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Level Reduction – Adjustment
Posted by rjschneider on January 21, 2023 at 1:59 amQuestion I had always wanted to ask about the difference in methods reducing three wire levels proscribes using the median between your upper and lower readings as opposed to the average.
I had very little formal math education, some high school, some jc, so I wouldn’t know the statistical
reasoning why the one method is proscribed, or given the tight range allowed in sightings, it would alter the final adjustment materially.
I’m not even sure where the answer to this could be found.
Has this even ever been asked before ?
Jon Payne replied 1 year, 5 months ago 6 Members · 10 Replies 
10 Replies

You should use the average of the upper, middle and lower readings in your BS/FS calculations. That average should be compared to the middle readings but the average is what is used for calculations.

For a dataset that is composed of three data points (high, middle, low wire readings), the median is the middle datapoint by default.
So using the median would be the same thing as just using the middle wire reading.
The average, on the other hand, allows for a blunder check (against that middle reading) and increased precision as well.
In straight statistics, the median (and mode) is far less sensitive to large outlier data points than the mean (average), so it is often used as a check for skewness of the data.
In a way, we are using the median in threewire levelling, but the way we have set up our data collection, the median should always equal or be very close to the mean. So if we pick up any “skew”, it just indicates a likely blunder.
And if we do not detect a blunder, using the mean will give us increased precision (not a ton, but we’re still using more datapoints from a nonskewed dataset rather than a single datapoint), so that’s what we use.
https://online.stat.psu.edu/stat200/lesson/2/2.2/2.2.4/2.2.4.1
“…people will come to love their oppression, to adore the technologies that undo their capacities to think.” Neil Postman 
I had always used the average in reducing readings, and one time it was declared rong wrong by the office. Did some reading – and I can’t remember which very old text I read – mentioned the the split between the upper and lower differences, with respect to the middle wire.
So that threw me, and I had always wondered since the ‘whys’ of it, which is where my question came from. I used ‘median’ to describe the split between upper/lower differences. Thank you.

I use only the main crosshair for the level
Top and bottom crosshairs are for distance
And the differences as a check

The statistically best value is the arithmetic mean (average) of the three readings.
There are several ways to express the disagreement of the measurements, all of which serve as blunder checks. You can compare differences TM vs MB, or (T+B)/2 vs M, and other ways.
Median of three values, as said above, is the middle value, not what is labeled in that diagram.
The value labeled incorrectly in the diagram as median is the mean of the top and bottom readings, and ignores the middle value, thus losing a trifle in statistical accuracy. If your reading accuracy is sigma, then averaging top and bottom gets you sigma/sqrt(2) whereas averaging three wires gets you sigma/sqrt(3) reducing your probable error to 82% of the first method or 58% of using one wire.
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The value labeled incorrectly in the diagram as median is the mean of the top and bottom readings
You’re correct. I applied the correction to only the split between the TB wire to balance the differences.
Ignored the middle reading. ugh.
Applying the difference 0.0025′ to the middle reading should have given me 4.1425′.
That was fun.
editing: That doesn’t seem right somehow. I think I like my answers better.

Applying the difference 0.0025′ to the middle reading should have given me 4.1425′.
That was fun.
editing: That doesn’t seem right somehow. I think I like my answers better.
R.J. I don’t think you need apply any amount in order to get the median value in a data set of three points. The median is the value with the same number of observed data points above and below it.
Looking briefly at three texts, they all show use of the average reading (Upper+Middle+Lower)/3 as the value used to calculate the next instrument height and next elevation.
It seems to me that using the average would help to reduce any bias in instrument person rounding when using an optical level. It should also help in compensating for imperfections in parts of the system (level adjustment, rod level adjustment, scale on the rod, etc..).

R.J. I don’t think you need apply any amount in order to get the median value in a data set of three points. The median is the value with the same number of observed data points above and below it.
Thank you, Jon. What I had originally applied to my two splits 0.98′ and 0.975′ to arrive at what
Bill93 has pointed out is simply the average of upper and lower wires. The middle wire was likely as valid a reading as the others, and needed to be taken into consideration. It’s a moot point though if the median of the two splits isn’t the correct reduction.
I used Median only because I didn’t have another word for it.
Still trying to wrap my head around the hypothetical; If I had to mean my two splits and apply it to the middle wire scenario

If I had to mean my two splits and apply it to the middle wire scenario
I’ve never seen or used that as a method, so I do not have any background on why the author of the older text you mentioned or the office personnel who said you were rong for using the average would suggest that method over averaging the three readings. As I said, the first three texts I grabbed showed using the average of the three readings.
I suspect the most likely sources of error would be a level rod not being perfectly plumb, scale error in the manufacture of the rod, or reading by a person if using optical equipment. Through the course of a level loop, I would expect these (with good procedures) to mostly cancel out.
Perhaps the older text and your office colleagues think that using the (swiping Bill’s suggested terminology) midpoint better addresses plumbing issues. Excluding the middle reading will pull the answer towards the larger difference between UM / ML each time (happens to be higher than the three point average in your example, but would vary throughout a level loop).
That seems to be accepting the U and L reading as being of greater weight (or more correct) than the middle reading. Just off the top of my head, I don’t think that would take into account the instrument operator (eyesight, parallax, bias in estimating readings).
So the question might become which source of potential error is greater – rod plumbing or instrument person. I don’t know which it is.
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