# Fraction truncation test

Minimal truncation of Fahrenheit temperature recordings would cause between .1C and .2C artificial cooling of temperatures before 1972.

It is known that at least 31% and probably about 50% of all temperature recordings in the pre 1972 Fahrenheit era were whole .0F.

This high proportion of rounding will average to insignificance if there is a 50/50 share of rounding up and rounding down. However, it is more likely that there will be some truncation of degrees by weather station observers.

The BoM protocol is to round .5 to the odd number but if observers weren't aware of or bothered with the bureau's instruction to log precise fractions, it is unlikely they would be aware of or observe this .5 odd number protocol. Many people believe, for example, that 61.5 rounds to 61 and 62.5 rounds to 62 - always down. "Bankers rounding" is known globally as rounding to the even number, American weather station observers are instructed to round to the even and the BoM .5 protocol is not taught in Australian schools.

This page examines different likely scenarios that would induce truncation of Fahrenheit temperature recordings and artificially lower temperatures before 1972.

The scenarios consider the influence of an ordered and random number of roundings if observers rounded down from .5. A truncation possibility is also examined that assumes the fraction above .5 most likely to be truncated down to .0 is .6.

Rounding down from .6 might be deliberate or accidental because of the very small scale of the thermometer, poor eyesight or adverse weather conditions causing observers to think .6 is .5 and mistakenly truncate to .0. Note that .6 and .7 at 4.74% and 4.25% were the least frequent fractions recorded in Australia's total Fahrenheit HQ temperature history.

Based on a 50% inferred estimate of .0F rounding among all Australian Fahrenheit temperatures before 1972, below is a range of temperatures in a hypothetical 30 day month with 10% fractional distribution compared to 50% rounded correctly, 50% rounded correctly including/plus rounding down of .5, and 50% rounded correctly including/plus .5 and .6.

The hypothetical can be extended into a warmer month:

The truncation of .5 and .6 can be emulated in a more realistic scenario such as all daily minima at Perth Regional Office 9034 during 1954. Perth Regional Office was one of the most accurate weather stations in WA during the Fahrenheit era with just 17.1% of all recorded minimum temperatures at .0F from 1897 to 1992.

This month has been randomly chosen simply to provide an array of temperatures that might normally be recorded, albeit following their conversion to Celsius and irrelevant to actual degrees that either occurred or were recorded at Perth Regional Office in 1954.

The effect of truncating can be tested randomly by simply rounding down all days that happened to be recorded as .0F. This is a lottery wherein the number and amount of truncation is left to chance but reflects the number of times the observer originally recorded .0F during the month, either correctly or due to rounding/truncation. The particular Celsius degree that converts back to a .0 Fahrenheit degree is truncated regardless of the size of the C fraction. This is not intended as a proper statistical analysis but simply to demonstrate what might happen if an utterly random amount of truncation was experienced.

The effect can also be gauged of truncating .5 and .6 in this scenario, in a scenario where 50% of all temperatures are rounded as occurred in Australia before 1972, and in a scenario where 100% of all daily temperatures were rounded up or down to .0F.

Annual average monthly temperatures in different scenarios at Perth Regional Office 9034 in 1954:

This analysis suggests:

• .1 to .2 degree reduction of the annual mean if .5, .6 and .0F coincidental degrees are truncated at random
• no change if either 50% or 100% of degrees are rounded correctly and exclusively
• .1 cooler if 50% are rounded including .5 truncation
• .2 cooler if both .5 and .6 are truncated

It is impossible to know the exact ratio of truncation among thousands of weather station observers around Australia before 1972 but likely trends can be estimated. Truncation of .5 or .6 would not be uniform and a national estimation of this bias would require a professional audit.

However, Perth Regional Office maintained more accurate temperature records than most other weather stations and it would not be unreasonable to expect similar or worse trends in less accurate rural stations.

The truncation scenarios presented might also be influenced by a small proportion of observers occasionally or frequently truncating down to .0 all Fahrenheit fractions within a degree up to .9, simply because they believed it was more accurate to round to the first number than log a degree they thought was too high. All scenarios would be more likely if day and night thermometer reading duties were shared among different people at the same weather station.

Monthly breakdown:

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