How to calculate an 8-hour TWA, and where the math goes wrong

This page explains what the rules and common practice call for. Every site is different, and what a survey or a filing needs at your facility is something we work out with you after we see it.

The math is simple enough to do on the back of an envelope. What goes wrong is almost never the arithmetic but the number you divide by.

The formula

To calculate an 8-hour time-weighted average (TWA), multiply each sample result by the time it covers, add those products together, and divide by 8 hours.

8-hour TWA = Σ (C × T) ÷ 8

Here Σ means the sum across all samples, C is the concentration measured on each sample, and T is how long that sample ran, in hours.

The top part of the formula is a dose, expressed in concentration-hours. Dividing by 8 converts that dose into the single steady concentration that would deliver the same dose over a full 8-hour shift.

Note what you divide by: 8, not the length of the shift. OSHA does it this way for its general air contaminants table (Table Z) and for most of its substance-specific standards. If the shift is 6 hours, the other 2 hours count as zero exposure and you still divide by 8.

That convention has a trap in it, which the second example below demonstrates.

Worked example 1: a standard 8-hour shift

Three consecutive samples on one worker exposed to lead cover the full shift with no gaps. Results are in micrograms per cubic meter of air (µg/m³), and the last column is each result multiplied by its hours.

Dividing the total by 8 gives the 8-hour TWA: 390.0 ÷ 8 = 48.75 µg/m³.

The permissible exposure limit (PEL) for lead is 50 µg/m³, so this result is under the PEL, by 2.5%. But lead also has an action level of 30 µg/m³. This result is well over it, which triggers periodic monitoring at least every 6 months, and training.

Medical surveillance comes with one more condition that is easy to miss. The lead standard applies it to workers exposed at or above the action level on more than 30 days in any consecutive 12 months, so exceeding it once is not enough. The standards for silica, hexavalent chromium, cadmium and benzene set the threshold at 30 or more days a year. So the result above puts this worker into periodic monitoring and training right away. Whether it puts them into medical surveillance depends on how many days a year the job looks like this one.

A result under the PEL does not end your obligations, so read a passing number for what it still requires.

→ Lead exposure monitoring

Worked example 2: a 12-hour shift

The operation is the same but the shift is longer, with three samples covering 12 hours:

The common mistake is to divide by the length of the shift:

459.0 ÷ 12 = 38.25 µg/m³. "Below 50, we're fine."

That is wrong on two counts, and the two correct ways to do the calculation show why.

Method A: the 8-hour TWA against the standard PEL. Dividing the total by 8 gives 459.0 ÷ 8 = 57.4 µg/m³, which is over the 50 µg/m³ PEL.

Method B: the daily average against lead's reduced limit. The lead standard lowers the PEL on longer shifts with a formula: maximum permissible limit = 400 ÷ hours worked. For 12 hours that is 400 ÷ 12 = 33.3 µg/m³. The daily average is 459.0 ÷ 12 = 38.25 µg/m³, which is over that reduced limit.

Both methods agree, and they agree exactly, as you can see by dividing each result by its limit: 57.4/50 = 1.147, and 38.25/33.3 = 1.147. They have to agree, because in mathematical terms, lead's formula is a cap on the daily dose. The cap is 50 µg/m³ × 8 h = 400 µg·h/m³, and this worker received 459.

What you cannot do is mix the two methods by dividing by 12 and then comparing to 50. That single step turns a result 15% over the limit into an apparent pass.

Most substances in Table Z have no adjustment formula for longer shifts at all, so for those you divide by 8 and compare to the PEL. Recommended limits from outside organizations are a different matter. ACGIH, the professional body that publishes the threshold limit values (TLVs), also publishes its own guidance on adjusting for shift length. Where a plant works longer shifts, the adjusted comparison is worth running for information.

→ OSHA PELs compared with recommended limits

For lead, full shift means 7 continuous hours

The lead standard is more specific than most, requiring full-shift personal samples of at least 7 continuous hours. Sampling for 2 hours and extrapolating the result across the shift does not meet it, however good the estimate is.

Read that alongside example 2, where a short sample taken during the S1 period and extrapolated over 12 hours would have given a wildly different number. It could be too high or too low, depending on which hour you picked.

An 8-hour TWA cannot show you meet a short-term limit

A short-term exposure limit (STEL) is a 15-minute average, while a PEL is an 8-hour average. They are different measurements that answer different questions, and the full-shift sample cannot stand in for the short-term one. Averaging over 8 hours is exactly what erases a 15-minute spike.

Consider a solvent transfer that puts 600 parts per million (ppm) in the air for 15 minutes, which is 0.25 hours, while the other 7.75 hours run at 5 ppm. Multiply each concentration by its time in hours, add the two products, then divide by 8:

• 0.25 h × 600 ppm = 150.0
• 7.75 h × 5 ppm = 38.75
• Total dose = 188.75 ppm·h
• 8-hour TWA = 188.75 ÷ 8 = 23.6 ppm

The 8-hour TWA is 23.6 ppm, and the 15-minute exposure was 600 ppm. If the substance has a STEL anywhere below 600, that worker is over the limit and the full-shift sample will never show it.

If a substance has a STEL or a ceiling limit, it needs its own sample, timed to the task. A ceiling limit is a concentration that must not be exceeded at any time. Timing the sample means knowing in advance when the task happens, which is one more reason field notes written on the day matter.

Noise does not use this math

Noise dose under OSHA's noise standard is calculated on a logarithmic scale, with a 5-decibel exchange rate, not with the straight multiply-and-add arithmetic above. A noise dosimeter result is a different kind of number, and none of the math on this page carries over to it.

→ Noise dosimetry · → Noise surveys · → Similar exposure groups