An angle is observed repeatedly using the same equipment and procedures producing the data below:32 ∘ 37' 00", 32 ∘ 37' 10", 32 ∘ 37' 10", and 32 ∘ 36' 55"Part A Calculate the angle is most probable value. Calculate the angle's most probable value
Calculate the standard deviation.
Calculate the standard deviation of the mean

a)  x_average = 34.51077°,   b)  σ = 2.69°, c)   σ = 2.5107°

Explanation:

a) The most likely value of a measure is

          x_average = ∑ / n-1

The angle measurements are in the Sixty Base System, to simplify the calculations let's reduce the measurements

     1° = 60 min

     1 min = 60 s

 Table 1

   Angle      angle converted    (x-x_average)   (x-x_average)²

     32               32,000                2,51077              6.3040

     37 00”        37,000                2,48923             6.1963

     32               32,000                2,51077              6.3040

     37 10”         37.00277            2.492                  6.2101

     32               32,000               2,51077               6.3040

     37 10”         37.1667              2.6559                7.0538

     32               32,000              2,51077               6.3040

     36 55”        36.9167             2.4059                5.7884

With this values in degree the calculation is easier

          x_average = (32 + 37 + 32 + 37.00277 + 32 + 37.1667 + 32 + 36.9167) / 8

         x_average = 276.08617 / 8

         x_average = 34.51077

b) We look for the standard deviation

               σ = √ ( -x_average)² / n-1

               σ = √ 50.4646 / (8-1)

               σ = 2.69

c) the deviation from the mean

               σ = | x -| / n

               σ = 20.08611 / 8

               σ = 2.5107

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