Center and Check Measurements

Different sensors may work on different numerical levels. In our small table, the first sensor reports values near 3, while the second reports values near 8. If a later calculation should describe change around each sensor's own level, we can subtract one mean from each column. This operation is called centering.

We begin with the raw table and the calibration from the preceding lessons:

[[ 2.  6.]
 [ 3.  8.]
 [ 4. 10.]]

Both arrays have shape (3, 2): axis 0 means observations and axis 1 means sensors. Centering should preserve that layout.

Find One Mean per Sensor

Each sensor occupies one column. We therefore average across the observation axis, axis=0:

[3. 8.]
(2,)

The reduction removes the observation axis. One result remains for each sensor, so the result has shape (2,).

We can check the first result by hand. The first sensor contains 2, 3, and 4, so its mean is

The second sensor contains 6, 8, and 10, whose mean is 8. The hand calculation confirms both the values and the axis choice. Using axis=1 instead would calculate one mean per observation, which answers a different question.

Subtract the Matching Mean from Each Column

The two means align with the two sensor columns. Broadcasting applies them to every observation row:

[[-1. -2.]
 [ 0.  0.]
 [ 1.  2.]]
(3, 2)

The first centered value is 2 - 3 = -1. The value beside it is 6 - 8 = -2. The middle observation equals the two means, so it becomes [0, 0]. The last observation lies above both means and becomes [1, 2].

The operation changes the values but not the axis meanings. Rows are still observations, columns are still sensors, and the result keeps shape (3, 2). The numbers now describe each reading's signed difference from its own sensor mean.

Keep an Axis When It Makes Alignment Easier to See

The shape (2,) broadcasts correctly across a (3, 2) table. Sometimes it is useful to keep the reduced observation axis as a length-one axis:

[[3. 8.]]
(1, 2)

Now the shape itself reads as one row by two sensors. The length-one observation axis can expand across the three observations during subtraction:

centered = calibrated - sensor_means

This produces the same (3, 2) values as before. keepdims=True is useful here because it makes the alignment visible. It is not a setting to add to every reduction. If the shorter shape (2,) is already clear, it is enough.

Check the Property Centering Should Create

After subtracting each sensor's mean, the mean of each centered column should be zero:

[0. 0.]
True

For this table, the result prints as exact zeros. With other floating-point values, a calculation that should be zero may leave a tiny value close to zero. np.allclose checks whether corresponding values are sufficiently near under a numerical tolerance. That makes it more suitable than exact equality for this floating-point property.

The check does more than repeat the code. It states the result we expect from centering: one near-zero mean per sensor. We should still check the result's shape and axis meaning, because near-zero numbers on the wrong axis would not answer the intended question.