### 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:

Vetrans |
Newbies |

195°25'38" |
195°25'25" 195°25'49" 195°25'11" 195°25'57" |

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:

Crew |
Method |
Distance (ft) |
Std Dev (ft) |

A | Steel tape | 356.89 | ±0.182 |

C | Construction TS | 356.72 | ±0.051 |

D | Geodetic TS | 356.69 | ±0.023 |

What is the most probable value of the distance?

Weights are inversely proportional to the standard deviation squared.

Crew |
W |
Rel W |

A | 30.2 | 1.0 |

B | 384.5 | 12.7 |

C | 1890.4 | 62.6 |

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