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INTRODUCTION

“Error analysis”is the study of uncertainties in physical measurements. There are no perfect measurements, and all measurements contain errors. If we measure something more than once, there is a good likelihood the subsequent measurements will be slightly different from the first measurement and from each other. Therefore, it is almost impossible to know the “true” value we are trying to determine.

The best way to come closer to the “true” value of a measurement is to take greater care in making the measurement, use more accurate measuring instruments, and design our experiments in such a way as to reduce the measurement’s errors. However, we must keep in mind that experimental errors will always be associated to experimental measurements

ERRORS

There are several types and sources of experimental errors:

The accuracy limitations of the measuring device: For example a digital scale may be accurate to 0.1 g, 0.01 g, 0.001 g, etc., a precision analytical balance may be accurate to 0.0002 g or

0.2 mg.

Instrument errors due to calibration of an instrument.

Reading errors: Incorrectly interpreting interpolated values; inaccurate readings due to

fluctuating instrument read-outs.

Uncontrolled environmental factors: Slight temperature or humidity variation in the laboratory may affect the measurements.

Over-simplifications or inherent limitations in the experimental design: For example, we may not be able to achieve a totally frictionless environment in a physics experiment.

Human errors: Reading the measurement incorrectly, or simply writing down the wrong number; reading a scale at angle may result in a “parallax error.”

Experimental error is the difference between a measurement and the true value. However, because we will know the true value, we have to make an estimation of this true value.

QUESTION 1

Imagine that your team has been assigned by a construction company to measure the dimensions of an empty lot in a subdivision. Briefly give examples of the 6 types/sources of errors described above and explain what you would do to minimize them.

Two important concepts: Accuracy and Precision.

Accuracy refers to how close a measured value is to the true or accepted value. As the true value of the quantity is often not known, it may be impossible to determine the accuracy of a measurement. Instead, we rely on estimated values.

Precision (also known as repeatability or reproducibility)refers to how closely repeated measurements agree with each other under identical experimental conditions.

These concepts are typically represented using the following bull’s eye diagrams:

QUESTION 2

A certain gas is known to have a pressure of 9 psi. We are asked to take measurements using two different sensors. These are the results that we obtained after measuring the pressure of the gas 7 different times.

Sensor 1: 9.8 10.010.3 9.7 9.9 10.1 9.8

Sensor 2: 9.3 9.5 8.6 9.0 8.9 9.2 8.5

Do you believe one of the sensors is more accurate than the other? What about precise? Explain your reasoning.

Percent error: This is the difference between the measured value and the real value divided by the real value. As we may not know the real value, most of the times we will use the accepted or estimated value. The percent error can be written as:

The |Measured Value – Accepted Value| means the absolute value between these two numbers; this means to omit the sign.

Example: Measuring a 1.5 V battery twice gives the following readings: 1.63 V and 1.37 V. Calculate the percent error of each measurement.

Mean and Standard Deviation: These are the two most used parameters at the time of evaluating experimental measurements.

Mean is the average of the measurements.It indicates the central value at which the values tend to congregate. If we called x each value, the mean is notated as x. It can be calculated as:

Standard Deviation is an indication of the spread of the individual measurements around the mean. Standard deviation is typically notated using the Greek letter Sigma (σ ). Standard deviation for a set of values (x) can be calculated as:

Let’s see what this means. The part (x – x ) indicates the difference between each one of the measurements and the mean. Each one of these number is squared. The symbol Σ means summation, so we will add those values. The total result will be then divided by the number of measurements minus 1 and finally, we calculate the square root of the result.

Example: Let’s calculate the mean and standard deviation of the gas pressure experiment using sensor 1. As you remember , the set of values is as follows:

Sensor 1: 9.8 10.0 10.3 9.7 9.9 10.1 9.8

IMPORTANT NOTE

Most of today’s handheld calculators and spreadsheet programs can perform calculations of mean, standard deviation and other statistical parameters. Other than for answering Question 3, you can use these tools for the rest of this lab as well as for calculations in future labs. If you are not familiar on how to do this, it is now a good time to dust off the calculator’s user manual or search for how to do these calculations in the spreadsheet you are using.

QUESTION 3

Calculate the mean and standard deviation from the values using Sensor 2. Show all your work step by step. Sensor 2values: 9.3 9.5 8.6 9.0 8.9 9.2 8.5

QUESTION 4

If we were able to perform totally accurate and precise measurements, what would be the standard deviation in these conditions?

QUESTION 5

By inspecting the mean and standard deviation values from Sensor 1 and Sensor 2, what can you say about the accuracy of each sensor. What about its precision?

PROCEDURE

Since this is the first lab experience, we will work with a set of given data rather than taking the data yourself.

The data in the table below corresponds to measurements taken during a free-fall experiment. At different time intervals the experimenter measures the distance of the object from its initial point as it falls down. The experiment is repeated 5 times (5 trials).

Distance (y) at different time intervals in free-fall experiment

Trial 1

Trial 2

Trial 3

Trial 4

Trial 5

Time (s)

Y1 (m)

Y2 (m)

Y3 (m)

Y4 (m)

Y5 (m)

0.00

0

0

0

0

0

0.50

1.0

1.4

1.1

1.4

1.5

0.75

2.6

3.2

2.8

2.5

3.1

1.00

4.8

4.4

5.1

4.7

4.8

1.25

8.2

7.9

7.5

8.1

7.4

QUESTION 6

Calculate the Mean (Y) and Standard Deviation for each time completing the table below.

Distance (y) at different time intervals in free-fall experiment

Trial 1

Trial 2

Trial 3

Trial 4

Trial 5

Mean

Std. Dev.

Time (s)

Y1 (m)

Y2 (m)

Y3 (m)

Y4 (m)

Y5 (m)

(Y (m))

σ

0.00

0

0

0

0

0

0.50

1.0

1.4

1.1

1.4

1.5

0.75

2.6

3.2

2.8

2.5

3.1

1.00

4.8

4.4

5.1

4.7

4.8

1.25

8.2

7.9

7.5

8.1

7.4

QUESTION 7

Using the spreadsheet or plotting program of your choice, plot (Y) vs. t (that is, plot (Y) in the vertical axis and t in the horizontal axis.

LABORATORY REPORT

Create a laboratory report using Word or another word processing software that contains at least these elements:

Introduction: what is the purpose of this laboratory experiment?

Description of how you performed the different parts of this exercise. At the very least, this part should contain the answers to questions 1-7 above. You should also include procedures, etc. Adding pictures to your lab report showing your work as needed always increases the value of the report.

Conclusion: What area(s) you had difficulties with in the lab; what you learned in this experiment; how it applies to your coursework and any other comments.