Cosine Similarity Calculator

Pearson is cosine after centring, and nothing else

They are taught in different courses and treated as unrelated. The formulas are identical up to one subtraction, and the tool shows all three quantities so the relationship is visible rather than stated.

Centre both vectors, take the cosine, and you have Pearson’s r. The dot product of centred vectors is the covariance times n; their norms are the standard deviations times √n; the n cancels.

On the shipped preset the raw cosine is 0.99689501 and Pearson 0.98414369. The centred cosine is 0.98414369 — identical to Pearson to the last digit, which is what makes the identity a check rather than a claim.

And the two can point in opposite directions. [1, 0, 1, 0] against [0, 1, 0, 1] has a cosine of exactly 0 — orthogonal, sharing nothing — and a Pearson of exactly −1, perfectly anti-correlated. Both are correct answers to different questions.

The choice is about what zero means. If zero means “none” — a word not present, an item not bought — centring invents a negative quantity that does not exist, and cosine is right. If zero is just a point on a scale — a temperature, a standardised score — Pearson is right.

Scaling is free, and that is the point

The invariance is exact rather than approximate, and the tool demonstrates it by scaling one vector a thousandfold and reporting the result.

Multiplying either vector by any positive number leaves the cosine identical. The scale factor appears in the dot product and in the norm and cancels exactly.

Which is why cosine is the default for text. A long document and a short one on the same topic have similar word PROPORTIONS and very different counts, and cosine sees only the proportions.

The second preset makes it concrete: [1, 2, 3, 4] and [10, 20, 30, 40] have a cosine of exactly 1 and a Euclidean distance of 49.2950. Identical by one measure, far apart by the other.

So when magnitude matters, cosine is the wrong tool. Comparing spend levels, absolute quantities or intensities needs Euclidean distance or a normalised difference — cosine will report two customers with the same purchase MIX as identical however differently they spend.

The range depends on whether the data can be negative

The textbook range is −1 to 1. Most real applications can only reach half of it, and reading a score without knowing which case you are in is a common error.

For non-negative vectors the cosine cannot go below 0. Word counts, ratings on a positive scale and purchase quantities all live in the non-negative orthant, where the largest possible angle is 90°.

So “0.3” on text is low but not negative-low, and the effective range being 0 to 1 rather than −1 to 1 changes how a threshold should be set.

Embeddings from a neural model do use the full range, since their coordinates can be negative — and a cosine of −0.4 between two embeddings is genuinely meaningful in a way it could never be between two term-frequency vectors.

The angle is often the more readable form. 0.99689501 sounds like near-identity; 4.5163° apart says the same thing in a unit people can picture.

On sparse data it is fast, and centring would destroy that

The practical reason cosine dominates text and recommendation work is computational as much as statistical.

The dot product only touches coordinates where BOTH vectors are non-zero. Two documents sharing twelve terms out of a 100,000-term vocabulary cost twelve multiplications.

Centring fills the vectors in. Subtracting a mean makes every zero into a non-zero, so a sparse vector becomes dense and the computation becomes 100,000 multiplications instead of twelve.

Which is why Pearson is rarely used on high-dimensional sparse data, even where it would be the more appropriate measure in principle.

Normalising the vectors once, in advance, makes it cheaper still: with unit-length vectors the cosine is just the dot product, which is why vector databases store embeddings pre-normalised.

What goes into the vector matters more than the measure

Most disappointing cosine results come from the vector construction rather than from the coefficient.

Raw counts let common words dominate. Two documents both full of “the” and “of” score high on nothing of interest, which is what TF-IDF weighting exists to fix.

Unscaled features do the same to numeric data. A coordinate measured in thousands swamps one measured in units, so the cosine becomes a statement about that one coordinate.

And the dimension order must match. The two vectors are paired coordinate by coordinate, so a misaligned vocabulary produces a meaningless number rather than an error.

For set-valued data, Jaccard is usually the better model, since it asks which items are shared rather than whether two count profiles point the same way.

Reporting a similarity

Three things, and the second is what makes a threshold meaningful.

Say whether it is cosine or a correlation. They differ by centring and can reach opposite verdicts, so the label is not decoration.

Say whether the data can be negative. It sets the achievable range, and a threshold of 0.3 means very different things on 0-to-1 and −1-to-1 scales.

And say how the vectors were built. Raw counts, TF-IDF, normalised features or embeddings give different scores on the same underlying objects, and the construction is where nearly all the judgement lives.

Sources and methodology

References for cosine similarity and its relatives.

• Salton, Wong & Yang, “A Vector Space Model for Automatic Indexing”, Communications of the ACM 18(11), 1975
• Manning, Raghavan & Schütze, Introduction to Information Retrieval, Cambridge, 2008
• Pearson, “On Lines and Planes of Closest Fit”, Philosophical Magazine 2(11), 1901
• Correlation Coefficient Calculator — the centred version of this formula
• Jaccard Index Calculator — for sets rather than count vectors
• How we calculate — methodology

Method. The cosine, Pearson’s r and the cosine of the mean-centred vectors are all computed from the same pass over the data, which is what makes the identity between the last two a per-input check rather than a claim made once — they agree bit-for-bit on every input. The scale-invariance demonstration is computed rather than asserted: the engine builds a copy of the second vector multiplied by a thousand and reports the cosine of that, which comes back identical to the original on every preset. The angle is taken from the cosine after clamping to [−1, 1], so floating-point drift at exactly parallel or antiparallel inputs cannot produce a NaN. The Euclidean distance is reported alongside precisely because cosine discards it. Fewer than two paired values, an all-zero vector and a non-finite coordinate all return no result. That engine is verified on every change against 100 assertions. The count and the per-case breakdown are published on the formula verification page.

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