### Solution: Random Errors

#### Problem (1)

Survey Crew A measured a distance multiple times: 118.54', 118.52', 118.48', 118.54', 118.53', 118.47'.

Determine

The most probable line length

Its standard deviation

The length's expected error

Compute all to 0.001'.

 Meas v v2 118.54 +0.027 0.000729 118.52 +0.007 0.000049 118.48 -0.033 0.001089 118.54 +0.027 0.000729 118.53 +0.017 0.000289 118.47 -0.043 0.001849 sums: 711.08 0.004734

#### Problem (2)

Two crews measured different distances multiple times. There results, in feet, are shown in the table below:

 Crew A Crew B Num of meas 4 12 Average 87.96 108.53 Standard deviation ±0.030 ±0.035

Precision?

Crew A had better precision because its standard deviation was lower.

Expected accuracy?

Must compute and compare EMPV for each crew

Crew B had better expected accuracy since its EMPV was lower.

#### Problem (3)

The zenith angle to the top of a flag pole was measured with these results: 37°18'55", 37°19'04", 37°19'09", 37°18'53", 37°19'02"

Determine

The most probable zenith angle

Its standard deviation

The angle's expected error

Compute all to 0.1".

Subtract 37°18' from each angle to work with just seconds.

 Angle Sec v v2 37°18'55" 55 -05.6 31.36 37°19'04" 64 +03.4 11.56 37°19'09" 69 +08.4 70.56 37°18'53" 53 -07.6 57.76 37°19'02" 62 +01.4 1.96 sums: 303 173.20

#### Problem (4)

The length and width of a building are measured in feet, summarized in the table below.

What are the building's area and expected area error in square yards?

 Length v v2 Width v v2 173.9 -0.33 0.109 89.6 -0.20 0.040 174.5 +0.27 0.073 90.1 0.30 0.090 174.3 +0.07 0.005 89.7 -0.10 0.010 sums: 522.7 0.187 269.4 0.140

The length sum has 4 sf, its average will have 4 sf.

The width sum has 4 sf, its average will have 4 sf.

Carry one more sf for each MPV to minimize intermediate rounding.

 Length: Width:

Each SD and EMPV are 3 sf, including an additional one for intermediate calculations.

Since length and width should both have 4 sf, Area = 5215 yd2.

Since additional sf were carried for L, W, and their EMPVs, the area error should be expressed to 2 sf: Error = ±10. yd2.

5215 yd2 ±10. yd2

#### Problem (5)

A lab technician was to determine the moisture content of a soil sample. She weighed the sample 4 times and obtained an average of 583.4 gr with a ±0.9 gr standard deviation. After the sample was dried for 24 hours at 400° F, she weighed it 6 times for an average of 552.9 gr and standard deviation of ±1.5 gr. What was the soil’s moisture content, and its expected error, in grams?

Because a subtraction is involved, must use Error of a Sum. Carry additional sf for the error comps.

30.5 gr ±0.8 gr

#### Problem (6)

Four different survey crews independently measured the elevation difference from point A to B. Because the have varying experience levels, not all the elevation differences are the same quality. Their results and expected uncertainties are listed in the table. What is the most probable elevation difference and its uncertainty?

Weights are inversely proportional to standard deviation squared. Weight can be "normalized" by dividing all the raw weights by the smallest one.

 Crew Elev Diff, m σ Wraw W W x m v v2 1 -5.36 ±0.05 400 5.76 -30.87 0.01 0.0001 2 -5.41 ±0.08 156.25 2.25 -12.17 -0.04 0.0016 3 -5.37 ±0.04 625 9.00 -48.33 0.00 0.0000 4 -5.43 ±0.12 69.44 1.00 -5.43 -0.06 0.0036 Sums 18.01 -96.80 0.0053

Elev Diff = -5.37 ft ±0.018 ft