Measurement of Length

OBJECTIVES

Demonstrate the specific knowledge gained by repeated measurements of the length and width of a table.

Apply the statistical concepts of mean, standard deviation from the mean, and standard error to these measurements.

Demonstrate propagation of errors by determining the uncertainty in the area calculated from the measured length and width.

CALCULATIONS

1. After all measurements are completed, perform the subtractions of the coordinate positions to determine the 10 values of the length Li, and the 10 values of width Wi. Record the 10 values of Li and Wi, in the appropriate table.

2. Use Equations 1 to calculate the mean length L‾, and the mean width W‾ and record their values in the appropriate table. Keep five decimal places in these results. For example, typical values might be L‾ = 1.37157 m and W‾ = 0.76384 m.

3. For each measurement of length and width, calculate the values of Li - L‾ and Wi - W‾ and record them in the appropriate table. Then for each value of the length and width, calculate and record the values of (Li, - L‾)² and (Wi - W‾)² in the appropriate table.

4. Perform the summations of the values of (Li - W‾)² and the summations of the values of (Li - L‾)² and record them in the appropriate box in the tables.

5. Use the values of the summations of (Li - L)² and of (Wi -W‾)² in Equations 2 to calculate the values of σ^L_n-1 and σ^W_n-1 unit and record them in the appropriate table.

6. Calculate L‾ -σ^L_n-1, L‾ + σ^L_n-1, W‾ - σ^W_n-1, W‾ + σ^W_n-1 and record the values in the appropriate table.

7. Use the values of σ^L_n-1 and σ^W_n-1 in Equations 3 to calculate the values of aL and xw and record them in the appropriate table.

8. Use the values of L‾ and W‾ in Equation 4 to calculate the value of A, the area of the table, and record it in the appropriate table. Use Equation 5 to calculate the value of α_A and record it in the appropriate table.

PRE-LABORATORY ASSIGNMENT

1. State the number of significant figures in each of the following numbers and explain your answer.
(a) 37.60
(b) 0.0130
(c) 13000
(d) 1.3400

2. Perform the indicated operations to the correct number of significant figures using the rules for significant figures.

3-6. Three students named Abe, Barb, and Cal make measurements (in m) of the length of a table using a meter stick. Each student's measurements are tabulated in the table below along with the mean, the standard deviation from the mean, and the standard error of the measurements.

Note that in each case only one significant figure is kept in the standard error z and this determines the number of significant figures in the mean. The actual length of the table is determined by very sophisticated laser measurement techniques to be 1.4715 m.

3. State how one determines the accuracy of a measurement. Apply your idea to the measurements of the three students above and state which of the students has the most accurate measurement. Why is that your conclusion?

4. Apply Equations 1, 2, and 3 to calculate the mean, standard deviation, and standard error for Abe's measurements of length. Confirm that your calculated values are the same as those in the table. Show your calculations explicitly.

5. State the characteristics of data that indicate a systematic error. Do any of the three students have data that suggest the possibility of a systematic error? If so, state which student it is, and state how the data indicate your conclusion.

6. Which student has the best measurement considering both accuracy and precision? State clearly what the characteristics are of the student's data on which your answer is based.