Centering with Reductions

Center an array by reducing one axis to compute means and broadcasting those means back across the original values. Use axis meaning to predict both intermediate and final shapes.

Centering subtracts a mean so that values describe differences from that mean. For a table of examples and features, the usual goal is to center each feature separately.

The source has shape (3, 2): three examples and two features. feature_means has shape (2,), one mean per feature. Broadcasting subtracts those two means from every row, so centered keeps shape (3, 2).

Compute the Means by Hand

The first feature mean is:

(1 + 3 + 5) / 3 = 3

The second feature mean is:

(10 + 14 + 18) / 3 = 14

Therefore:

feature_means = [3.0, 14.0]

Subtracting these values from each row gives:

[[-2, -4],
 [ 0,  0],
 [ 2,  4]]

Read Centered Values as Differences from the Mean

Before centering, the first feature contains [1.0, 3.0, 5.0]. After subtracting its mean, it contains [-2.0, 0.0, 2.0]. These values have a direct interpretation:

• -2.0 means two units below the feature mean;
• 0.0 means equal to the feature mean;
• 2.0 means two units above the feature mean.

Centering changes the reference point, but it does not change differences between values. The distance from 1.0 to 5.0 is 4.0, and the distance from -2.0 to 2.0 is also 4.0. It also does not make different feature scales equal; that requires an additional scaling step.

The centered feature means are zero, apart from possible tiny floating-point rounding differences in other examples.

Choose the Axis from the Intended Groups

Reducing axis=0 combines examples and leaves one mean for each feature:

(examples, features) -> (features,)

Reducing axis=1 combines features and leaves one mean for each example:

(examples, features) -> (examples,)

The correct axis follows from what should be centered. Feature centering uses one mean per feature. Row centering uses one mean per row.

Keep a Row Mean as a Column

To center every row separately, keep one mean beside each row. Without keepdims=True, data.mean(axis=1) has shape (examples,). Broadcasting aligns that flat result with the final, feature axis—not automatically with the example axis. It fails when the two lengths differ and can express the wrong calculation when they happen to be equal.

Preserve the reduced axis with length one:

For data.shape == (3, 2), row_means.shape is (3, 1). The length-one feature axis expands across each row, and row_centered.shape remains (3, 2). For the example data:

row_means =
[[ 5.5],
 [ 8.5],
 [11.5]]

row_centered =
[[-4.5, 4.5],
 [-5.5, 5.5],
 [-6.5, 6.5]]

Feature centering and row centering answer different questions. Choose between them from the meaning of the axes, not from which expression happens to run.

Check the Result

Do not stop after the subtraction runs. Verify the property that centering was supposed to create:

print(centered.mean(axis=0))

For feature-centered data, each feature mean should be zero or very close to zero. This check tests the meaning of the operation, not only its shape.