me 101 measurement demonstration (md 1) introduction

ME 101 Measurement Demonstration (MD 1)

File: Measurement Demonstration (MD1) - Instructions.doc 1

INTRODUCTION This laboratory investigation involves making both length and mass measurements of a population, and then assessing statistical parameters to describe that population. For example, one may want to determine the diameter and mass of a certain product that is mass-produced to insure these products meet the intended specifications. The tools used to measure length will include a linear scale and micrometer.

DEFINITIONS Precision - A measure of agreement between repeated measurements (repeatability).

Error - The difference between the true value and the measured value.

Accuracy - Agreement between measured and true values (the absence of error).

Resolution - The smallest step or interval that we can measure or distinguish on the object or parameter being measured.

LENGTH MEASUREMENTS Length measurements require some scale of comparison for reporting the length determinations, and for this measurement to be meaningful, the unit of measure should be some widely accepted standard. For example, with the System International (SI) units the Meter is the base unit for the measurement of length. For the United States Customary System (USCS) of units, the inch is a derived unit of measurement. One example of a scale for making linear measurements is the steel scale as depicted in Fig. 1 below.

Figure 1: Steel ruler with Metric (SI) and English (USCS) scales

When making length measurements with a ruler it is important to understand the design of these instruments to avoid measurement error. Most specifically the steel ruler that you will be using in lab has two scales, on one side SI in centimeters and the other USCS in inches. Please note the smallest subdivision of the SI scale is 1 mm, while the USCS scale is resolved into 1/32" increments. For this lab we will use the SI scale. It is important to not measure an object from the edge of the ruler because often the ruler’s edge does not mark the zero point or the edge could be damaged.



For the most accurate results when utilizing a linear scale for length measurements, students are urged to use the following procedures:

a) Line up one edge of the object to be measured with one of the internal whole inch tic marks (e.g. 2, 3, …10).

b) Next, compare the remaining object measured with one of the internal whole inch tic marks (e.g. 2, 3, …10).

c) Report your measurement to the nearest 1/32 of an inch, being certain to double check and verify the correct number of whole inch increments (see Fig. 2).

Figure 2: Correct alignment of object to be measured with steel ruler A micrometer is a device for measuring diameters of machined objects or the thickness of a material. Figure 3 depicts a typical micrometer with the identification of various parts. As with any measurement instrument, careful use and handling is essential to make accurate measurements and avoid damage that renders it incapable of performing its intended function. The correct procedures for using this instrument are as follows:

a) Be certain the “locknut” (locking screw) is loose prior to making any adjustments with the “thimble.”

b) Rotate the “thimble” in a counterclockwise direction (when viewed from the “ratchet” end) until the “anvil” and “spindle” are separated a distance greater than the length or thickness to be measured.

c) Place the micrometer on the object to be measured and rotate the “ratchet” knob in the clockwise direction until both the “anvil” and “spindle” are in contact with the object. Continue tightening the “thimble” until the “ratchet screw” friction joint begins to slip. Be careful to turn only the “ratchet screw” while making this adjustment. Do not turn the “thimble” itself as this may result in damage to the instrument!

d) After closing the “spindle” to the correct position, set the "locknut" and remove the micrometer from the object. Read the Vernier scale to determine the final measurement to the nearest 0.001 in. Procedures for reading the scale are to look at the tic marks on the "sleeve" that are closest to the "thimble." The major marks indicate 0.1 increments while the finest tic marks indicate 0.025 inch increments. Add to the sleeve measurement the number of 0.001 inch increments as read from the “thimble” at the line on the “sleeve” parallel to the axis of the “sleeve.” Figure 4 below provides an example.



Figure 3: Identification of micrometer parts

Figure 4: Reading the Vernier scale on a micrometer DESCRIPTIVE STATISTICS Statistics are parameters used to describe the magnitude and variability that occurs within a sample of observations from a population. The most common statistical parameters that engineers encounter are the mean and standard deviation of a population. When utilizing these parameters to describe a population, we must make a simple assumption, and that is that the population being described is distributed "normally" (Gaussian distribution). You may have encountered the term "normal" in your prior academic career. As often times grades may be assigned in accordance with a "bell shaped" curve – many students receive a letter grade of "C" while a significantly small proportion of students get a letter grade of "A." Figure 5 depicts a



"standard normal distribution." Please note that two parameters are used to describe the shape and location of this curve, and they are "mean" and "standard deviation." The "mean" describes the position of the curve on the horizontal axis, while the "standard deviation" is an indication of the spread of the distribution. If the horizontal axis is scaled in increments of 1s, then the central portion of the curve from -1.0 to 1.0 contains 68% of the entire area under the curve. Correspondingly, as we move to +2s and +3s, the corresponding area under the curve is 95% and 99.7% of the total area, respectively.

Figure 5: Standard Normal Distribution

While the Greek characters µ and σ are used to denote the population mean and standard deviation, respectively, it must be recognized that the true values of either parameter may never be known. In reality we must sample the populations, making measurements of the parameters of interest. From this data we can then estimate the values for mean and standard deviation, and as these calculated parameters are not exact, we adopt new variables to represent these values. The new variables x and S are used to denote estimates for the mean and standard deviation, respectively, as calculated from a sample of the population. Figure 6 below illustrates the relationship of the variables in question.

Sample Population

Mean µ

Variance S2 σ2

Standard Deviation S σ

x

Figure 6: Descriptive statistics variables



To calculate the mean from a sample of the population, you simply sum the measurement values, and then divide by the number of observations as shown in the equation below,

1

n

iix

Xn==∑

where xi is the observation i's value, n is the total number of observations from the population sample. As you may recall from our previous discussion, the standard deviation is a measure of the spread of the normal distribution. We estimate this value, again using the same sample observation values as before, with the following equation,

( )

( )

22 2

2 1 11

1 1

n nn

i iii ii

n x xx xS

n n n= ==

⎛ ⎞−− ⎜ ⎟⎝ ⎠= =

− −

∑ ∑∑

Please note that we divide the sum of square of differences between the individual observation values by n-1 to get an unbiased estimate of the variance (square of the standard deviation). At this point you might be asking why divide by n-1 versus n by itself? The justification is that we want an unbiased estimate of the variance, and by using n-1 we slightly overestimate the variance. The final step is to take the square root of S2 to get the standard deviation of S.



Laboratory Procedures Part 1: Measurements of the washer 1) Determining the inner diameter, outer diameter, and thickness of washers

a) At each station there are 10 washers, each of these washers have an identifier on them. It

is imperative that you note the identifier with each measurement that is taken. Make sure to record each measurement into the proper section of the provided table.

b) Using the ruler measure the I.D. (inner diameter) and O.D. (outer diameter) of each of the washers and record the values. Make all of initial measurements in inches.

c) Using the micrometer, measure the thickness of each of the washers and record the values. The micrometers used in this lab measure in inches.

2) Determining the masses of the washers a) Using the digital scales find the mass of each washer in grams. b) Determine the mean and standard deviation of the 10 mass measurements.

3) Homework: Calculating the density of the washers and comparing it to an expected value a) Determine the mean and standard deviation for the I.D., O.D., and thickness for the set of

10 washers. You may use your calculator or a computer program (such as Microsoft EXCEL) to help calculate the mean and standard deviation. However, do each calculation longhand at least one time and include it with your lab report.

b) Using the I.D., O.D., and thickness measurements, calculate the volume of each washer in cubic centimeters.

c) Determine the density of the ten washers using what you know about each washer’s mass and volume. (Hint: density = mass/volume)

d) Determine the mean and standard deviation of the washer densities. Part 2: Homework: Concluding Questions

1. Explain in your own words what standard deviation is and why it is useful when

describing experimental data.



2. What range for mass of the washers would you expect 68% of the samples to fall into?

What about 95%? (Hint: use the standard deviation calculated for mass measurements)

3. Do you think that a sample population of two washers would have been adequate to generate the descriptive statistics for the entire population? Explain

4. For what reasons would someone generally NOT want to use the very end of a ruler for taking measurements.

5. Assume that the actual density of the washers is 7.81 g/cm3. What is the percent error of your calculated mean? What sources of error exist in this experiment?



Name Section

Washer Inner Diameter

Outer Diameter Thickness Volume Mass Density

(units) ( ) ( ) ( ) ( ) ( ) ( )

1

2

3

4

5

6

7

8

9

10

Mean

Standard Deviation

Hand Calculations for Standard Deviation:

Homework:Part I: Washer Measurements

me 101 measurement demonstration (md 1) introduction

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