18 formulas from statistics, machine learning and LLMs. Each in plain words, every symbol explained, and a worked example, most with numbers from our own tools. For the words around them, see the glossary.
Add up the values and divide by how many there are.
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Worked example
Five delivery times (made-up), 28, 35, 31, 42, 39 minutes: (28 + 35 + 31 + 42 + 39) ÷ 5 = 175 ÷ 5 = 35 minutes.
How far values typically sit from their mean: average the squared distances (dividing by n − 1), then take the square root.
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Worked example
The same times: distances from 35 are -7, 0, -4, 7, 4; their squares add to 130; 130 ÷ 4 = 32.5; √32.5 = 5.70 minutes.
How many standard deviations a value sits above (or below) the mean.
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Worked example
A 42-minute delivery, with mean 35 and standard deviation 5.70: (42 − 35) ÷ 5.70 = 1.23, about one and a quarter standard deviations slow.
How much a sample’s mean would wobble from sample to sample: the spread divided by the square root of the sample size.
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Worked example
One roll of a die has a spread of 1.71. Averages of 30 rolls: 1.71 ÷ √30 = 0.31, and the central limit theorem simulator’s 1,000 averages had a spread of 0.30.
An estimate, plus and minus about two standard errors: a range made by a method that catches the true value 95% of the time.
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Worked example
In the A/B test calculator’s example, B converts 4.52% and A 4.00%: a difference of 0.52 points, with a standard error of 0.286. So 0.52 ± 1.96 × 0.286 = -0.04 to +1.08 points, which includes zero.
Is B’s rate really different from A’s? The difference, divided by how much it would wobble if there were no difference.
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Worked example
400 of 10,000 against 452 of 10,000: z = 1.82, so the p-value, the chance of a gap this big if the versions were the same, is 0.069: not significant at 0.05.
A prediction as a starting value plus a fixed amount for each unit of the feature.
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Worked example
A made-up line, ŷ = 20 + 3x: at x = 4, ŷ = 20 + 3 × 4 = 32. With more features, each gets its own β.
The share of the target’s variation the model explains: 1 minus its squared errors over the squared distances from the mean.
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Worked example
The overfitting playground’s degree-3 curve, on its 200 new points: R² = 0.77. Degree 15’s wild curve scores far below zero: worse than predicting the mean every time.
The share of all predictions that are right.
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Worked example
The confusion matrix’s churn model at 0.5: (45 + 583) ÷ 1,000 = 62.8%. Saying nobody leaves scores 93.1%, which is why accuracy misleads when one class is rare.
Of everything flagged positive, the share that really is.
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Worked example
The churn model at 0.5 calls 393 customers and 45 of them were leaving: 45 ÷ (45 + 348) = 11.5%.
Of everything really positive, the share the model finds.
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Worked example
69 customers left and the model at 0.5 flagged 45 of them: 45 ÷ (45 + 24) = 65.2%.
Precision and recall in one number, their harmonic mean, which stays low if either is low.
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Worked example
With precision 0.115 and recall 0.652: 2 × 0.115 × 0.652 ÷ (0.115 + 0.652) = 0.19, much nearer the lower of the two.
How wrong predicted probabilities are; a confident wrong answer costs the most.
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Worked example
A customer who left (y = 1): predicted 0.8, the loss is −log 0.8 = 0.223; predicted 0.2, it’s −log 0.2 = 1.609, seven times as much.
Turns any list of scores into probabilities that add up to 1, giving the biggest score the biggest share.
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Worked example
Made-up scores 2, 1, 0.1 become 0.659, 0.242, 0.099. A language model does this over its whole vocabulary, 151,936 tokens for Qwen 3, at every step.
Divide every score by T first: below 1 sharpens the choice, above 1 flattens it.
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Worked example
The same scores, 2, 1, 0.1: at T = 0.5 they become 0.864, 0.117, 0.019; at T = 2, 0.502, 0.304, 0.194.
How alike two lists of numbers are by the angle between them: multiply them position by position, add up, divide by both lengths.
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Worked example
Two made-up vectors, [3, 4] and [4, 3]: (3×4 + 4×3) ÷ (5 × 5) = 0.96. bge-small-en-v1.5 rates “How do I get a refund?” and “Can I get my money back?” at 0.87 across its 384 numbers.
Score each token’s key against a query, scale, turn the scores into shares, and mix the values by those shares.
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Worked example
In lesson 3’s worked example, “bank” gives 37.0% of its mix to “river”, the most of any token.
A key and a value, for every layer, every key-value head and every number in a head, at so many bytes each, for every token.
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Worked example
Qwen 3 (8B), from its config: 2 × 36 × 8 × 128 × 2 = 1,47,456 bytes a token; for 32,768 tokens, about 4.8 GB.
Every formula starts as a sentence. Read that, and the symbols are just short names for its parts.
The worked examples use numbers from our own tools where there is one, so you can open the tool and see the same number.
Interviewers rarely want the derivation. They want you to say what each part does and why it’s there.
