Adding decibels: the formula and a calculator in Excel

Why you cannot add decibels directly

A decibel value is a logarithm: it compares a sound with a reference on a log scale. Logarithms do not add like ordinary numbers, so two 85 dB machines are not 170 dB. What does add up is the sound energy (intensity) of independent sources. Convert each level to energy, add, then go back to decibels.

This holds for incoherent sources, such as different machines, voices or vehicles. Two loudspeakers playing the same signal in phase can add up to 6 dB, but that is a special case. For noise from separate sources, energy addition is the rule.

The decibel addition formula in Excel

Total dB = 10 × LOG10(10^(L1/10) + 10^(L2/10) + … + 10^(Ln/10))

• Convert each level in a helper column: =10^(B2/10). The bench grinder at 91 dB has an energy of about 1.26 billion relative units.
• Add the energies and convert back: =10*LOG10(SUM(C2:C7)) = 94.2 dB.
• Without a helper column: =10*LOG10(SUMPRODUCT(10^(B2:B7/10))) gives the same result in one cell.

The parentheses matter: 10^B2/10 raises 10 to the power 91 and then divides by 10.

Worked example: adding six machine noise levels

Six fictional workshop machines, each measured alone at the operator's position (yellow inputs). Column C converts each level into energy. Rows 9 to 13 add them correctly, show two wrong methods and switch off the grinder. Rows 15 to 17 check the rules of thumb, and rows 19 to 21 subtract background noise from a measurement.

Worked example: adding six machine noise levels

	A	B	C
1	Source	Level (dB)	Energy 10^(L/10)
2	Lathe	84.0	251,188,643
3	Air compressor	82.0	158,489,319
4	Bench grinder	91.0	1,258,925,412
5	Band saw	88.0	630,957,344
6	Extraction fan	76.0	39,810,717
7	Hydraulic press	85.0	316,227,766
8
9	Total of all six sources		94.2
10	Wrong: plain sum of the dB values		506.0
11	Wrong: average of the dB values		84.3
12	Total without the bench grinder		91.5
13	Drop when the grinder is switched off		2.8
14
15	Two equal sources: 85 dB + 85 dB		88.0
16	10 dB apart: 85 dB + 75 dB		85.4
17	Four equal sources of 80 dB: L + 10·log n		86.0
18
19	Measured with the machine running		87.0
20	Background, machine off		82.0
21	Machine alone (background subtracted)		85.3

Fictional levels. The formula assumes incoherent sources, which holds for different machines. Educational only: exposure limits vary, so check the noise rules that apply where you work.

Rules of thumb for adding two levels

The increase depends only on the difference between the two levels. Add this amount to the louder one:

• 0 dB apart: +3.0 dB (85 + 85 = 88.0)
• 1 dB apart: +2.5 dB
• 2 dB apart: +2.1 dB
• 3 dB apart: +1.8 dB
• 4 dB apart: +1.5 dB
• 6 dB apart: +1.0 dB
• 10 dB apart: +0.4 dB (85 + 75 = 85.4)

For n equal sources the total is L + 10 × log10(n): four machines of 80 dB give 86.0 dB, and doubling the number of equal sources always adds 3 dB.

Which source to fix first

Because the scale is logarithmic, the loudest source dominates. Remove the energy of the 91 dB grinder and recalculate: =10*LOG10(SUM(C2:C7)-C4) = 91.5 dB, a drop of 2.8 dB from one machine. Removing the 76 dB extraction fan instead changes the total by less than 0.1 dB. Noise control starts with the loudest machine, not with the most numerous ones.

Subtracting background noise

The same idea works backwards. Measure with the machine running, switch it off, measure the background, and subtract the energies: =10*LOG10(10^(C19/10)-10^(C20/10)). A reading of 87 dB with a background of 82 dB means the machine alone produces 85.3 dB.

When the two readings differ by less than about 3 dB the result is unreliable, so measure closer to the machine. When they differ by more than 10 dB, the background can be ignored.

dB(A) and noise exposure

Workplace noise is usually measured in A-weighted decibels, dB(A), which follow the sensitivity of the human ear. dB(A) values add with the same formula, as long as all of them are A-weighted and measured at the same position. Exposure over a shift is a time-weighted energy average of the same kind, and many occupational rules use 85 dB(A) over eight hours as an action or exposure level, with a 3 dB or 5 dB exchange rate depending on the rule. These limits are cited here in general terms, so check the rules that apply where you work. Safety teams often report noise exposure next to injury rates such as the TRIR.

Common mistakes

• Adding the dB values. 84 + 82 + 91 + 88 + 76 + 85 = 506 dB is meaningless. Add energies, then convert back: 94.2 dB.
• Averaging the dB values. The plain average is 84.3 dB. The energy average, the total minus 10 × log10(6), is 86.5 dB, because loud sources weigh more.
• Missing parentheses. 10^B2/10 is not 10^(B2/10). The first divides an enormous number by 10.
• 20 instead of 10. 20 × log10 converts a pressure ratio into dB. For adding energies or intensities, the factor is 10.
• Mixing weightings or positions. dB(A) and dB(C), or levels measured at different distances, cannot be added as if they were the same thing.
• Subtracting a background that is too close. With less than about 3 dB between the total and the background, the machine level is too uncertain to report.