Graunt of London is known as the father of vital statistics due to his studies on statistics of births and deaths. Median is one of the most important topics in statistics. The median is a measure of central tendency, which denotes the value of the middle-most observation in the data. In this article, we will discuss how to find the median for grouped and ungrouped data.

    What is Median?

    Median is the most middle value in a given set of data.

    How to calculate Median step by step?

    You can use the following steps to calculate the median.

    For ungrouped data:

    Step 1. Arrange the given values in ascending order.

    Step 2. Find the number of observations in the given set of data. It is denoted by n.

    Step 3. If n is odd, the median equals the [(n+1)/2]th observation.

    Step 4. If n is even, then the median is given by the mean of (n/2)th observation and [(n/2)+1]th observation.

    For grouped data:

    Step 1. Make a table with 3 columns. First column for the class interval, the second column for frequency, f, and the third column for cumulative frequency, cf.

    Step 2. Write the class intervals and the corresponding frequency in the respective columns.

    Step 3. Write the cumulative frequency in column cf. It is done by adding the frequency in each step.

    Step 4. Find the sum of frequencies, ∑f. It will be the same as the last number in the cumulative frequency column.

    Step 5. Find n/2. Then find the class whose cumulative frequency is greater than and nearest to n/2. This is the median class.

    Step 6. Now we use the formula

    Median=l+(n2−cff)×h

    l = lower limit of the median class

    n = no. of observations

    cf denotes the cumulative frequency of the class preceding the median class

    f = frequency of median class

    h = class size (assuming classes are of equal size)

    Formula

    For ungrouped data:

    Median = [(n+1)/2]th observation, if n is odd.

    Median = mean of (n/2)th observation and [(n/2)+1]th observation, if n is even.

    For grouped data:

    Median=l+(n2−cff)×h

    Solved Example

    Example: The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set

    (A) Is increased by 2

    (B) Is decreased by 2

    (C) Is two times the original median

    (D) Remains the same as that of the original set

    Solution:

    Given n = 9 Median = 20.5

    Median term = [(n+1)/2]th term

    = [(9+1)/2]th term = 10/2)th term = 5th term

    Given the largest 4 observations are increased by 2. Since the median is the 5th term, there will be no change in it.

    Hence option (D) is the answer.

    Related Links:

    Properties Of Median

    What is a Mean and Median?

    Statistics Problems

    Mean, Median and Mode

    3,751

    Frequently Asked Questions

    What do you mean by median in Statistics?

    The middlemost value in a given set of data is called the median.

    How to find the median of ungrouped data?

    We have to arrange the set of data in ascending order. Denote the number of observations in the given set of data by n. If n is odd, the median is the [(n+1)/2]th observation. If n is even, then the median is given by the mean of (n/2)th observation and [(n/2)+1]th observation.

    Give the formula for the median of grouped data.

    The median for grouped data is given by the equation, median = [l + ((n/2) – cf)/f)h ], where cf is the cumulative frequency, l is the lower limit of median class, n is the number of observations, f is the frequency of median class, h is the class size (assuming equal size classes).

    Test your Knowledge on Median

    Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

    Select the correct answer and click on the “Finish” button

    Check your score and answers at the end of the quiz

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    Write the formula to find the median of a grouped data and explain the alphabet in it.

    Click here👆to get an answer to your question ✍️ Write the formula to find the median of a grouped data and explain the alphabet in it.

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    Write the formula to find the median of a grouped data and explain the alphabet in it.

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    Updated on : 2022-09-05

    For grouped data-

    Solution Verified by Toppr Median=l+ f 1 ​ 2 N ​ −f 0 ​ ​ ×h Whereas

    l= Lower limit of edian class

    N= Total frequency f 0 ​

    = Frequancy of class preceding the median class

    f 1 ​

    = Frequency of median class

    h= Width of median class

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    स्रोत : www.toppr.com

    Write the formula to find the median for grouped data and explain each term.

    Write the formula to find the median for grouped data and explain each term.. Ans: Hint: Median of a grouped data is given by \\[median = l + \\left( {\\dfrac{{\\dfrac{n}{2} - cf}}{f}} \\right) \\times h\\]. This formula of median for grouped data is...

    Write the formula to find the median for grouped data and explain each term.

    Answer Verified 179.1k+ views 6 likes

    Hint: Median of a grouped data is given by

    median=l+ ⎛ ⎝ ⎜ n 2 −cf f ⎞ ⎠ ⎟ ×h median=l+(n2−cff)×h

    . This formula of median for grouped data is similar to the formula of mode in an interval. Here,

    l l

    represent the lower limit of the median class,

    n n

    represents the total sum of all frequencies,

    cf cf

    represent the cumulative frequency before the median class,

    f f

    represent the frequency of the median class and

    h h

    represents the class width.

    Complete step-by-step answer:

    The formula for median in a grouped data is

    median=l+ ⎛ ⎝ ⎜ n 2 −cf f ⎞ ⎠ ⎟ ×h median=l+(n2−cff)×h , where l l

    represent the lower limit of the median class,

    n n

    represents the total sum of all frequencies,

    cf cf

    represent the cumulative frequency before the median class,

    f f

    represent the frequency of the median class and

    h h

    represents the class width.

    Median of a data set represents the middle value, when the data is arranged in ascending order. It generally helps in determining the most likely value of the data set, but only if the distribution of data is continuous. The value of the median should have the highest frequency for a symmetrical distribution of data. However, if the distribution of the data is unsymmetrical or skewed, median doesn’t represent the most likely value as there will be an extremely high or an extremely low value as compared to the rest of data.

    Note: The median of a given set of discrete data can be found out by simply arranging it into ascending order and then picking out the middle value. However for a grouped data, we have to use the formula of median used above. It is best for a student to be well versed with these formulae.

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