z-scores, the empirical rule, and Chebyshev

Position in SD units, comparing values across groups, what a bell shape buys you, and the coefficient of variation.

A value in the units of its own spread

A z-score says how many standard deviations a value sits from the mean:

$z = \frac{x - x ˉ}{s}$

Positive means above the mean, negative below; the size says how unusual. Because $z$ has no units, it lets you compare a value from one group with a value from another group that has a different mean and a different spread.

Worked example — which branch's slow account is worse?

Northfield Credit wants to know which of two overdue accounts is the more unusual for its branch.

Branch A — this account took 52 days to collect; the branch's collection times average 45 days with SD 5. $z_{A} = (52 - 45) /5 = 1.40$ .
Branch B — this account took 60 days; the branch averages 50 with SD 8. $z_{B} = (60 - 50) /8 = 1.25$ .

B's account is slower in days, but A's is the more unusual for its branch: 1.40 SDs above its mean against 1.25. Relative position is what $z$ measures, and the larger $z$ is the more extreme one. The same logic ranks two negative z-scores: the one closer to zero (the less negative) is the better relative position, because it is nearer its own mean.

What a bell shape buys you — the empirical rule

If a distribution is roughly bell-shaped (symmetric, unimodal, tails that thin out), then approximately

68 % of values lie within 1 SD of the mean,
95 % within 2 SD,
99.7 % within 3 SD.

For Branch A, the 95 % band is $45 \pm 2 \times 5$ = 35 to 55 days. An account at 52 days is inside it; one at 60 days ( $z = 3$ ) would be outside 99.7 % of the branch's collections — worth a phone call.

Any shape at all — Chebyshev

Without a bell, the empirical rule is not available. Chebyshev's inequality holds for every distribution: the fraction of values within $k$ SDs of the mean is at least $1 - 1/ k^{2}$ . For $k = 2$ that is at least 75 %; for $k = 3$ at least 88.9 %. It is a floor, not an estimate — much weaker than the empirical rule's 95 %, which is the price of assuming nothing about shape.

Spread relative to size — the coefficient of variation

An SD of 5.29 hours is large for a firm averaging 20 hours a month and small for one averaging 400. The coefficient of variation $C V = s / \overset{x}{ˉ}$ removes the scale: for Bramble & Voss (L3.3), $C V = 5.2915/44 = 0.1203$ , about 12 %. Compare CVs when two series are in different units or on different scales — dollar returns against percentage returns, a small branch against a large one.