the glossary.

A float32 exponent runs from −126 to +127, and storing a signed number would mean either a sign bit of its own or two’s complement — both of which break ordering. Instead 127 is added first, so the whole range lands in 0…255 and the stored bits rise monotonically with the value. The payoff is that two positive floats can be compared as if they were plain integers, bit pattern against bit pattern, which is why hardware comparison is one instruction. The two ends of the stored range, all-zeros and all-ones, are reserved for the special values.

A standard deviation of 6 means nothing until you know whether the mean is 40 or 4,000. Dividing by the mean turns it into a percentage and makes it portable: the team in the essay below runs at about 15%, and that number can be set beside a team scoring on a completely different scale. It is also the term that carries the whole cost of an experiment — the required sample size grows with its *square*, so halving a team’s variability cuts the wait to a quarter.

The whole of the format is a convention: the bits are a plain integer, and the reader agrees where the point sits — Q16.16 means sixteen bits of whole part and sixteen of fraction, a step of 2⁻¹⁶ everywhere on the line. That uniformity is the point. Money is counted in fixed-point (or in integer cents) because a cent must weigh the same at ten and at ten million, and a format whose step grows with the value cannot promise that. What it gives up is range: the same bits that buy a constant step near zero run out early at the top.

Also called the significand. In IEEE 754 the mantissa is stored normalised, as a fraction after an implied leading 1, which is why a 23-bit float32 field buys 24 bits of precision: the first bit is never written down because it is never anything else. This is also why precision is a fixed *number of digits* rather than a fixed step size — the mantissa says how many digits, the exponent says where they sit. Doubling a value leaves the mantissa alone and adds one to the exponent, so the gap to the next representable number doubles with it.

It is a value, not an error: the operation returns, and the program keeps going. NaN is contagious — every arithmetic operation touching one produces another — so a single meaningless step at the top of a long calculation is still visible at the bottom rather than being quietly absorbed. It is also the one value not equal to itself, and x != x is the honest test for it, because equality with a thing that is not a number cannot be true. That inequality is a consequence of the definition, not a quirk, and it is why a NaN sorts strangely and why a lookup keyed on floats can lose an entry to one.

Its usefulness here is not that velocity is bell-shaped; it is that *averages* are, near enough, almost regardless of what they average. Two numbers then describe the whole picture, and distances read off in widths: about 68% of the bell lies within one width of the centre, 95% within 1.96 of them. That second number is where a detection threshold comes from, and it is the only reason a rule like "under 7 points, say nothing" can be derived rather than guessed.

Named by Lepper, Greene and Nisbett in 1973, who took preschoolers already choosing to draw in free play and split them three ways: promised a certificate before drawing, given one unexpectedly afterwards, or given nothing. Two weeks later only the promised group drew less, and drew worse — and since that group and the surprise group both ended up holding the same certificate, the reward is not what did the damage. The promise is. The name is precise in an uncomfortable way: the problem is not too few reasons but too many. Given an external reason good enough to explain the behaviour, the internal one becomes unnecessary, and what is unnecessary fades.

It is the square root of the sample variance, and it answers the question the variance cannot: how far off is a normal reading. For the team in the essay below it is 6.23 points against a mean of 40, which says that a sprint landing six points away from average is not an event — it is Tuesday. Everything downstream depends on it: the standard error, the detection threshold, and the number of sprints an experiment needs.

Squaring is what makes the deviations stop cancelling out, and it is also why the result is in squared units — points squared, which nothing in the world is measured in. That is the only real reason the standard deviation exists: it is the square root, taken to get back to the unit you started in. The n−1 is not a fudge; using n would understate the spread, because the sample mean sits closer to its own data than the true mean does.

A sprint is short and, crucially, always the same length, which is what makes the counts from different sprints comparable at all. It is also the sample size of every argument built on velocity: a quarter of six sprints is six data points, and six is a small number to reason from. Any question of the form "did we get faster?" is really a question about how many sprints have been observed.

The distinction from the standard deviation is the one people skip and then get wrong: the standard deviation describes how scattered the individual readings are, while the standard error describes how unreliable their average is. Averaging steadies things, but only as fast as √n — four times the data for half the wobble, which is why measuring a small effect gets expensive so quickly. It is the number that decides whether a difference between two averages means anything.

Story points are deliberately not a unit of time: the team compares a piece of work against work it has done before and assigns a number, so the scale is local to that team and means nothing across teams. The trouble starts when the number is lifted out of planning, where it does useful work, and into performance measurement, where it becomes a target the same people who assign it can move. Even setting that aside, a points total is a noisy measurement, which is the subject of the essay below.

Below the smallest normal value the exponent has nowhere left to go, and without subnormals the line would jump straight from that value to zero — a gap larger than the gap between any two neighbours above it. So the all-zero exponent is reserved: the leading 1 is no longer implied, and the mantissa is read as a plain fraction. The numbers reach nearer to zero, and pay for it, losing a bit of precision for every power of two they descend until nothing is left. The property this buys is that a − b == 0 if and only if a == b, which is not true in a format that flushes to zero.

You cannot drive the risk to zero, only choose it: setting it at 5% is what puts the threshold 1.96 widths out from the centre, and demanding 1% pushes the threshold further still. Every threshold is therefore a decision about how often you are willing to be fooled, made before the data arrives. Guarding against this error alone is not enough — pushing the threshold out to be safe makes the opposite mistake more likely.

The two errors trade against each other: a threshold set far out to avoid false alarms is exactly a threshold a real effect struggles to clear. An effect sitting precisely on the threshold is caught half the time, which is the uncomfortable fact that makes "we hit the number" so weak a statement. Deciding to catch a real effect 80% of the time — the usual choice — is what pushes the required difference well past the threshold, and with it the amount of data an honest answer needs.

ULP(v) = 2^(E−p): it depends on the exponent, which is to say on how large the value is. This is the single fact that makes floating-point behave the way it does. Precision is *relative* — the ratio ULP(v)/v is roughly 2^−p everywhere, so a float keeps about the same number of significant digits at 0.001 and at a billion — and it is the right unit for talking about error: "correct to within one ULP" is a claim about the format, while "correct to within 0.0001" is a claim that stops being true as the numbers grow.

Velocity is measured after the fact, not planned: it is whatever the team actually closed. Because it is a count over a fixed window it fluctuates for reasons that have nothing to do with capability — someone off sick, a story that turned out easier than it looked, a sprint that swallowed an incident. That fluctuation is the reason a quarter is a *series* and not a figure, and the reason any claim about velocity going up has to clear the noise before it means anything.