Mean, Median, Mode Calculator - Ungrouped & Grouped Data

Mean, Median, and Mode Calculator

Use this unified calculator to find the mean, median, and mode for both ungrouped (raw) data and grouped (frequency distribution) data.

Quick Start

Choose your data type below, enter your values, and click Calculate:

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How to Use This Calculator

For Ungrouped (Raw) Data

Step 1: Select “Ungrouped (Raw Data)” as your data type

Step 2: Enter your data values separated by commas (e.g., 10, 15, 20, 25, 30)

Step 3: Click “Calculate”

Results will show:

• Number of observations (N)
• Sum of values
• Mean (average)
• Sorted values (in ascending order)
• Median (middle value)
• Mode (most frequent value)

For Grouped (Frequency Distribution) Data

Step 1: Select “Grouped (Frequency Distribution)” as your data type

Step 2: Choose frequency distribution type:

• Discrete: For individual class values (e.g., 10, 15, 20)
• Continuous: For class intervals (e.g., 10-20, 20-30, 30-40)

Step 3: Enter class values or intervals separated by commas

Step 4: Enter the corresponding frequencies separated by commas

Step 5: Click “Calculate”

Results will show:

• Number of observations (N)
• Mean of frequency distribution
• Median of grouped data
• Mode of frequency distribution
• Frequency distribution table

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Formulas & Theory

Mean (Average)

The mean is the sum of all values divided by the number of values.

For Ungrouped Data

$$\overline{x} = \frac{\sum x_i}{n} = \frac{x_1 + x_2 + \cdots + x_n}{n}$$

Where:

• $x_i$ = individual data values
• $n$ = number of observations
• $\sum x_i$ = sum of all values

Example: For data {10, 15, 20, 25, 30}: $$\overline{x} = \frac{10 + 15 + 20 + 25 + 30}{5} = \frac{100}{5} = 20$$

For Grouped Data

$$\overline{x} = \frac{1}{N}\sum_{i=1}^{k}f_i x_i$$

Where:

• $x_i$ = class midpoint (for continuous) or class value (for discrete)
• $f_i$ = frequency of class i
• $N = \sum f_i$ = total number of observations
• For continuous distributions, $x_i$ = (lower limit + upper limit) / 2

Median (Middle Value)

The median is the value that divides the data into two equal halves when arranged in order.

For Ungrouped Data

Procedure:

1. Arrange data in ascending order
2. If n is odd: Median = middle value
3. If n is even: Median = average of two middle values

Examples:

• Data {10, 15, 20, 25, 30}: Median = 20 (middle value, n=5 is odd)
• Data {10, 15, 20, 25}: Median = (15 + 20)/2 = 17.5 (n=4 is even)

For Grouped Data

$$\text{Median} = l + \left(\frac{\frac{N}{2} - F_<}{f}\right) \times h$$

Where:

• $l$ = lower boundary of the median class
• $N$ = total number of observations
• $F_<$ = cumulative frequency before the median class
• $f$ = frequency of the median class
• $h$ = class width (for continuous distributions)

Step-by-step:

1. Find $N/2$
2. Locate the class with cumulative frequency ≥ N/2 (this is the median class)
3. Apply the formula above

Mode (Most Frequent Value)

The mode is the value that appears most frequently in the data.

For Ungrouped Data

Simply identify which value appears most often.

Examples:

• Data {10, 15, 20, 20, 25, 20}: Mode = 20 (appears 3 times)
• Data {10, 15, 15, 20, 25, 25}: Bimodal - modes are 15 and 25
• Data {10, 15, 20, 25, 30}: No mode (all appear once)

For Grouped Data

$$\text{Mode} = l + \left(\frac{f_m - f_1}{2f_m - f_1 - f_2}\right) \times h$$

Where:

• $l$ = lower boundary of the modal class (class with highest frequency)
• $f_m$ = frequency of the modal class
• $f_1$ = frequency of the class before modal class
• $f_2$ = frequency of the class after modal class
• $h$ = class width

Note: The modal class is the class interval with the highest frequency.

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Worked Examples

Example 1: Ungrouped Data

Data: Test scores for 7 students: 65, 72, 68, 75, 72, 80, 70

Solution:

Step 1: Find Mean $$\overline{x} = \frac{65 + 72 + 68 + 75 + 72 + 80 + 70}{7} = \frac{502}{7} = 71.71$$

Step 2: Find Median Arrange in order: 65, 68, 70, 72, 72, 75, 80

• n = 7 (odd), so median = middle value = 72

Step 3: Find Mode

• 65 appears 1 time
• 68 appears 1 time
• 70 appears 1 time
• 72 appears 2 times (most frequent)
• 75 appears 1 time
• 80 appears 1 time

Mode = 72

Interpretation: On average, students scored 71.71. The middle score was 72, and the most common score was 72.

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Example 2: Grouped Data (Continuous)

Problem: The following table shows the number of hours students studied per week. Calculate mean, median, and mode.

Study Hours	10-15	15-20	20-25	25-30	30-35
Number of Students	5	12	18	10	5

Solution:

Step 1: Create Calculation Table

Class	Midpoint (x)	Frequency (f)	f×x	Cumulative Frequency
10-15	12.5	5	62.5	5
15-20	17.5	12	210	17
20-25	22.5	18	405	35
25-30	27.5	10	275	45
30-35	32.5	5	162.5	50
Total		50	1,115

Step 2: Calculate Mean $$\overline{x} = \frac{\sum f_i x_i}{N} = \frac{1,115}{50} = 22.3 \text{ hours}$$

Step 3: Calculate Median

• N/2 = 50/2 = 25
• Cumulative frequency ≥ 25 is 35 (in class 20-25), so median class = 20-25
• l = 20, F_< = 17, f = 18, h = 5

$$\text{Median} = 20 + \left(\frac{25 - 17}{18}\right) \times 5 = 20 + \left(\frac{8}{18}\right) \times 5 = 20 + 2.22 = 22.22 \text{ hours}$$

Step 4: Calculate Mode

• Modal class = 20-25 (highest frequency = 18)
• l = 20, f_m = 18, f_1 = 12, f_2 = 10, h = 5

$$\text{Mode} = 20 + \left(\frac{18 - 12}{2(18) - 12 - 10}\right) \times 5 = 20 + \left(\frac{6}{14}\right) \times 5 = 20 + 2.14 = 22.14 \text{ hours}$$

Results:

• Mean = 22.3 hours (average study time)
• Median = 22.22 hours (middle value)
• Mode = 22.14 hours (most common study time)

Interpretation: Students study an average of 22.3 hours per week, with a median of 22.22 hours and most common value around 22.14 hours.

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Key Differences: Ungrouped vs. Grouped Data

Aspect	Ungrouped Data	Grouped Data
Data Format	Individual raw values	Classes with frequencies
Information Loss	None - exact values known	Some - exact values not known
Mean Calculation	Simple: sum all, divide by count	Weighted by frequencies using midpoints
Median Finding	Find exact middle value(s)	Estimate using class formula
Mode Identification	Find most frequent single value	Find class with highest frequency
When to Use	Small datasets, all values available	Large datasets, data already summarized
Accuracy	Exact values	Approximate values

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When to Use Ungrouped vs. Grouped

Use Ungrouped Data Calculator When:

• You have a small dataset with all individual values
• You need exact statistical measures
• Data hasn’t been summarized into groups
• Examples: Quiz scores for a small class, daily temperatures for a week

Use Grouped Data Calculator When:

• You have a large dataset already organized into classes
• Only frequency distribution is available (not raw data)
• Data ranges across many values
• Need to summarize large amounts of information
• Examples: Income distribution in a country, product weights from manufacturing, test score ranges across many students

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Common Mistakes to Avoid

❌ WRONG: Using mean when data has extreme outliers ✓ RIGHT: Consider median as well when outliers present

❌ WRONG: Calculating mode for continuous grouped data without determining modal class first ✓ RIGHT: Always identify the class with highest frequency first

❌ WRONG: Using ungrouped data formula on grouped data with class intervals ✓ RIGHT: Use grouped data formula that accounts for class width

❌ WRONG: Assuming mode must exist ✓ RIGHT: Some datasets have no mode (all values appear once)

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Interpretation Guide

What the Mean Tells You

• Average value of the dataset
• Affected by extreme values (outliers)
• Best used when data is relatively symmetric

What the Median Tells You

• Middle value - 50% of data above, 50% below
• Robust to outliers (not affected by extreme values)
• Better for skewed distributions

What the Mode Tells You

• Most common value
• Useful for categorical data and discrete values
• Can be multiple modes or no mode

Use Together for Better Understanding

Comparing all three measures reveals distribution shape:

• Mean ≈ Median ≈ Mode: Symmetric distribution
• Mean > Median > Mode: Right-skewed (positively skewed)
• Mean < Median < Mode: Left-skewed (negatively skewed)

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Related Calculators & Tutorials

Explore related concepts:

• Variance and Standard Deviation
• Quartiles, Percentiles, Deciles
• Five-Number Summary
• Central Tendency Tutorial

More Descriptive Statistics:

• Descriptive Statistics Calculators
• Summary Statistics
• Skewness and Kurtosis