Mean Median Mode Formula

Mean, Median, Mode Formula (Grouped & Ungrouped)

Mastering statistics requires a clear understanding of the mean, median, and mode formulas. Whether you are dealing with a simple list of numbers (ungrouped data) or large frequency tables (grouped data) in CBSE Class 9 and 10, this guide provides the ultimate master table, simple explanations, their differences, specific uses, and step-by-step solved examples.

Class: 9 & 10
Topic: Statistics
Exams: CBSE · ICSE · State Boards

Master Table of Mean, Median, and Mode Formulas

Use this cheat sheet to quickly reference the correct formula based on whether your data is raw (ungrouped) or sorted into frequency tables (grouped).

Measure Ungrouped Data (Raw Data) Grouped Data (Class Intervals)
Mean (Average) Sum of all values / Total number of values x̄ = Σ(fixi) / Σfi
Median (Middle) If n is odd: ((n+1)/2)th term

If n is even: Average of (n/2)th & ((n/2)+1)th terms

 l + [ (n/2 − cf) / f ] × h
Mode (Most Frequent) The number that repeats the most times l + [ (f1 − f0) / (2f1 − f0 − f2) ] × h

Infographic showing mean, median, and mode formulas with grouped and ungrouped data layout.

1. Mean Formula (Explanation & Examples)

A. Ungrouped Mean

The ungrouped mean is simply the arithmetic average. You add up all the numbers in your list and divide by how many numbers there are.

Mean = Sum of observations / Total number of observations

Example 1: Ungrouped Mean

Ravi's scores in 5 math tests are: 12, 15, 18, 20, and 25. What is his mean score?

Step 1: Find the sum: 12 + 15 + 18 + 20 + 25 = 90
Step 2: Count the tests: n = 5
Step 3: Divide: 90 / 5 = 18.
Answer: Mean = 18

B. Grouped Mean (Direct Method)

When data is organized into class intervals (e.g., 10-20), we use the midpoint of the interval (class mark, $x_i$) and multiply it by how many items are in that interval (frequency, $f_i$).

Mean (x̄) = Σ(fixi) / Σfi

Example 2: Grouped Mean

Find the mean: Classes (0-10, 10-20, 20-30) and Frequencies (5, 8, 7).

Step 1 (Find midpoints xi):
0-10 → 5
10-20 → 15
20-30 → 25
Step 2 (Multiply fi × xi):
5 × 5 = 25
8 × 15 = 120
7 × 25 = 175
Step 3 (Sums): Σfixi = 25 + 120 + 175 = 320. Σfi = 5 + 8 + 7 = 20.
Step 4 (Divide): Mean = 320 / 20 = 16.
Answer: Mean = 16

2. Median Formula (Explanation & Examples)

A. Ungrouped Median

The median is the exact middle value of a list. Rule #1: Always sort your numbers from smallest to largest first!

Example 3: Ungrouped Median

Case 1 (Odd): Find the median of 15, 12, 20, 18, 25.

Sorted: 12, 15, 18, 20, 25.
There are 5 numbers. The middle one is the 3rd number.
Median = 18.

Case 2 (Even): Add '28' to the list. Find the median of 12, 15, 18, 20, 25, 28.

There are 6 numbers. The middle is between the 3rd and 4th numbers (18 and 20).
Average them: (18 + 20) / 2 = 38 / 2 = 19.
Median = 19.

B. Grouped Median (Class 10 Method)

To find the median of grouped data, calculate the Cumulative Frequency (running total) to find the "median class" where the middle value sits.

Median = l + [ (n/2 − cf) / f ] × h

Example 4: Grouped Median

Find the median: Classes (0-10, 10-20, 20-30) | Frequencies (5, 8, 7).

Step 1 (Find cf):
0-10: cf = 5
10-20: cf = 5 + 8 = 13
20-30: cf = 13 + 7 = 20
Total n = 20. Half of n (n/2) = 10.Step 2 (Identify Median Class): The cf just greater than 10 is 13, so the median class is 10-20.
l (lower limit) = 10.
cf (cumulative freq. of previous class) = 5.
f (frequency of median class) = 8.
h (class size) = 10.Step 3 (Calculate):
Median = 10 + [ (10 − 5) / 8 ] × 10
Median = 10 + [ 5 / 8 ] × 10
Median = 10 + 6.25 = 16.25.
Answer: Median = 16.25

3. Mode Formula (Explanation & Examples)

A. Ungrouped Mode

The mode is simply the number that appears the maximum number of times in a dataset.

Example 5: Ungrouped Mode

Find the mode of: 3, 5, 7, 5, 9, 5, 11.

Look at the numbers. '5' appears three times, while every other number appears only once.
Answer: Mode = 5

B. Grouped Mode (बहुलक का सूत्र)

To find the mode of a frequency table, first identify the "Modal Class" (the group with the highest frequency), and plug it into this formula:

Mode = l + [ (f1 − f0) / (2f1 − f0 − f2) ] × h

Example 6: Grouped Mode

Find the mode: Classes (0-20, 20-40, 40-60, 60-80) | Frequencies (5, 12, 10, 3).

Step 1: Highest frequency is 12. Modal Class is 20-40.
l = 20
f1 (freq. of modal class) = 12
f0 (freq. of previous class) = 5
f2 (freq. of next class) = 10
h = 20Step 2 (Calculate):
Mode = 20 + [ (12 − 5) / (2(12) − 5 − 10) ] × 20
Mode = 20 + [ 7 / (24 − 15) ] × 20
Mode = 20 + [ 7 / 9 ] × 20
Mode = 20 + (140 / 9) = 20 + 15.55 = 35.55.
Answer: Mode = 35.55

4. Differences, Use, and Examples

The three measures of central tendency often produce different answers on the exact same data. This is because each measure responds differently to the shape of the data. Here is a breakdown of their differences and when to use each one.

Aspect Mean Median Mode
What it measures Average of all values The exact middle value The most frequent value
Affected by outliers? Yes, strongly No No
Best for symmetric data? ✔ Yes ✔ Yes ✔ Yes
Best for skewed data? ✘ No ✔ Yes Sometimes
Best for categorical data? ✘ No ✘ No ✔ Yes

When to Use Mean, Median, or Mode

  • Use the Mean when: The data is roughly symmetric without major outliers. Example: Average daily temperature over a month, or a student's average test score.
  • Use the Median when: The data is skewed or contains outliers. A single extreme value would distort the average. Example: Household income, real estate prices. A few billionaires can pull the mean income far higher than what most people earn, making the median a much better representative.
  • Use the Mode when: The data is categorical and you want to find the most common choice. Example: The most popular shoe size sold in a store, or the favorite color of a class.

Example 7: One Dataset, Three Answers

Consider the dataset: {4, 8, 8, 11, 13, 14, 16, 18, 22}

Mean Calculation:
Mean = (4 + 8 + 8 + 11 + 13 + 14 + 16 + 18 + 22) / 9
Mean = 114 / 9 ≈ 12.67Median Calculation:
There are 9 items (odd). Median is the ((9+1)/2)th value = 5th value.
The 5th value in the ordered list is 13.Mode Calculation:
The most frequent value is 8 (it appears twice).

Conclusion: The mean (12.67), median (13), and mode (8) all tell a completely different story about the exact same data!

5. The "Magic" Shortcut: Empirical Formula

If an exam question gives you the Mean and Median, but asks for the Mode, do not calculate the whole table again! For moderately skewed unimodal distributions, the three measures are connected by Karl Pearson's Empirical relation:

Empirical Formula
Mode = 3 Median − 2 Mean

Example 8: Using the Empirical Formula

If the Mean of a dataset is 24 and the Median is 26, calculate the Mode.

Mode = 3(26) − 2(24)
Mode = 78 − 48 = 30
Answer: Mode = 30

Top Queries Answered (FAQs)

What is the mean median mode formula in simple words?
Mean is the "average" (total divided by count). Median is the "exact middle" number when lined up in order. Mode is the number that "repeats the most". For grouped data, specific interval formulas are used.
What is the difference between mean and median?
The mean is the arithmetic average of all values, using every data point, which makes it vulnerable to extreme outliers. The median is the physical middle value when the data is ordered, rendering it completely unaffected by extreme high or low values. For skewed datasets (like income), the median is usually the better representative.
What does बहुलक का सूत्र (bahulak ka sutra) mean?
"बहुलक का सूत्र" is simply Hindi for the "Mode formula". In CBSE class 10 Hindi medium, it is written exactly the same mathematically: l + [ (f1 - f0) / (2f1 - f0 - f2) ] × h.
If I only have the mean and median formula, can I find the mode?
Yes! You use the Empirical relationship: Mode = 3(Median) - 2(Mean). You do not need the large, complicated formulas if you already have two of the final answers.