Covariance Distance Explorer
RafikiArdhi · Research tools
Covariance Distance Explorer
Calculate Mahalanobis distance from a mean vector and covariance matrix. Explore how variance and correlation change multivariate distance, with transparent validation and no data sent anywhere.
Input data
Use the longer worked example, enter values manually, or open a CSV containing observations. CSV mode calculates a mean and sample covariance from selected numeric columns, then lets you choose a row to measure against that reference.
<button class="tab" id="manualTab" role="tab" aria-selected="true" aria-controls="manualPanel">Manual / example</button>
<button class="tab" id="csvTab" role="tab" aria-selected="false" aria-controls="csvPanel">Open CSV</button>
Example: a soil sample represented by two standardised attributes.
Choose covariance when you have Σ, or precision when you already have the inverse covariance matrix Σ⁻¹. The tool calculates the distance using the selected matrix and displays its inverse for reference.
The matrix must be square, symmetric, and positive definite. Rows and columns must follow the same variable order as x and μ.
<button id="calculate">Calculate distance</button>
<button id="example" class="secondary">Load example</button>
<button id="clear" class="secondary">Clear</button><p>Choose a local CSV file with a header row and numeric columns. The file is read in your browser and is not uploaded.</p>
<label for="csvFile">CSV file</label>
<input id="csvFile" type="file" accept=".csv,text/csv">
<div id="csvStatus" class="status" style="margin-top:12px">No file loaded.</div>
<label for="csvColumns">Variables to include</label>
<select id="csvColumns" multiple size="6" aria-describedby="csvHelp"></select>
<small id="csvHelp">Select at least two numeric variables. Hold Ctrl (Windows) or Command (Mac) to select multiple columns.</small>
<label for="csvRow">Target observation</label>
<select id="csvRow"></select>
<div class="actions">
<button id="useCsv">Use selected CSV data</button>
<button id="downloadDistanceMatrix" class="secondary">Download distance matrix CSV</button>
</div>
<p class="muted">Selected columns define the reference mean and sample covariance. The chosen row becomes the target vector. Rows with missing or non-numeric selected values are excluded from the reference calculation.</p><h2>Result</h2>
<span class="pill" id="state">Ready to calculate</span>
<div class="metrics">
<div class="metric"><span>Mahalanobis distance</span><strong id="distance">—</strong></div>
<div class="metric"><span>Squared Mahalanobis distance</span><strong id="squared">—</strong></div>
<div class="metric"><span>Euclidean distance</span><strong id="euclidean">—</strong></div>
<div class="metric"><span>Squared Euclidean distance</span><strong id="euclideanSquared">—</strong></div>
<div class="metric"><span>Variables</span><strong id="dimension">—</strong></div>
<div class="metric"><span>Matrix condition estimate</span><strong id="condition">—</strong></div>
</div>
<div id="interpretation" class="status">Enter a target, mean, and covariance matrix, then calculate.</div>
<div class="result-section">
<h3>Difference vector (x − μ)</h3>
<div id="delta" class="muted">—</div>
</div><h2>What is being calculated?</h2>
<p>Mahalanobis distance measures how far a vector is from a reference mean while accounting for covariance among variables.</p>
<p><strong>Squared Mahalanobis distance: D² = (x − μ)ᵀ P (x − μ)</strong>, where P is the precision matrix (Σ⁻¹).</p>
<p><strong>Squared Euclidean distance: E² = Σᵢ(xᵢ − μᵢ)²</strong>; Euclidean distance is √E².</p>
<p class="muted">Euclidean distance treats each variable axis equally in the supplied units. Mahalanobis distance accounts for covariance and variable scales. Neither distance is a probability by itself.</p>