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For equal prior probabilities, $b=\frac{1}{2}{a}^T({\mu }_1+{\mu }_2)$. The projected score is normal under either class because it is a linear image of a multivariate normal vector, and its variance isMoreover $a^T{\mu }_1-a^T{\mu }_2={\delta }^2$, so the midpoint threshold lies ${\delta }^2/2$ from either projected mean. HenceStandardizing the two normal distributions, with $\Phi$ the standard normal distribution function, givesThus the equal-covariance Gaussian classification error is $p(\delta )=\Phi (-\delta /2)$. The Mahalanobis distance $\delta$ between the class means is their separation in units of within-class noise. It yields $p\to 1/2$ as $\delta \downarrow 0$ and $p\to 0$ as $\delta \to \infty$. The stipulated $\delta >0$ means the two class means are distinct.

Cluster analysis groups observations without given class labels. It starts by deciding what similarity means, usually through a dissimilarity matrix. Euclidean distance is natural for comparable continuous coordinates; a Mahalanobis distance can account for differing scales and correlations when a meaningful nonsingular covariance estimate is available. Scaling and variable selection can change the groups. Binary, ordinal or categorical data may need a different dissimilarity, rather than arbitrary numerical codes and Euclidean distance.

Agglomerative hierarchical clustering begins with singleton groups, repeatedly merges the pair with smallest intergroup dissimilarity, and records the merges in a dendrogram. The height represents the algorithm's merge criterion. Cutting the dendrogram gives partitions at different resolutions, and merges cannot subsequently be reversed. In single-linkage clustering, the criterion is the nearest cross-pair distance; this detects chains and nonconvex groups, but a thin bridge can connect otherwise separated clouds. Complete-linkage clustering uses the farthest cross-pair distance and tends to produce compact groups, at the cost of sensitivity to extreme points. Average-linkage clustering uses the mean cross-pair distance, lying between these two criteria for a given pair of clusters.

For squared Euclidean geometry, Ward clustering merges the pair that least increases the within-group sum of squares. Expanding deviations about the merged mean givesThis favors low within-group dispersion, but interpreting it as a sum-of-squares increase requires Euclidean data. Different software may plot a rescaled height, so the vertical units of a dendrogram from Ward clustering must be stated.

A nonhierarchical alternative is k-means clustering. For a prescribed $K$, minimizeAt fixed assignments, expanding squared distances or differentiating shows that the minimizing $m_k$ is the group mean. At fixed centroids, each point is assigned to a nearest centroid. Alternating these operations does not increase $W_K$; with a consistent tie rule and nonempty groups it reaches a locally stable partition. Initialization matters, so use multiple starts, and give an explicit rule for empty groups. It favors roughly spherical clusters and is sensitive to outliers and scaling. It cannot be expected to reproduce a nonconvex single-linkage partition.

The sketch displays a constructed point configuration and the complete-linkage merges computed from its actual Euclidean distances. A cut between the within-pair and between-pair merge heights yields three groups. Such visual separation is useful evidence, not proof of a uniquely correct number of populations. Select a resolution using subject knowledge, stability under resampling, separation diagnostics or the change of $W_K$ with $K$. Clustering always gives a partition even for unstructured data, and different distance or linkage choices can yield different sensible summaries. Clustering is an exploratory construction whose conclusions depend on the geometry and the chosen resolution.

Write $x_r,x_t$ for two rows of the data matrix and $\delta =x_r-x_t$. Three possible dissimilarities for continuous variables are the following.

The Euclidean distance isIt is simple and has the usual geometric interpretation when coordinates have comparable units. Its disadvantage is sensitivity to the numerical scales: a change of units or a high-variance variable can dominate it. Strongly correlated coordinates can also give repeated weight to essentially the same information. Squared Euclidean distance is useful in some methods, but is not itself a metric because it need not satisfy the triangle inequality.

The standardized Euclidean distance iswhere $s_j^2$ is the sample variance of coordinate $j$ across individuals. It measures differences in standard-deviation units and is invariant under positive coordinate rescalings when the scales are estimated consistently. It is useful for differently measured variables, but requires nonzero sample variances, ignores correlations between coordinates and can give excessive influence to a low-variance noisy variable. With fixed positive scales it is a Euclidean distance after a diagonal change of coordinates, so it is a metric.

The Mahalanobis distance iswhere $S$ is the pooled sample covariance matrix of all individuals. If $S=C{C}^T$ is its Cholesky decomposition, then $d_M=\Vert C^{-1}\delta {\Vert }_2$: it is Euclidean distance in whitened coordinates. It adjusts for both scales and correlations, avoiding duplicate weight for highly correlated variables, and is invariant under nonsingular affine transformations when the sample covariance matrix is transformed accordingly. No multivariate normal distribution assumption is required merely to define the distance. However, $S$ must be a positive-definite matrix; when there are too few observations or dependent coordinates, its inverse is unavailable. Estimation of $S^{-1}$ can also be unstable and sensitive to outliers. These trade-offs determine whether covariance adjustment is preferable to the simpler distances.

For rows $x_i,x_j$, three useful choices are Euclidean distance, standardized Euclidean distance, and Mahalanobis distance:The Euclidean choice is simple, rotation invariant and appropriate when the original coordinate scales carry comparable meaning. It is sensitive to scale changes, may be dominated by a high-variance variable and gives correlated or duplicate variables repeated influence. Squared Euclidean dissimilarity is sometimes used algorithmically, but removing the square root generally loses the triangle inequality.

The standardized Euclidean choice removes marginal units using fixed positive sample scales $s_r$. It gives equal marginal weight to variables, but ignores correlations and can magnify noise when an estimated variance is small. Constant variables require omission or an explicitly chosen positive external scale.

The Mahalanobis choice incorporates the full positive-definite sample covariance $S$, so it is Euclidean distance after whitening. It accounts for both scale and redundancy and is invariant under an invertible change of coordinates when $S$ is transformed consistently. However estimating $S$ requires enough observations and can be sensitive to extreme values; inversion is unstable near singularity and unavailable when $S$ is singular. With fixed positive scales or positive-definite $S$, all three square-root expressions are genuine metrics on the measurement space. Their different geometries encode different notions of which individuals should be considered similar.