Mean, median, and mode are the three measures of central tendency used in statistics to summarise a dataset. Mean is the arithmetic average. Median is the middle value when data is arranged in order. Mode is the value that appears most frequently. Each has a separate formula for ungrouped and grouped data, and all three are connected through the empirical relationship: 3 Median = Mode + 2 Mean.
The mean median mode formula is one of the highest-scoring topics in CBSE Class 10 Statistics. Board exams ask students to calculate all three measures from frequency tables, apply the empirical formula to find a missing value, and identify which measure suits a given situation. This page covers every formula you need: ungrouped and grouped data, the empirical relation with solved examples, the full variable key for each formula, and a Class 10 board exam question guide.
Key Takeaways
| Measure |
Ungrouped Formula |
Grouped Formula |
CBSE Class |
| Mean |
x̄ = Σx / n |
x̄ = Σfx / Σf |
Class 9, 10 |
| Median |
((n+1)/2)th term (odd n) |
L + ((n/2 − CF) / f) × h |
Class 9, 10 |
| Mode |
Most frequent value |
L + ((fm − f1) / (2fm − f1 − f2)) × h |
Class 10 |
| Empirical Relation |
3 Median = Mode + 2 Mean |
Same formula applies |
Class 10 |
Mean Formula: Ungrouped and Grouped Data
The mean is the most widely used measure of central tendency. It gives the average value of a dataset and works best when no extreme values distort the data.
Mean Formula for Ungrouped Data
x̄ = Σx / n
Where:
- Σx = sum of all observations
- n = total number of observations
Example: A cricketer scores 12, 34, 45, 50, and 24 in five ODI matches. Mean = (12 + 34 + 45 + 50 + 24) / 5 = 165 / 5 = 33
Mean Formula for Grouped Data
x̄ = Σfx / Σf
Where:
- f = frequency of each class
- x = midpoint of each class interval
- Σf = total frequency (equal to n)
- fᵢ = frequency of the i-th class
- xᵢ = midpoint of the i-th class
Median Formula: Ungrouped and Grouped Data
The median formula for Class 10 is one of the most frequently tested formulas in CBSE Statistics. Median gives the middle value after arranging data in order. It does not get pulled by outliers the way mean does.
Median Formula for Ungrouped Data
First arrange all values in ascending or descending order.
- If n is odd: Median = ((n + 1) / 2)th term
- If n is even: Median = average of (n/2)th term and ((n/2) + 1)th term
Example: Dataset: {5, 3, 8, 7, 2, 4, 6}, Ordered: {2, 3, 4, 5, 6, 7, 8} n = 7 (odd), so Median = ((7+1)/2)th = 4th term = 5
Median Formula for Grouped Data
Median = L + ((n/2 − CF) / f) × h
Where:
- L = lower boundary of the median class
- n = total number of observations
- CF = cumulative frequency of the class before the median class
- f = frequency of the median class
- h = class width
The median class is the class in which the (n/2)th observation falls.
Mode Formula: Ungrouped and Grouped Data
The mode formula for Class 10 grouped data is the formula students most frequently search for before board exams. For ungrouped data, mode is simply the value that appears most often and no calculation is needed.
Mode Formula for Ungrouped Data
Scan the dataset and identify the value with the highest frequency. If two values tie, the dataset is bimodal. If three or more values tie, it is multimodal. If no value repeats, the dataset has no mode.
Example: Shoe sizes: {7, 8, 9, 7, 8, 7, 10, 7, 9} Size 7 appears 4 times, which is the most frequent. Mode = 7
Mode Formula for Grouped Data
Mode = L + ((fm − f1) / (2fm − f1 − f2)) × h
Where:
- L = lower boundary of the modal class
- fm = frequency of the modal class (highest frequency)
- f1 = frequency of the class before the modal class
- f2 = frequency of the class after the modal class
- h = class width
The modal class is the class interval with the highest frequency. The mode formula class 10 appears in almost every CBSE board paper. Memorise the variable key alongside the formula. Boards often test whether students can correctly identify f1 and f2.
Relation Between Mean Median and Mode
The empirical formula for mean median mode connects all three measures. It is a direct board exam formula tested in Class 10 and appears in both 2-mark and 3-mark questions.
3 Median = Mode + 2 Mean
This can also be written as:
- Mode = 3 Median − 2 Mean
- Mean = (3 Median − Mode) / 2
- Median = (Mode + 2 Mean) / 3
This formula is very practical. If you know any two of the three measures, you can calculate the third.
Forward Example: Find Mode, given Mean and Median
Mean = 26, Median = 28. Find Mode. Mode = 3 Median − 2 Mean = 3(28) − 2(26) = 84 − 52 = 32
Backward Example: Find Mean, given Mode and Median
Mode = 50, Median = 46.5. Find Mean. 2 Mean + Mode = 3 Median 2 Mean + 50 = 3 × 46.5 2 Mean + 50 = 139.5 2 Mean = 89.5 Mean = 44.75
Both question types appear in CBSE Class 10 board papers. Practise both directions.
When to Use Mean Median or Mode
Students often lose marks choosing the wrong measure in word problems. The choice depends on the data type and distribution.
| Measure |
Best Used When |
Example Situation |
| Mean |
Data is evenly distributed, no extreme outliers |
Average score in a class test |
| Median |
Data is skewed or has extreme values |
Average income in a city with a few very high earners |
| Mode |
Data is categorical or you need the most common value |
Most popular shoe size, most sold product |
A quick rule: if the question says "most common" or "most frequent," use mode. If it says "middle value," use median. If it says "average," use mean.
Measures of Central Tendency Formula: Solved Examples
These solved examples cover all three measures of central tendency formula types tested in the 2026 CBSE board exam.
Example 1: Mean (Ungrouped Data)
Find the mean of: {22, 25, 30, 28, 24, 26, 27} Sum = 22 + 25 + 30 + 28 + 24 + 26 + 27 = 182 n = 7 Mean = 182 / 7 = 26
Example 2: Median (Ungrouped Data)
Find the median of: {5, 3, 8, 7, 2, 4, 6} Ordered: {2, 3, 4, 5, 6, 7, 8} n = 7 (odd) Median = ((7+1)/2)th = 4th term = 5
Example 3: Mode (Ungrouped Data)
Find the mode of shoe sizes: {7, 8, 9, 7, 8, 7, 10, 7, 9} Frequency: 7 appears 4 times, 8 appears 2 times, 9 appears 2 times, 10 appears 1 time. Mode = 7
Example 4: All Three from One Dataset
Dataset: 17, 22, 17, 18, 17, 20, 18, 25, 17
Mean: Sum = 171, n = 9, Mean = 171 / 9 = 19
Median: Ordered: 17, 17, 17, 17, 18, 18, 20, 22, 25 n = 9 (odd), Median = 5th term = 18
Mode: 17 appears 4 times. Mode = 17
Example 5: Empirical Relation
The mode is 65 and the median is 61.6. Find the mean. 2 Mean = 3 Median − Mode 2 Mean = 3(61.6) − 65 = 184.8 − 65 = 119.8 Mean = 59.9
Mean Median Mode Formula Class 10: Board Exam Question Types
Class 10 Statistics carries significant marks in the 2026 CBSE board paper. These are the question types that appear most often.
Find mean using grouped data: given a frequency table, calculate Σfx and divide by Σf.
Find median using grouped data: identify the median class, then apply L + ((n/2 − CF) / f) × h.
Find mode using grouped data: identify the modal class, then apply L + ((fm − f1) / (2fm − f1 − f2)) × h.
Use the empirical formula: given two measures, find the third using 3 Median = Mode + 2 Mean.
Find missing frequency: given mean or median and a frequency table with one unknown frequency, solve for it using the formula.
The empirical formula question and the missing frequency question are the two highest-probability 3-mark questions in the Statistics chapter every year.