How do you calculate mean absolute deviation step by step?

Step 1: Calculate the mean of your dataset (sum all values, divide by count). Step 2: For each data point, subtract the mean and take the absolute value of the result. Step 3: Sum all those absolute deviations. Step 4: Divide by the number of data points. That final result is the MAD. Example: for {2, 4, 6, 8, 10}, mean = 6, deviations = {4, 2, 0, 2, 4}, MAD = (4+2+0+2+4)/5 = 12/5 = 2.4.

What is the difference between MAD from the mean and MAD from the median?

MAD from the mean uses the arithmetic mean as the central point. MAD from the median uses the statistical median instead. The median-based MAD is more robust to outliers because the median itself is not influenced much by extreme values. In datasets with significant skew or outliers, MAD(median) often gives a better sense of typical spread.

What is the median absolute deviation?

Median absolute deviation (also abbreviated MAD) refers to the MAD from the median. It is calculated as the median of {|xi - median|} across all data points. This is distinct from the mean absolute deviation from the median that this calculator computes. This calculator shows the mean of absolute deviations from the median, not the median of them.

Mean Absolute Deviation Calculator

Q: What is the difference between MAD and standard deviation?

Both measure spread, but standard deviation squares the deviations before averaging (making it more sensitive to outliers), then takes the square root. MAD takes absolute values instead of squaring, so it weights all deviations equally regardless of their size. MAD is more robust to outliers. For a normal distribution, standard deviation = MAD / 0.7979.

Q: What is a good MAD value?

There is no universal good or bad MAD value - it depends on the scale and context of your data. A MAD close to zero means data points are tightly clustered around the mean. A large MAD relative to the mean indicates high spread. Compare MAD as a percentage of the mean (coefficient of dispersion = MAD/mean x 100%) for context.

Q: Can MAD be zero?

Yes. If all data points are identical (zero spread), MAD equals zero because every deviation from the mean or median is zero. MAD can never be negative since it is an average of absolute values, which are always non-negative.

Q: What is the practical use of mean absolute deviation?

MAD is widely used in forecasting to measure forecast error (Mean Absolute Error is the same concept applied to predictions vs actuals). It is used in finance to measure portfolio return dispersion, in quality control to assess process consistency, and in machine learning as a loss function. It appears in tracking signals used to monitor whether a forecasting model has become biased.

Compute mean absolute deviation from the mean and from the median. Enter any comma-separated dataset for instant results.

MAD What is Mean Absolute Deviation?

Mean absolute deviation (MAD) is one of the most intuitive measures of statistical dispersion. It answers the question: on average, how far do individual data points stray from the centre of the dataset? Unlike variance (which squares the differences and loses interpretability) or standard deviation (which requires knowing variance), MAD is calculated directly in the original units of the data and is immediately understandable.

There are two versions of MAD depending on the choice of central reference point. MAD from the mean measures average absolute deviation from the arithmetic mean, making it sensitive to all data points equally. MAD from the median measures average absolute deviation from the median, making it more robust in the presence of outliers or skewed distributions because the median itself is resistant to extreme values.

MAD has wide practical applications. In forecasting, it appears as Mean Absolute Error (MAE), a common metric for evaluating how accurate predictions are. In finance, MAD quantifies the spread of portfolio returns. In quality control, it tracks process consistency. In machine learning, the L1 loss function is equivalent to minimising MAD from the mean. In robust statistics, MAD from the median is used as a robust estimator of scale.

For datasets following a normal distribution, MAD from the mean relates to the standard deviation by: standard deviation = MAD / 0.7979. This means you can use MAD as an accessible approximation when standard deviation feels abstract or when working with small samples where squaring deviations may amplify noise too aggressively.

This calculator computes both types of MAD simultaneously so you can compare them. A large discrepancy between the two typically signals the presence of outliers in your data, since outliers pull the mean but not the median.

Formulas and Step-by-Step

MAD from the Mean:

MAD(mean) = (1/n) ∑ |x_i − &bar;x|

x_i = Each data point

&bar;x = Arithmetic mean = (∑ x_i) / n

n = Number of data points

MAD from the Median:

MAD(median) = (1/n) ∑ |x_i − M|

M = Median (middle value when sorted; average of two middles for even n)

Computing the Median:

Sort values: x₍₁₎ ≤ x₍₂₎ ≤ ... ≤ x_(n)

If n is odd: M = x_((n+1)/2)

If n is even: M = (x_(n/2) + x_{(n/2 + 1)}) / 2

Relationship to standard deviation for normal data:

σ = MAD / 0.7979 ≈ 1.2533 × MAD

How to Use This Calculator

Enter your data — type your numbers into the text area, separated by commas. For example: 4, 8, 15, 16, 23, 42. Spaces around commas are optional.
Click Calculate — both MAD values, the mean, and the median are computed and displayed instantly.
Interpret the results — MAD from mean is the standard MAD; MAD from median is the robust version. If they differ significantly, outliers may be present.
Use the note — the grey note below the results confirms which dataset was used and the step-by-step computation.

Example Calculations

Example 1 — Lost Numbers from LOST

Dataset: 4, 8, 15, 16, 23, 42

Mean = (4+8+15+16+23+42)/6 = 108/6 = 18.0

Absolute deviations from mean: |4-18|=14, |8-18|=10, |15-18|=3, |16-18|=2, |23-18|=5, |42-18|=24

MAD(mean) = (14+10+3+2+5+24)/6 = 58/6 = 9.6667

Median = (15+16)/2 = 15.5. MAD(median) = (11.5+7.5+0.5+0.5+7.5+26.5)/6 = 54/6 = 9.0

MAD(mean) = 9.6667 · MAD(median) = 9.0000 · Mean = 18.0000 · Median = 15.5

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Example 2 — Evenly Spaced Data

Dataset: 2, 4, 6, 8, 10 (arithmetic sequence)

Mean = 30/5 = 6. Deviations: 4, 2, 0, 2, 4. MAD(mean) = 12/5 = 2.4

Median = 6 (middle value of 5). MAD(median) = same = 2.4

Both MADs equal because symmetric, evenly spaced data has mean = median

MAD(mean) = 2.4000 · MAD(median) = 2.4000 · Mean = 6.0000 · Median = 6.0

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Example 3 — Skewed Data with Outlier

Dataset: 10, 20, 30, 40, 50, 60, 70 (linear spread)

Mean = 280/7 = 40. Deviations: 30, 20, 10, 0, 10, 20, 30. MAD(mean) = 120/7 = 17.1429

Median = 40 (4th of 7). MAD(median) = same = 17.1429

MAD(mean) = 17.1429 · MAD(median) = 17.1429 · Mean = 40.0000 · Median = 40.0

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Example 4 — Fibonacci Sequence

Dataset: 1, 1, 2, 3, 5, 8, 13, 21 (Fibonacci numbers)

Mean = 54/8 = 6.75. Deviations: 5.75, 5.75, 4.75, 3.75, 1.75, 1.25, 6.25, 14.25. Sum = 43.5. MAD(mean) = 5.4375

Median = (3+5)/2 = 4. MAD(median) = (3+3+2+1+1+4+9+17)/8 = 40/8 = 5.0

MAD(mean) > MAD(median): the large value 21 pulls the mean but not the median

MAD(mean) = 5.4375 · MAD(median) = 5.0000 · Mean = 6.7500 · Median = 4.0

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Frequently Asked Questions

What is mean absolute deviation (MAD)?+

Mean absolute deviation is a measure of statistical dispersion that tells you, on average, how far each data point is from the centre of the dataset. It is calculated by taking the absolute value of each deviation from the mean (or median), then averaging those absolute values. Unlike variance, MAD is in the same units as the original data, making it directly interpretable.

How do you calculate MAD step by step?+

Step 1: Calculate the mean. Step 2: For each data point, subtract the mean and take the absolute value. Step 3: Sum all those absolute deviations. Step 4: Divide by n. For {2, 4, 6, 8, 10}: mean = 6, absolute deviations = {4, 2, 0, 2, 4}, sum = 12, MAD = 12/5 = 2.4.

What is the difference between MAD from mean and MAD from median?+

MAD from the mean uses the arithmetic mean as the central point. MAD from the median uses the statistical median. The median-based MAD is more robust to outliers because the median is not influenced much by extreme values. In skewed datasets or data with outliers, MAD(median) often better captures typical spread.

What is the difference between MAD and standard deviation?+

Both measure spread. Standard deviation squares deviations (making it more sensitive to outliers) then takes the square root. MAD takes absolute values instead of squaring, weighting all deviations equally. MAD is more robust to outliers. For normal data: standard deviation = MAD / 0.7979.

What is a good MAD value?+

There is no universal good or bad MAD - it depends on the scale of your data. A MAD close to zero means data is tightly clustered. For context, compute the coefficient of dispersion: MAD/mean x 100%. Values below 10% typically indicate low spread; above 30% indicates high spread relative to the mean.

Why use MAD instead of standard deviation?+

MAD is preferred when your data has outliers or is not normally distributed, because it is more robust. Standard deviation penalises large deviations disproportionately due to squaring. In financial risk, MAD treats upside and downside deviations equally. In robust statistics, MAD(median) is a key tool for outlier detection.

How does MAD relate to forecast error?+

Mean Absolute Error (MAE) in forecasting is equivalent to MAD applied to forecast errors (predicted minus actual values). MAE is widely used to evaluate the accuracy of regression models and time-series forecasts. A lower MAE means predictions are on average closer to actual values.

Can MAD be zero?+

Yes. If all data points are identical, MAD equals zero because every deviation from the mean or median is zero. MAD can never be negative since it is an average of absolute values, which are always non-negative. A MAD of zero means perfect uniformity - all values are the same.

How does MAD relate to the normal distribution?+

For a normal distribution, the expected MAD from the mean equals sigma x 2/sqrt(2pi) = sigma x 0.7979. So sigma = MAD / 0.7979 = MAD x 1.2533. This lets you estimate standard deviation from MAD for approximately normal data without computing squares.

What is the practical use of MAD?+

MAD is used in forecasting as Mean Absolute Error, in finance to measure portfolio return dispersion, in quality control to assess process consistency, and in machine learning as the L1 loss function. It also appears in tracking signals used to detect whether a forecasting model has become biased over time.

Mean Absolute Deviation Calculator

MAD What is Mean Absolute Deviation?

Formulas and Step-by-Step

How to Use This Calculator

Example Calculations

Example 1 — Lost Numbers from LOST

Dataset: 4, 8, 15, 16, 23, 42

Example 2 — Evenly Spaced Data

Dataset: 2, 4, 6, 8, 10 (arithmetic sequence)

Example 3 — Skewed Data with Outlier

Dataset: 10, 20, 30, 40, 50, 60, 70 (linear spread)

Example 4 — Fibonacci Sequence

Dataset: 1, 1, 2, 3, 5, 8, 13, 21 (Fibonacci numbers)

Frequently Asked Questions

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