Pacific  Academy,  Encinitas  |  Physics  2015-­‐2016  |  Ms.  Abigail  Dommer  |  www.paecourses.org/physics  

 

LAB  1:  THE  SIG  FIG  LAB   1    

Lab 1: Significant Figures & Error

The Sig Fig Lab

Objective

To learn how to use significant figures, error, and statistics to evaluate a data set.

Introduction

Scientists evaluate the validity of their data by using significant figures and error. The error and precision of measurements is often the driving force behind technological development and rising equipment costs. This lab will introduce you to the proper way to take measurements, evaluate error, and use statistics to analyze a data set.

Accuracy & Precision. Accuracy and precision are often confused to be the same term. However, as scientists, it is important to be able to use them correctly. Accuracy refers to how close your measured value is to the actual value. Precision refers to the repeatability of the measurement using a given instrument or technique, regardless of how close the value is to the true value.

Significant Figures. Significant figures are the number of digits used to express a measurement or a calculation, which represent how precise the measurement is.1 For example, if I want to calculate the molarity of sodium in a solution where I dissolved 4.56 mol NaCl into 500 mL of distilled water, I would take mol/L = 4.56/0.5 = 9.12 M. However, since I only know the volume of the water to one significant figure (500), I can only report the molarity to one significant figure. Therefore, the molarity of the solution is 9 M instead of 9.12 M. The rules for significant figures are outlined below:

Determining Uncertainty. The uncertainty of a measurement is determined in a variety of ways, depending on the instrument used to take the measurement. For example, various glassware have uncertainties associated with them, provided by the manufacturer. An analytical balance may be accurate to ±0.0001 decimal places.

Significant Figures Rules2

1) Zeros within a number are always significant. Both 4308 and 40.05 contain four significant figures.

2) Zeros that do nothing but set the decimal point are not significant. Thus, 470,000 has two significant figures. However, 470,000. has 6 significant figures because there is a terminating decimal point.

3) Trailing zeros that aren't needed to hold the decimal point are significant. For example, 4.00 has three significant figures.

If you cannot find the uncertainty of an instrument, it is useful to determine the smallest increment the instrument can measure, and then divide that value by two.2 Why? Consider a scale that is accurate to 0.1g. The actual value of the mass could be up to 0.05 g greater than or less than the digitally reported value of the scale, or it would report a value higher or lower, based on rounding.

To determine the uncertainty of a ruler, you would simply determine the limiting increment on the ruler. If you measure the length of a toothpick using this ruler, which reports values to the nearest decimeter, you would report the uncertainty of the length of the toothpick to be ±0.01 cm.

Reporting uncertainty should always be done with your best judgment. It’s your level of certainty. If you are trying to measure the diameter of a spherical object with a ruler, your uncertainty will definitely not be as small as 0.01 cm. You may only be confident to 0.1 or even 0.5 cm. Make sure you are fully confident in your reported values. It is always better to report a value as less certain than more certain than it actually is.

Reporting Uncertainty.3 The error associated with some quantity X is always reported as

 𝑋 = 𝑋!"#! ± ∆𝑋

where Xbest is your best estimate of the actual value, and ∆𝑋 is the uncertainty of that value. Using this notation, it is assumed that the actual value lies somewhere between (Xbest - ∆𝑋) and (Xbest + ∆𝑋). All reported values should use the notation described by (1). Error bars should be used to express uncertainty on a graph. Additionally, the uncertainty should always be reported to only one significant digit.

Propagating Error.3

Addition & Subtraction. When two values x and y are added, the final uncertainty of q, ∆𝑞, is the square root of the sum of the squares of each uncertainty, ∆𝑥 and ∆𝑦.

𝑥!"#$ ± ∆𝑥 + 𝑦!"#$ ± ∆𝑦 = 𝑞!"#$ ± ∆𝑞

∆𝑞 = ∆𝑥! + ∆𝑦!

Multiplication & Division. To calculate the uncertainty associated with multiplication and division of values, a similar formula is used, except all uncertainties are replaced with the relative errors, or ∆!

!!"#$.

𝑥!"#$ ± ∆𝑥 𝑦!"#$ ± ∆𝑦 = 𝑞!"#$ ± ∆𝑞

(1)

   

 

3  

∆𝑞 = 𝑞!"#$∆𝑥𝑥!"#$

!

+∆𝑦𝑦!"#$

!

Rounding. There are several ways to round an answer that results in a half (0.5) increment. Traditionally, 0.5 is rounded up, but statistically, this creates a positive bias in the numbers. For the purposes of this class, we will use an unbiased rounding system called round-to-even. If an answer comes out to be 3.45, but it must be rounded to 2 digits, 3.45 should be rounded to the nearest even number, which is 3.4. Similarly, 3.35 would be rounded to 3.4. Additionally: 6.75 à 6.8, 6.65à6.6, 0.5à0, 1.5à2.

Statistics. The standard deviation of a data set is a measure of how disperse the data is. It is represented by the Greek sigma, σ. The standard deviation of a data set is calculated as the square root of the variance of that data set. The variance is calculated by determining the mean of the squared deviations from the mean of each value, expressed below:

𝑉𝑎𝑟 𝑋 = 𝜎! =Σ(𝑥! − 𝑥)!

𝑁

where X is the data set, xi is the ith value in the data set, 𝑥 is the mean of all values in the data set, and N is the total number of values measured.

In a Normally distributed (Gaussian) curve, 96% of the data fall within the two standard deviations of the mean, as shown above. Gaussian curves generally represent highly variable values within a population, such as human height and exam scores.

Procedure

A. Taking Measurements.

1) Measure and record the diameter of various spherical objects provided to you with (1) a tape measurer and (2) a ruler, making a note of the error associated with the measurement in your notebook.

2) Calculate Volume (ruler) and Volume (tape) and record it in your notebook with correct error. 3) Measure their masses using (1) an analytical balance and (2) a triple beam balance, and record

the error. 4) Using these values, calculate the density of the objects. Which values should you use to calculate

the density of the objects and why?

5) Combine your data with the rest of the class’ data for Object #1. Determine the mean, standard deviation and variance of the values.

6) How accurate were your measurements? How precise were your measurements?  

 

Object #

Diameter (ruler) /cm (±  𝐸𝑅𝑅𝑂𝑅  )

Diameter (tape m.) /cm (±𝐸𝑅𝑅𝑂𝑅)

Mass (triple b.) /g (±𝐸𝑅𝑅𝑂𝑅)

Mass (scale) /g (±𝐸𝑅𝑅𝑂𝑅)

Volume (ruler) (±𝐸𝑅𝑅𝑂𝑅)

Volume (tape) (±𝐸𝑅𝑅𝑂𝑅)

Density (±𝐸𝑅𝑅𝑂𝑅)

1  

B. Measuring population variation. 1) Measure & plot variation of a single trait in a population of your choice (marble circumference,

potato chip diameter, student height, etc.). The population must have at least 50 members. 2) Determine the error in your measurements, the standard deviation of the sample, the variance,

and the mean. 3) Plot your results using a histogram with error bars.

Discussion Questions

1) Which method of measuring the diameter of the spheres was the most accurate? Which method for measuring mass was most accurate? Why?

2) How close were your measurements to those of the rest of the class? How do you know? 3) Was a histogram the best way to represent the variation in your chosen population in Part B? 4) What kind of distribution is represented by your data set in Part B? Are the measurements normally

distributed? Should they be? Why or why not? 5) Are there any outliers in your sample for Part B?

References

1. Significant Figures and Units, Washington University in St. Louis Online Tutorials. http://www.chemistry.wustl.edu/~coursedev/Online%20tutorials/SigFigs.htm (accessed Aug 22, 2015).

2. Significant Figures, Purdue University Topic Review for General Chemistry. http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch1/sigfigs.html (accessed Aug 22, 2015).

3. Uncertainties & Error Analysis Tutorial, Washington University in St. Louis Department of Physics. http://physics.wustl.edu/introphys/Phys117_118/Lab_Manual/Tutorials/ErrorAnalysisTutorial.pdf (accessed Aug 22, 2015)

Table 1. Suggested table format for the lab notebook.