Averaging Readings: Mean, Median and Weighted

Why this matters

Add the readings, divide by how many there are: that is the average most people learned, and it is the right answer less often than it feels. It is right when every reading is a fair, equal sample of one quantity. It is wrong when one reading is bad, because a single bumped gauge drags the whole answer with it. And it is wrong when the readings stand for unequal shares of the thing being measured: three registers moving different amounts of air, two flow rates that each ran for a different length of time. Then the plain mean answers a question nobody asked. This card is about choosing between the three averages a tech actually needs, the mean, the median and the weighted mean, with worked examples where the wrong choice produces a confident wrong number.

How a meter's own averaging, or a long logging window, can hide a short event inside a reading is a separate problem, owned by Averaging and Spikes: What Your Meter Hides and How Averaging Hides a Fault. This card starts after the readings are written down.

The plain mean: equal samples of one quantity

The mean (arithmetic average) is the sum divided by the count. It is the right average when three conditions hold: the readings measure the same quantity, each represents an equal share of it, and the scatter between them is random rather than one bad value.

Five voltage readings taken a minute apart on a steady circuit: 118.2, 118.6, 118.4, 118.5 and 118.3 V. Sum: 592.0 V. Mean: 592.0 / 5 = 118.4 V. Every reading is within 0.2 V of the mean, the spread is small and even, and averaging does what it promises: it smooths random reading noise so the result is steadier than any single reading.

The median: when one reading may be bad

The median is the middle value once the readings are sorted. With an even count it is the mean of the two middle values. It ignores how far an extreme reading sits from the rest, which is exactly its use.

A tech takes five static water-pressure readings at a hose bib, a few minutes apart: 61, 62, 61, 63 and 88 psi. The 88 came right after someone slammed a valve in the house, or the gauge was knocked.

• Mean: (61 + 62 + 61 + 63 + 88) / 5 = 335 / 5 = 67 psi.
• Median: sorted, 61, 61, 62, 63, 88. The middle value is 62 psi.

The mean has been pulled at least 4 psi above every reading but one, so it describes no reading that was actually taken. The median sits where the system actually is.

Two cautions. A median does not tell you why the odd reading happened, and the odd reading is sometimes the finding: a pressure spike after a valve closes is water hammer, and a tech who medians it away has deleted the evidence. Record every reading, use the median to describe the steady state, and write down the outlier and its suspected cause beside it. And a median of three readings is weak; with so few, one bad value is a third of your data. Take five or more when you expect noise.

The weighted mean: readings that stand for unequal shares

When each reading represents a different amount of the thing you care about, each one is multiplied by its share (its weight) before averaging, and the sum is divided by the total weight rather than the count.

weighted mean = sum of (reading x weight) / sum of weights

The weight is whatever the reading stands for: airflow for a temperature, time for a rate, area for a velocity.

SUPPLY AIR TEMPERATURE, THREE REGISTERS

register   temp F   airflow CFM   temp x CFM
A            55        400          22,000
B            58        300          17,400
C            64        100           6,400
total                  800          45,800

weighted   45,800 / 800          = 57.25 F
plain      (55 + 58 + 64) / 3    = 59.0 F

FILL RATE, TWO STAGES

stage   rate GPM   minutes   gallons
1          8          10        80
2          2          30        60
total                 40       140

true average   140 gal / 40 min  = 3.5 GPM
plain mean     (8 + 2) / 2       = 5.0 GPM

Worked example: supply temperature by airflow

The register table above comes from a question a tech actually asks: what is the average supply-air temperature this system is delivering into the house? Register C is at the end of a long run and only gets 100 CFM; its 64 F air has picked up heat in the duct.

The plain mean, 59.0 F, gives register C the same say as register A, which moves four times the air. The weighted mean, 57.25 F, counts each register's temperature by the air it carries, and it is the temperature you would get if all 800 CFM were mixed in one place. If the question is "what is the system delivering," 57.25 F is the answer; 59.0 F overstates the supply temperature by 1.75 F, and on a cooling check that small difference shifts the sensible heat estimate noticeably.

Check the weighted mean by a sanity rule: it must lie between the lowest and highest reading, and closer to the readings with the biggest weights. 57.25 F lies between 55 and 64, closer to 55, the register moving the most air. A weighted mean outside that range, or pulled toward the lightest weight, has its weights backwards.

The weakness is the weights themselves. If the register airflows are estimates, the weighted mean is only as good as they are, and where they are unknown the honest move is to measure them, not to fall back on the plain mean as though equal weights were neutral. Equal weights are a choice too.

Worked example: averaging two flow rates the wrong way

A tank fills at 8 GPM for 10 minutes while a supply valve is wide open, then at 2 GPM for 30 minutes after someone throttles it. Someone asks the average fill rate, to predict the next fill.

• The plain mean of the two rates: (8 + 2) / 2 = 5.0 GPM.
• The real average: total gallons over total time. Stage 1 delivered 8 x 10 = 80 gal. Stage 2 delivered 2 x 30 = 60 gal. Total 140 gal in 40 minutes. 140 / 40 = 3.5 GPM.

The plain mean treats the two rates as if each ran for the same time. They did not: the slow rate ran three times as long. A prediction built on 5.0 GPM says a 140 gal fill takes 140 / 5.0 = 28 minutes; it took 40. The time-weighted average is the rate that, held steady, would have delivered the same water in the same time, and that is the only "average rate" that predicts anything.

The trap has a mirror image. If the two rates each ran for the same VOLUME rather than the same time, the weight is the time each took, and the right average is total volume over total time again. Drain 60 gal at 6 GPM (10 minutes) and another 60 gal at 2 GPM (30 minutes): 120 gal over 40 minutes is 3.0 GPM, while the plain mean of 6 and 2 says 4.0. Whichever way the problem is posed, the safe method is the same: convert every stage to an amount and a time, add the amounts, add the times, divide. Then the weighting cannot be wrong, because you never chose one.

Averaging percentages: the denominator is a weight

A percentage is a ratio, and averaging ratios without their denominators is the same mistake as averaging rates without their times. Two storage tanks on one job: a 40 gal tank reads 50 percent full and an 80 gal tank reads 80 percent full.

• Plain mean of the percentages: (50 + 80) / 2 = 65 percent.
• What is actually there: 0.50 x 40 = 20 gal, and 0.80 x 80 = 64 gal. 84 gal in 120 gal of capacity, 84 / 120 = 70 percent.

The bigger tank holds twice the capacity, so its percentage should count twice as much, and the plain mean ignores that. The fix is the one from the fill-rate example: turn each percentage back into its amount and its base, total both, and divide. A percentage averaged across groups of different sizes (rooms, tanks, crews, months with different job counts) always needs that step, and Percent Change and Percentage Points Done Right covers the related trap of comparing percentages built on different bases.

Area weighting, in one paragraph

A duct traverse takes velocity readings across a duct face, and each reading stands for the patch of area around it. On a grid laid out so every point represents an equal area, a plain mean of the velocities is a correct area-weighted mean. On any other grid, each velocity is weighted by its patch. Average the velocities, not the velocity pressures they were read from: velocity goes as the square root of velocity pressure, and averaging the pressures first and taking one square root comes out higher than the true mean velocity. How to lay out the grid and convert the result to CFM is covered in How to Average a Duct Traverse into CFM.

Which average, in practice

• Repeated readings of one steady quantity, small even scatter: plain mean.
• Repeated readings with a suspect value, or noise with occasional jumps: median, with the outlier recorded and explained.
• Readings that each stand for a different share (air, time, area, volume, units of work): weighted mean, with the weights written beside the readings.
• Rates over stages: never average the rates; total the amounts and the times and divide.

And one rule above all of these: if the spread of the readings is large compared with the precision you need, no average fixes it. Readings of 40, 55 and 70 psi do not average to "about 55 psi"; they say something is changing while you watch. Find out what, before reporting any single number. An average describes a quantity that holds still long enough to be measured, and a large spread is the readings telling you this one does not.

How to show your average on a report

Write the readings, the method and the result together: "five readings, 61 to 63 psi with one spike to 88 psi after a valve closed; median 62 psi." Or: "supply temps 55, 58 and 64 F at 400, 300 and 100 CFM; airflow-weighted mean 57.3 F." The next reader can redo it, see the spike, and judge for themselves whether the method suited the readings, which is more than a bare "average 67 psi" ever lets them do.

References

• See related: Averaging and Spikes: What Your Meter Hides; How Averaging Hides a Fault; How to Average a Duct Traverse into CFM; Percent Change and Percentage Points Done Right