G. Weight a Minute...

1. Everything's the Same

So far, when discussing the mean, standard deviation, and error in the mean, we've only dealt with equal quality measurements. All such examples used measurements made by the same personnel, using the same equipment, under the same conditions (the three error sources), the mean was simply the measurements sum divided by their number.

But what if the set consists of varying quality measurements? For example, a distance measured three times using a Total Station (TS) (242.15, 241.16, 242.14) and three more times using a fiberglass tape (242.05, 242.29, 242.38). It's reasonable, just on their spread alone, to assume the TS measurements are more accurate than the tape's. However, adding them and dividing by 6 treats the measurements the same:

(242.15+241.16+242.14+242.05+242.29+242.38) / 6 = 242.495

2. One of These Things is Not Like the Other

To combined different quality measurements, we use weights. A weight is a multiplier applied to an individual measurement based on its relative accuracy compared to other measurements.

Assume the TS's measurements are twice as good as the tape's - the TS's weight is 2, the tape's is 1. The effect would be adding each TS measurement twice, each tape measurement once for a total of 9 measuremetns:

(242.15+242.15+241.16+242.16+242.14+242.14+242.05+242.29+242.38) / 9 = 242.180

Or we could multiply each measurement by its weight, add them, then divide by the weight total:

[2(242.15)+2(241.16)+2(242.14)+1(242.05)+1(242.29)+1(242.38)] / [2+2+2+1+1+1] = 242.180

The general equation for a weighted mean is

Equation (G-1)

And the weighted standard deviation:

Equation (G-2)

3. How to Determine Weights?

Weight selection must be done carefully to correctly reflect measurement quality. Sometimes a rigorous numeric method can be used, other times may require judgement calls based on the surveyor's experience and expertise. Weight selection is discussed in greater detail in Chapter E of XVIII Least Squares Lite.

Regardless the method used,

• weights are a reflection of random errors - the smaller the random error, the higher the weight.
• weights are relative - they indicate how much stronger or weaker one type of measurement is against other types

Providing sufficient measurements are made, one way to assign weights is by considering standard deviation. In general , a measurement weight is inversely proportional to the standard deviation squared:

Equation (G-3)

4. Examples

a. Angle measurement

The same angle is measured multiple times by a seasoned survey crew and a crew of newly graduated techicians. The same equipment is used under similar conditions. Their results are:

Assuming the vetran crew is consistently four times more accurate than the techs, what is the most probable value of the angle?

Because the degree and minute portion of the measurements don;t change, we can work with just the seconds.

Ave_Sec = [4(38+41+36) + 1(25+49+11+57)] / [4(3) = 1(4)] = 602/16 = 37.625

 M_W = 195°25'37.6"

b. Distance measurement

Three different survey crews used three different ways to determine an unknown distance. Each crew measured a sufficient number of times that they were able to compute their standard deviation.  Their results are shown in the following table: 

What is the most probable value of the distance?

Weights are inversely proportional to the standard deviation squared.

Because weights are relative, we can divise the initial weights by the lowest one t reduce their size. Tis does not affect the weighted mean.

M_W = [1.0(356.89)+12.7(356.72)+62.6(356.69)] / [1.0+12.7+62.6]

M_W = 356.70