Mean Absolute Deviation Calculator - Free MAD Calculator Online

The Mean Absolute Deviation (MAD) is a statistical measure that calculates the average distance between each data point and the mean of the dataset. Unlike standard deviation, MAD provides a more intuitive and less complex way to understand data variability, making it particularly useful for beginners in statistics.

What is Mean Absolute Deviation?

Mean Absolute Deviation is defined as the average deviation of data points from a center point, typically the mean. It measures how spread out the data values are from the average, giving you a clear picture of data variability. The MAD is expressed in the same units as the original data, making it easier to interpret than variance or standard deviation.

How to Find Mean Absolute Deviation

Calculating MAD involves a straightforward three-step process:

Step 1: Calculate the Mean

Find the average of all data points by summing all observations and dividing by the sample size:

Mean (x̄) = (x₁ + x₂ + x₃ + ... + xₙ) / n

Step 2: Find Absolute Deviations

Calculate the absolute difference between each data point and the mean. This involves:

• Subtracting the mean from each data value

• Taking the absolute value (removing negative signs) of each difference

Step 3: Calculate the Average of Absolute Deviations

Sum all the absolute deviations and divide by the number of data points:

MAD = Σ|xᵢ - x̄| / n

Where:

• xᵢ = individual data values

• x̄ = mean of the dataset

• n = number of data points

• |xᵢ - x̄| = absolute deviation

Mean Absolute Deviation Formula

For Ungrouped Data:

MAD = (1/n) Σ|xᵢ - μ|

For Grouped Data:

MAD = Σf|x - xᵢ| / Σf

Where f represents the frequency of each data point.

Solved Examples

Let's work through the specific examples mentioned in your query:

Example 1: Dataset {4, 5, 4, 8, 6, 10, 2, 5, 3, 1}

Step 1: Calculate the mean
Mean = (4 + 5 + 4 + 8 + 6 + 10 + 2 + 5 + 3 + 1) ÷ 10 = 48 ÷ 10 = 4.8

Step 2: Find absolute deviations

• |4 - 4.8| = 0.8

• |5 - 4.8| = 0.2

• |4 - 4.8| = 0.8

• |8 - 4.8| = 3.2

• |6 - 4.8| = 1.2

• |10 - 4.8| = 5.2

• |2 - 4.8| = 2.8

• |5 - 4.8| = 0.2

• |3 - 4.8| = 1.8

• |1 - 4.8| = 3.8

Step 3: Calculate MAD
MAD = (0.8 + 0.2 + 0.8 + 3.2 + 1.2 + 5.2 + 2.8 + 0.2 + 1.8 + 3.8) ÷ 10 = 20 ÷ 10 = 2.0

Example 2: Dataset {6, 4, 4, 12, 2, 20, 12, 8, 24, 24}

Step 1: Calculate the mean
Mean = (6 + 4 + 4 + 12 + 2 + 20 + 12 + 8 + 24 + 24) ÷ 10 = 116 ÷ 10 = 11.6

Step 2: Find absolute deviations and calculate MAD
Sum of absolute deviations = 76
MAD = 76 ÷ 10 = 7.6

Example 3: Dataset {5, 9, 1, 5, 2, 5, 5, 5, 9, 3}

Step 1: Calculate the mean
Mean = (5 + 9 + 1 + 5 + 2 + 5 + 5 + 5 + 9 + 3) ÷ 10 = 49 ÷ 10 = 4.9

Step 2: Calculate MAD
After finding all absolute deviations: MAD = 2.29

Additional Examples:

Dataset {2, 20, 4, 16, 10, 12, 8, 4}:
Mean = 9.5, MAD = 5.0

Dataset {2, 14, 10, 8, 20, 20, 18, 8}:
Mean = 12.5, MAD = 5.5

Dataset {8, 2, 12, 10, 2, 6, 12, 20}:
Mean = 9, MAD = 4.5

Dataset {6, 2, 8, 4, 8, 6, 8, 8}:
Mean = 6.25, MAD = 1.75

Dataset {2, 8, 6, 8, 6, 8, 10, 12}:
Mean = 7.5, MAD = 2.5

Dataset {12, 10, 12, 6, 8, 4, 2, 12}:
Mean = 8.25, MAD = 3.25

Mean Absolute Deviation Calculator for Grouped Data

For grouped data, the calculation involves using frequencies and class midpoints. The formula becomes:

MAD = Σf|xᵢ - x̄| / Σf

Where:

• f = frequency of each class

• xᵢ = class midpoint

• x̄ = weighted mean

• Σf = total frequency

Free Online MAD Calculators

Several free online tools are available for calculating MAD:

• Statistics Kingdom MAD Calculator - Provides step-by-step calculations

• Alcula Statistics Calculator - Simple interface for MAD computation

• Cuemath MAD Calculator - Includes detailed explanations and examples

• BYJU'S MAD Calculator - Fast calculation with clear results

These calculators allow you to input your data separated by commas and instantly receive the MAD value along with detailed step-by-step solutions.

When to Use MAD

MAD is particularly useful when you need a simpler, more intuitive measure of variability that's less sensitive to outliers compared to standard deviation. It's ideal for:

• Introductory statistics education

• Quick variability assessments

• Situations where you need results in the same units as your data

• When dealing with datasets that contain outliers

The Mean Absolute Deviation provides a straightforward way to understand how much your data points typically deviate from the average, making it an essential tool for statistical analysis and data interpretation.

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