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z-score (standard score)

How many standard deviations a value is from the mean: (x - mean) / SD.

A z-score translates a raw value into "how many standard deviations away from the mean is this?". The formula is

z=x−μσz = \frac{x - \mu}{\sigma}

where xx is the value, μ\mu the mean and σ\sigma the standard deviation. With sample data you use xˉ\bar{x} and ss instead. A z-score of 0 is exactly average, +2+2 is two SDs above, −1-1 is one SD below. The units cancel out, so an order value in dollars and a weekly order count become comparable numbers on one scale. That is why z-scores are also used for standardisation before clustering or regression.

The second use is reading a value against the normal curve. If the variable is roughly normal, the z-score tells you what share of values lie below it via the standard normal table (or NORM.S.DIST(z, TRUE) in a spreadsheet). A few anchors worth remembering:

z
Share below
Share above
−1
15.9%
84.1%
0
50%
50%
1
84.1%
15.9%
1.96
97.5%
2.5%
2
97.7%
2.3%

Many test statistics are z-scores in disguise. The two-proportion z-test in an A/B test takes the observed difference in conversion rates and divides it by its standard error: it asks how many standard errors the result is from "no difference". That is why ∣z∣>1.96|z| > 1.96 corresponds to p<0.05p < 0.05 in a two-tailed test.

One caveat: turning a z-score into a percentile only works when the data are close to normal. For skewed metrics such as spend per customer, a z-score still measures distance in SD units, but the table's percentages will be off.

Example

Reading one order. Suppose CatChow's order values are roughly normal with \mu = \50andand\sigma = $12$.

  • An order of $74: z=(74−50)/12=24/12=2.0z = (74 - 50) / 12 = 24 / 12 = 2.0. The table gives 97.7% below z=2z = 2, so only about 2.3% of orders are this large or larger.
  • An order of $41: z=(41−50)/12=−9/12=−0.75z = (41 - 50) / 12 = -9 / 12 = -0.75. The table gives about 22.7% below, so roughly one order in four or five is smaller.

Comparing apples and kibble. Last week the growth team saw two jumps and asked which one is more unusual:

  • Orders: 1,360 against a usual weekly mean of 1,200 with SD 80, so z=(1,360−1,200)/80=160/80=2.0z = (1{,}360 - 1{,}200) / 80 = 160 / 80 = 2.0.
  • Average order value: $53 against a usual $50 with SD $2, so z=(53−50)/2=3/2=1.5z = (53 - 50) / 2 = 3 / 2 = 1.5.

In raw terms "+160 orders" and "+$3" can't be compared. In z-scores the order count is further from normal: a week this high happens about 2.3% of the time, versus about 6.7% for the AOV jump. The order spike is the one to investigate first.

Common mistakes

  • Reading percentiles off the normal table for skewed data. A z of 2 on 90-day spend does not mean "top 2.3%"; with a long right tail the real share can be quite different.
  • Mixing up SD and standard error in the denominator. For a single value divide by the SD; for a sample mean or a difference in an A/B test divide by the standard error.
  • Letting outliers set the scale. One huge order inflates both the mean and the SD, so other unusual values end up with modest z-scores (masking). Robust rules such as the IQR fences are safer for flagging outliers.
  • Treating |z| > 2 as "something happened". If you scan 50 dashboard metrics every week, a couple will cross 2 by chance alone.
  • Forgetting the sign. A z of −2 is just as unusual as +2, only in the other direction; decide in advance whether you care about both tails.

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