Standard deviation in weather and climate

The city and all its numbers below are hypothetical, chosen to show the method. They do not describe any real place.

Normals, anomalies and standardised anomalies

Climate statistics start from a baseline. Meteorological services usually use a 30-year "climate normal"; the World Meteorological Organization's current standard period is 1991–2020. The normal gives the average for each place and time of year, and the same 30 years give a standard deviation.

A temperature anomaly is the observed value minus the normal. A standardised anomaly divides that by the standard deviation, which is simply a z-score:

anomaly = observed − normal standardised anomaly = (observed − normal) / SD

Suppose our city's July days have an average high of 24.0 °C with a standard deviation of 3.5 °C across all July days in the baseline, and the 30 July monthly averages have the same mean, 24.0 °C, but an SD of only 1.1 °C.

A hot day versus a hot month

The day is the bigger anomaly in degrees, 7.0 against 2.6, but the month is the rarer event. A day 2 SD above normal should turn up on about 2.3% of July days, roughly 0.7 days in an average July. A month 2.36 SD above normal should happen in about 0.9% of Julys, or about once in 110 years. The "share beyond" column comes from the normal distribution; real temperature data is only approximately normal, so treat the tail figures as rough.

Try it: raw score calculator

The calculator opens with its own defaults; enter a mean of 24.0, an SD of 1.1 and a z-score of 2 to find the 26.2 °C "two standard deviations above normal" month, or use your own station's normal and SD.

Open the full raw score calculator to turn any z-score or percentile back into a temperature.

A month exactly 2 SD above normal (26.2 °C here) has a 2.3% chance under the normal model, about once in 44 years. That is why a forecaster calling a month "two standard deviations above normal" is saying something strong: most people alive will see only a handful of such months in one place.

Records are a different measure. If every year's July came from the same distribution, the chance that a given July is the hottest of the last 30 is 1 in 30, whatever the SD. A record tells you a value is the highest in the list; the standardised anomaly tells you how far beyond the usual range it sits. A record can be a narrow one, 0.1 °C above the previous best, or a value several SDs above anything before it. The second is the more striking event, and only the SD can show the difference.

Why daily temperatures vary more than monthly averages

An average of many values varies less than the values themselves. If each July day were an independent draw with SD 3.5 °C, the standard deviation of a 31-day average would be

3.5 / √31 = 0.63 °C

That is the standard error of a mean, the same √n rule the standard deviation of the sample mean calculator uses. But our city's monthly SD is 1.1 °C, well above 0.63. The reason is that weather has memory. A heat wave lasts several days; a blocking pattern can hold for weeks. Consecutive days are correlated, so a month contains fewer independent pieces of information than 31.

You can put a number on it. If the monthly SD is 1.1 °C, the month behaves as if it held (3.5 ÷ 1.1)² = 10.1 independent days rather than 31. The exact figure differs from place to place and season to season, but the pattern holds in general: averaging reduces spread, and persistence limits how much.

Comparing places and seasons in SD units

Degrees are a poor yardstick across climates. A +2.6 °C month is exceptional where monthly averages barely move and routine where they swing widely. Dividing by the local SD puts them on one scale:

The tropical month is only 1.2 °C warm yet as unusual as our 2.6 °C July. The continental January matches that 2.6 °C but is a roughly one-in-seven event, because winter months there vary much more. This is also why maps of standardised anomalies look different from maps of raw anomalies: the raw map highlights high-variability regions, the standardised map highlights genuinely unusual ones. A reminder from the weather example on the use cases page applies here too: never divide a temperature SD by its mean, because the zero of the Celsius scale is arbitrary.

What a shifting mean does to the tails

Standard deviation also explains why a small warming of the average produces a large rise in extremes. Keep the SD fixed and move the mean up by one SD. A value that used to be +2 SD is now only +1 SD above the new mean:

The further into the tail, the bigger the multiplier. This is a property of the bell curve, not a forecast for any location; real shifts in mean and spread vary by place and season. The standard deviation graph lets you move a curve and watch the tail areas change.

One practical consequence: anomalies and SDs depend on the baseline. Moving from an older 30-year normal to a more recent one raises the normal in a warming climate, so the same observed month shows a smaller anomaly. When comparing anomaly figures from different sources, check the baseline period before comparing the numbers.