A score of 130 where the mean is 100 and the standard deviation is 15 sits exactly two standard deviations above average. That converts to the 97.7th percentile, with 2.28 per cent of the distribution above it and a two tailed p-value of 0.0455.
Change the standard deviation to 30 and the same 130 becomes one standard deviation out, which is ordinary. The raw gap did not move; the context did.
What the standard deviation is doing
The standard deviation is the typical distance from the mean, in the same units as the data. It converts an absolute gap into a relative one, which is what makes values from different scales comparable.
A z-score of 2 means the same thing whether the underlying numbers are exam marks, blood pressures or rainfall totals. That is its whole purpose: strip the units, keep the unusualness.
The percentages worth remembering
For data that is roughly bell shaped, about 68 per cent falls within one standard deviation of the mean, about 95 per cent within two and about 99.7 per cent within three. So two standard deviations out is around a 1 in 40 event on one side, and three is rare enough that it usually prompts a question about the data rather than about the world.
Where it stops working
Those percentages assume a roughly symmetric, bell shaped distribution. Applied to strongly skewed data, they mislead badly in one direction, because a long tail puts far more of the distribution beyond two standard deviations on one side than the normal figures suggest.
Income is the standard example. Its mean sits well above its median, and a z-score treats the tail as if it were symmetric when it is nothing of the kind. Checking the mean against the median first, as in the piece on averages, is the fastest way to know whether z-scores are appropriate at all.
Unusual is not the same as important
A small p-value says a result would be uncommon if nothing were going on. It does not say the effect is large, or that it matters, or that the study measured what it claimed to. With a big enough sample, trivial differences become statistically detectable, which is why effect size deserves as much attention as significance.
The z-score calculator converts a value into its standard deviation distance and percentile, the descriptive statistics calculator supplies the mean and standard deviation to feed it, and the percentile rank calculator answers the ranking question directly when the data is not bell shaped.