Did you know that two equal-length sound readings of 60 and 70 decibels do not have a true energy-equivalent average of 65 decibels? A pupil may add them and divide by two as if they were temperatures or quiz marks. The arithmetic is easy, but the result answers a different question from the one a noise study usually asks.
A decibel (dB) is a logarithmic expression of a ratio, often used to describe sound levels. A time-equivalent sound level, commonly written Leq, represents the constant level with the same total sound energy over the measurement period under the relevant weighting and measurement conventions. To combine or energy-average sound levels, convert the decibel values back to their linear energy ratios, average those with the correct durations, and convert back to decibels. Do not simply take their ordinary arithmetic mean.
The difference starts with the scale, not with the calculator
Every increase of 10 dB represents a tenfold increase in the corresponding sound-energy or intensity ratio under the usual logarithmic definition. That does not mean the human ear experiences exactly ten times the loudness. Perceived loudness, sound energy, sound pressure and safety exposure are related but distinct quantities.
The US Occupational Safety and Health Administration’s technical manual on noise explains that decibels are logarithmic and that decibel readings should not be combined by ordinary arithmetic addition. Its combination and averaging guidance provides the physical basis for the school example below. The example illustrates arithmetic on measured levels; it is not a workplace exposure compliance assessment.
A fully worked example: ten minutes at each level
Invented school corridor project: A group uses a suitable instrument to record a steady A-weighted sound level of 60 dBA during one 10-minute interval and 70 dBA during the next 10-minute interval. Assume the measurements are comparable, the levels are constant within each interval, the A-weighting and measurement basis are the same, and each interval lasts exactly 10 minutes. What constant energy-equivalent level represents the full 20 minutes?
First express each level on the linear energy-ratio scale. For 60 dBA, the corresponding ratio is 1060/10 = 1,000,000. For 70 dBA it is 1070/10 = 10,000,000. Because the intervals are of equal duration, average those ratios: (1,000,000 + 10,000,000) ÷ 2 = 5,500,000. Now convert back: 10 × log10(5,500,000) ≈ 67.4 dBA.
A basic reasonableness check follows. The result must lie between 60 and 70 dBA because it summarises two positive equal-duration intervals. It should lie closer to 70 than to 60 because the 70 dBA interval carried ten times the sound-energy ratio of the 60 dBA interval. The ordinary arithmetic mean, 65 dBA, does not reflect that energy balance. The result is about 67.4 dBA, not 65 dBA and certainly not 130 dBA.
Why adding 60 and 70 is another mistake
Sound levels cannot be combined by claiming 60 + 70 = 130 dB. Even two equal independent sources each at 60 dB, measured at the same receiving position under assumptions that permit sound intensities to add, would produce roughly 63 dB in total rather than 120 dB. An extra 3 dB is approximately a doubling of acoustic energy, not a simple doubling of the decibel number. The exact result depends on assumptions about sources, measurement conditions and coherence.
This is a key mathematical transfer: the visible numbers are logarithms. To combine the underlying physical quantities, work with those physical ratios before returning to the logarithmic display. Just because a scale displays 60 and 70 does not mean its values behave like counts of chairs or litres of water.
Different durations change the answer
Now keep the same two sound levels but change the invented time record. Suppose the corridor remained at 60 dBA for 15 minutes and 70 dBA for 5 minutes. The second level still contributes more energy per minute, but it now occurs for less time. A fair energy-equivalent calculation weights each ratio by duration.
Compute the duration-weighted ratio: [(15 × 1,000,000) + (5 × 10,000,000)] ÷ (15 + 5) = (15,000,000 + 50,000,000) ÷ 20 = 3,250,000. Convert back: 10 × log10(3,250,000) ≈ 65.1 dBA. The shorter high-level interval produces a lower overall equivalent level than the equal-duration case, about 67.4 dBA. The answer is not the ordinary average of the two displayed readings.
These calculations are valid for the simplified constant-level intervals and a consistent equivalent-energy basis. Real sound-level meters may record continuously varying readings and compute an LAeq,T directly. A device configured to calculate a different metric, using thresholds or an exchange-rate policy, may not produce the same displayed number. Read the instrument’s labels and settings before comparing its output with the worked example.
A five-question diagnosis before students start computing
- What is the quantity? dB, dBA, peak level, LAeq,T and an occupational time-weighted exposure are not automatically interchangeable.
- Are the intervals equally long? If not, a simple equal-weight average of ratios is inappropriate.
- Do the readings share a consistent basis? A-weighted and unweighted sound-pressure readings should not be casually mixed.
- Is the instrument reliable for this purpose? A phone app may lack calibration or sufficient documentation for quantitative sound-level claims.
- What is the question? A rough comparison, an energy-equivalent summary and a legal exposure assessment demand different methods.
A student who knows the logarithmic formula but forgets the time weighting needs a duration lesson. A student who multiplies dB by minutes and divides by total minutes needs a units-and-scale lesson. A student who uses readings from two different uncalibrated apps needs a measurement-quality lesson. Each error suggests a different repair.
Progressive practice that makes the idea stick
Stage 1: practise ratios without a calculator
Ask: if a sound-energy level increases from 50 to 60 dB on the same basis, what happens to the energy ratio? It increases tenfold. If it rises from 60 to 70 dB, it increases tenfold again. After these comparisons, pupils should recognise why ordinary averaging underweights the stronger interval.
Stage 2: repeat the checked 60/70 example
Provide the linear values 1,000,000 and 10,000,000 on a worksheet. Let the learner average them first and use the logarithm step only after explaining the units. If a calculator is available, verify 10 × log10(5,500,000) ≈ 67.4. Ask for a sentence explaining why the result is closer to 70.
Stage 3: change only duration
Use the 15-minute and 5-minute intervals above. Without calculating, ask whether the overall level should be higher or lower than 67.4 dBA. Then compute 65.1 dBA and explain the direction of change. This tests reasoning rather than formula copying.
Stage 4: an independent fresh example
Give two equal-duration readings of 50 and 60 dBA, and ask for the equivalent-energy average. Ratios are 100,000 and 1,000,000; average = 550,000; 10 × log10(550,000) ≈ 57.4 dBA. The 10 dB separation is the same as before, so the energy-average position relative to the endpoints is also the same. Students can use this pattern as a self-check without pretending that every noise scenario is identical.
Stage 5: recognise when the answer cannot be computed honestly
Show a project sheet with readings ’60 dB for a while’ and ’70 dBA later’, no durations and no instrument information. The correct response is not to invent two equal intervals. The learner should list the missing duration and measurement-basis details, then request a consistent data record. Accurate analysis includes knowing when a numerical answer is unsupported.
A school survey should measure sound, not private conversations
If pupils survey school noise, use an appropriate sound-level measurement method and obtain school permission. They do not need to record other people’s speech to estimate sound levels. This is why the separate Core Aim guide to corridor noise surveys without recording conversations is a useful companion. Measuring aggregate sound levels and collecting identifiable voice recordings are different activities.
Also choose observation periods carefully. A reading during assembly and one from a quiet classroom answer different contextual questions. Do not present two selected intervals as if they were a representative full school day. For a better project, record times, location, activity, instrument settings, and the limitations of the sample.
How parents and teachers can recognise real progress
A teacher can ask students to solve one equal-duration example, then an unequal-duration example, then inspect a flawed worksheet that averaged dB directly. The learner should be able to explain the logarithmic scale, identify the missing weights and show the calculations with appropriate units. A parent can ask a child, ‘Why is 67.4 closer to 70?’ and listen for the tenfold-energy explanation rather than a memorised answer.
The observable milestone is an independent check: the student states the metric and assumptions, obtains a result between the endpoints when appropriate, and refuses an unsupported average when key information is absent. Following the Clementi Mathematics reference’s diagnosis-first, guided-to-independent learning approach, correct the misconception before assigning large question sets. The broader eduKate Education learning hub links this habit to other evidence-based learning skills.
What the formula does not promise
An estimated average from a classroom exercise cannot determine whether a space is safe for hearing, whether a school meets an occupational standard, or whether every listener had the same exposure. Those decisions require appropriate measurement protocols, frequency and time weighting, instrument calibration, exposure criteria and professional interpretation. Do not use this article as medical or occupational-safety advice.
The simplified dBA examples assume measurement consistency. In acoustics, adding distinct simultaneous sources has additional caveats; averaging readings over time is not the same operation as estimating how many people talking at once affect sound at a microphone. Keep the physical question explicit.
Frequently asked questions
Why is 60 and 70 dB not simply 65 dB on average?
The numerical midpoint is 65, but a decibel is a logarithm of a ratio. Equal time at 70 represents more energy than equal time at 60, so an energy-equivalent summary is about 67.4 dB under the stated assumptions.
Can I add 60 dB and 60 dB to get 120 dB?
No. Under appropriate independent-source assumptions, doubling equal sound energy raises the level by about 3 dB, not by adding the displayed numbers.
What if my meter already shows LAeq?
Use the meter’s documented metric and interval settings. Do not replace a properly calculated energy-equivalent result with an arithmetic average of snapshots.
Do these numbers prove that a classroom is too noisy?
No. A safety or educational judgement needs context, representative calibrated measurements and the applicable criteria. This lesson is about interpreting the scale correctly.
The core aim
The real achievement is not mastering one exotic formula. It is recognising that numbers inherit meaning from the scale and measurement process that produced them. A student who asks what a unit represents before combining it can carry that judgement into science, health, engineering and the everyday claims made by charts and devices.