Median Formula for Class 10: Examples and Steps

The median is the middle value after data has been arranged in order. For ungrouped data, use the middle observation (or average the two middle observations); for grouped data, the Class 10 formula is Median = l + [(N/2 − cf)/f] × h, where the symbols refer to the median class and its frequency information. The right method depends on how the data is presented.

What the median tells you

The median is a measure of central tendency: it gives a central point in a set of values. Half the observations lie at or below it, and half lie at or above it, allowing for ties. Unlike the mean, the median is not calculated by adding every value and dividing by the number of values. Instead, it depends on the position of values once they have been ordered.

This positional idea is useful when data includes an unusually high or low value. For example, a few very large incomes can pull the mean upward, while the median still identifies the value around the middle of the ordered list. The median does not describe every feature of a data set, but it can be a helpful summary when the distribution is uneven or contains extreme values.

Median formula for ungrouped data

Ungrouped data is a list of individual observations rather than a frequency table. First arrange the values from smallest to largest. If the number of observations, n, is odd, the median is the observation in position (n + 1)/2. If n is even, take the average of the observations in positions n/2 and (n/2) + 1.

Odd number of observations: Median = value of the [(n + 1)/2]th observation.
Even number of observations: Median = [value of the (n/2)th observation + value of the ((n/2) + 1)th observation] ÷ 2.

Positions are counted after sorting, starting at 1. Do not confuse the position number with the value in that position: if the median is in the fourth position, the median is the data value occupying that position, not automatically the number 4.

Example 1: an odd number of values

Find the median of 7, 3, 9, 5, 11, 2, 6. Arrange the observations in ascending order: 2, 3, 5, 6, 7, 9, 11. There are seven observations, so n = 7 and the middle position is (7 + 1)/2 = 4. The fourth value is 6, so the median is 6.

There are three values below 6 and three above it. This equal split is a quick way to check that the correct position was selected. Sorting in descending order would also work, provided the middle position is counted consistently; ascending order is usually less likely to cause mistakes.

Example 2: an even number of values

Find the median of 12, 8, 15, 10, 9, 14. In order, the values are 8, 9, 10, 12, 14, 15. Here n = 6, so the two middle positions are 6/2 = 3 and 3 + 1 = 4. Their values are 10 and 12. Therefore, the median is (10 + 12)/2 = 11.

The median does not have to be one of the original observations. In this example, 11 is between the two middle values, even though 11 does not appear in the list. Averaging the middle pair is the standard definition for an even-sized set; choosing only the lower or higher middle value would give a different result.

Median from a discrete frequency table

A discrete frequency table lists distinct values and how many times each occurs. To find its median, add the frequencies to get the total number of observations, then use cumulative frequency to locate the middle position or positions. Cumulative frequency is a running total of the frequencies from the smallest value upward.

For example, suppose the values 1, 2, 3, 4, and 5 have frequencies 2, 3, 5, 4, and 1. The total is N = 2 + 3 + 5 + 4 + 1 = 15. The median position is (15 + 1)/2 = 8. The cumulative frequencies are 2, 5, 10, 14, and 15. The eighth observation falls in the group with value 3, because the cumulative total rises from 5 to 10 there. The median is therefore 3.

For an even total, identify both middle positions and their corresponding values, then average those values. This approach works because the frequency table represents the same ordered observations as a list, but records repeated values compactly. Check that the final cumulative frequency equals N; otherwise, a frequency may have been omitted or added incorrectly.

Median formula for grouped data

Grouped data is presented in class intervals, such as 0–10, 10–20, and 20–30, with a frequency for each interval. Since the exact individual observations are not shown, the median is estimated using the median class and the grouped-data formula:

Median = l + [(N/2 − cf)/f] × h

l is the lower boundary of the median class.
N is the total frequency, found by adding all class frequencies.
cf is the cumulative frequency of the class immediately before the median class.
f is the frequency of the median class.
h is the width of the class interval.

To identify the median class, calculate N/2 and find the first cumulative frequency that is equal to or greater than N/2. The class corresponding to that cumulative frequency is the median class. The value of cf in the formula is the cumulative frequency before that class, not the cumulative frequency at the end of the median class.

Example 3: grouped-data calculation

Consider this grouped distribution: 0–10 has frequency 5; 10–20 has frequency 9; 20–30 has frequency 12; 30–40 has frequency 8; and 40–50 has frequency 6. The total frequency is N = 5 + 9 + 12 + 8 + 6 = 40, so N/2 = 20. The cumulative frequencies are 5, 14, 26, 34, and 40. Since 26 is the first cumulative frequency to reach or exceed 20, the median class is 20–30.

For that class, l = 20, cf = 14, f = 12, and h = 10. Substituting gives Median = 20 + [(20 − 14)/12] × 10 = 20 + 5 = 25. This is an estimate based on the grouped distribution, not necessarily the exact median of the original observations. The calculation assumes observations are spread evenly within the median class.

How to choose the right method

Look at the way the question presents the information before applying a formula. A raw list calls for sorting and locating the middle. A table of individual values with frequencies calls for cumulative frequency to locate the middle observation. Class intervals call for the grouped-data formula. These formats may all describe data, but their calculations are not interchangeable.

Read the table headings. Separate individual values from interval boundaries, and identify which column contains frequency.
Find the total. For a frequency table, add the frequencies to obtain N. Do not add the data values in place of the frequencies.
Locate the middle. Use the middle position for a list or discrete table; use N/2 to identify the median class in grouped data.
Apply the suitable calculation. Average two middle values when the count is even, or substitute the grouped values carefully into the formula.
Check whether the result makes sense. The median should lie within the data range and, for grouped data, within the median class.

Common errors and how to avoid them

A frequent mistake is using the grouped-data formula for a simple list. The formula with l, cf, f, and h is intended for grouped continuous intervals; it is unnecessary when every observation is available. Another common error is forgetting to sort ungrouped values. The original order has no bearing on which value is in the middle.

Using the wrong middle position: For an odd count, use (n + 1)/2. For an even count, use both n/2 and n/2 + 1.
Confusing frequency with cumulative frequency: In the grouped formula, f is the frequency of the median class, while cf is the cumulative frequency before it.
Choosing the wrong median class: Locate the first cumulative frequency that reaches or passes N/2. Do not choose a class merely because its frequency is largest.
Using a class width incorrectly: Find h by subtracting the lower boundary from the upper boundary. A class 20–30 has width 10, not 30.
Skipping the final check: A grouped estimate should fall in its median interval. If it does not, revisit the class selection and substitutions.

Pay attention to boundary conventions if the question uses intervals such as 10–19 and 20–29 for measurements recorded to whole numbers. Some textbooks first convert such classes to continuous boundaries, while other questions already provide continuous intervals such as 10–20 and 20–30. Follow the convention shown in the question or taught in the course, and use consistent boundaries when identifying l and h.

Median, mean, and mode: what is different?

The mean is the sum of all observations divided by their count. The median is the middle of ordered observations. The mode is the value that occurs most often. These measures answer different questions, so one is not universally better than the others. For example, in the values 2, 3, 3, 4, 28, the mean is 8, the median is 3, and the mode is 3. The high value 28 pulls the mean upward, while the median marks the central position.

A question asking for the median should not be answered with the mean simply because the mean formula is more familiar. Likewise, a median is not always the most frequent value: it may not occur in the data at all, as the even-list example with median 11 shows. In a symmetric data set, the measures may be close or even equal, but that is not a rule to rely on when solving a problem.

Frequently Asked Questions

What is the median formula for Class 10?
For an odd number of ungrouped observations, the median is the value at position (n + 1)/2; for an even number, average the values at positions n/2 and n/2 + 1. For grouped data, use Median = l + [(N/2 − cf)/f] × h.

How do I find the median if the number of values is even?
Sort the values, identify the two middle observations, and calculate their average. For example, if the middle values are 10 and 12, the median is (10 + 12)/2 = 11.

What does cf mean in the grouped median formula?
cf means the cumulative frequency of the class just before the median class. It is the number of observations accounted for before entering the median class, not that class’s own frequency.

How do I find the median class?
Add the frequencies to find N, calculate N/2, and read the cumulative-frequency column from the top. The first class whose cumulative frequency is at least N/2 is the median class.

Can the median be a number that is not in the data?
Yes. When there is an even number of individual observations, the median is the average of the two middle values and may fall between them. A grouped-data median is also generally an estimate within an interval.

Is the grouped median exact?
Usually not. Grouped data does not reveal the individual values within each interval, so the formula estimates the median by assuming observations are distributed evenly across the median class.

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