A Look at Biases in Density Measurements

Introduction

Density, a measurement of the mass of a substance divided by its volume, and specific gravity, a unitless ratio of the density of a substance to the density of water, are common measurements performed in the brewery. Several different tools can be used to perform these measurements, yet the choice of which tool to use during production is often guided by the speed with which results can be obtained rather than by accuracy and precision. In the absence of a comparison of the different tools, we felt that we were also selecting the tool based on tradition (e.g., we have always done it that way) rather than making an educated decision.

Therefore, we set out to compare four common tools used to measure the density and/or specific gravity of wort and beer. Our comparison was guided by the speed of the process and correlated with the accuracy and precision of the measurement. Unfortunately, the measurement of density, as is the case with most measurements, is hindered by biases or errors that occur during the actual measurement (1). These biases or errors can be classified as systematic, random, and operator errors (also known as gross error) (2).

Systematic Error

Systemic error is predictable and is the result of an inherent bias in the measurement procedure. It can occur because of a bias built into the instrument or the process that occurs during the measurement. Since this type of error is predictable, it is possible to identify and significantly reduce or eliminate systematic error. There are several subtypes of errors that can contribute to the overall systematic error. For example, instrument errors typically arise from problems with calibration that can be corrected, although other errors associated with the quality of the instrument can also cause this type of problem. Observational errors typically imply the improper use of an instrument in a systematic way. The result would be a bias that can be eliminated once the proper use of the instrument is identified. Finally, environmental errors arise from temperature, humidity, and other controllable variables that affect the measurement.

Random Error

Random error is an unpredictable bias that occurs in all measurements. This type of bias can be identified and measured, but it cannot be removed. Static or noise in a signal from an instrument is a good example of a random error. For example, if an instrument’s measurement is based on or derived from voltage that unevenly fluctuates with time, the measurement will also fluctuate. Most modern instruments are designed to minimize this type of random error, but the error cannot be eliminated completely.

Operator Error

The last major class of error in a measurement is operator error. This error arises because of the performance of the person making the measurement. Large operator errors can be due to a worker’s inexperience with the procedure or lack of proper technique. With repeated proper training, this type of error can be minimized. However, it can never be completely eliminated. There is always some amount of operator error inherent during a measurement.

Investigation

As a result, every measurement will have some bias or error. This reduces the precision of the measurement and increases the potential that multiple measurements will not “agree” with each other.

With these error types in mind, we investigated four of the most common measurement instruments and methods for determining wort and beer density during the brewing process. We examined the accuracy (how close the measurements were to the actual value) and precision (how close the measurements were to each other) of each method versus the speed at which the measurement could be obtained. The magnitude of the errors observed during the process helped determine the overall quality of each test instrument and method.

Materials and Methods

To reduce errors that would occur between different people performing the measurements, only one person was selected to measure density using each test instrument. Approximately 800 mL of a series of sucrose solutions was prepared with densities in the range of 1.010 to 1.060 g/mL by obtaining the appropriate mass of sucrose from an analytical balance to six significant figures and dissolving it in the required mass of distilled water measured in a graduated cylinder to four significant figures. Therefore, the final calculated densities, reported to four significant figures, were considered to represent the actual value of the density for each of the solutions. Enough of each sucrose solution was prepared that it could be used across all instruments for comparison.

The density of each of the solutions, plus a distilled water sample, was determined using a common triple-scale wort hydrometer, a pycnometer, a refractometer (Abbe 3L, Bausch and Lomb), and a densitometer (model DMA 35 basic, Anton-Parr). Ten replicates of each measurement were taken to assist with statistical analysis. The best operating procedure identified in the literature was further developed for each instrument using a trial-and-error approach until the greatest precision, as measured by the %RSD (percent relative standard deviation) for each measurement, was obtained (3). (Note, many step-by-step instructions can be found on the Internet for the process used to measure the density of a liquid. Specific directions can also be found as a supplement to the ASBC Methods of Analysis at asbcnet.org.) Any modification of the methods to lower the amount of operator bias were minor or specific to the person performing the measurement. For example, instead of using a 25-mL pycnometer, as was recommended by the procedure we initially identified, a 100-mL pycnometer was used. The larger volume increased the precision of the measurement.

The hydrometer employed in this study was typical of those found in homebrewing. It was a wide-range hydrometer capable of measuring densities from 0.990 to 1.160 g/mL to four significant digits. The ASBC Method of Analysis Beer-3b for using a hydrometer was followed (4), with particular attention paid to maintaining the temperature of the sample at exactly 20.0°C. While some hydrometers indicate they should be read at the top of the meniscus, the one employed here was read at the bottom of the meniscus. The value obtained from the hydrometer was then converted to the specific gravity by dividing by the gravity of distilled water determined using the same hydrometer at 20.0°C.

The pycnometer procedure also involved standard protocols for the use of this instrument (e.g., see http://www.academia. edu/13163735/density_determination_by_pycnometer). A cleaned pycnometer was dried overnight in a warm oven and allowed to attemperate to room temperature in a desiccator before use. The sample and pycnometer were carefully maintained at 20.0°C with the use of a water bath during the measurement procedure. The density of the solution was determined by dividing the mass of the liquid, obtained by subtracting the mass of the empty pycnometer from that of the full pycnometer, by the volume of the sample.

The densitometer (model DMA 35 basic, Anton-Parr) was also used to determine the density of the solutions. ASBC Method of Analysis Beer-2b (5) was followed with special care to maintain the sample and rinse solutions at exactly 20.0°C over the course of the measurements. The densitometer was filled and evacuated three times prior to each measurement to ensure the instrument itself was also at a stable temperature. Density values were recorded to four decimal places.

The manufacturer’s procedure for the use of the refractometer (Abbe 3L, Bausch and Lomb) was followed for the measurement of the index of refraction of the samples. The samples and the prism stage were maintained at 20.0°C using a recirculating water bath. The values obtained from the refractometer were then correlated to the degrees Plato (°P) using published data (6). For comparison to the other density measurements, these values were converted to specific gravity using the published formula: SG = 1 + (°P/{258.6 – [(°P/258.2) × 227.1]}) (7).

Each sucrose solution was prepared at least four times, and 10 measurements of each solution were performed using each of the four methods. All replicate data were retained unless an individual measurement was able to be eliminated using a standard Q test (8). Then, the mean (average), standard deviation, %RSD, and percent error for each solution within each set of solutions were determined.

Results and Discussion

Six solutions were prepared from the appropriate mass of sucrose and water such that their calculated specific gravities were 1.010, 1.020, 1.030, 1.040, 1.050, and 1.060. In addition, distilled water was included as a standard solution. The specific gravity values determined using the four measurement devices indicated in the methodology are listed in Table 1. Although there were four trials conducted for each of these sucrose solutions, only the values from the last trial are shown. The first three trials were used for development of the protocol for each instrument.

The values in Table 1 indicate that the hydrometer used in this study was biased to read values greater than the calculated specific gravity. This systematic error was likely due to the use of a standard hydrometer that was not calibrated. The pycnometer and refractometer provided values that were slightly above the calculated specific gravity, and the densitometer provided values that appeared to be slightly below the calculated specific gravity of the solutions.

The %RSDs for each instrument, as shown in Table 1, indicate that the densitometer provided the most precise measurement of specific gravity. In each case, the %RSD was lower for the densitometer than for the hydrometer, pycnometer, and refractometer. Thus, the densitometer provided a more precise measurement.

The percent error of each measurement is listed in Table 2. The percent error in the measurement is taken as the difference between the measurement and the actual value divided by the actual value multiplied by 100. This value can be used as a measure of the accuracy of the specific method. Note, the method that was the most accurate was the refractometer. However, this could be a function of the calculations to convert from refractive index to degrees Plato to density more than a specific function of the accuracy of the measurement. Moreover, the use of the refractometer with fermenting wort or beer samples would be very specific for those samples based on the amount of ethanol in the samples. In other words, the refractometer, while accurate as a measuring device for density of wort, would not be suitable for use during fermentation or postfermentation without significant additional study. In any case, the next most accurate measurements were provided by the densitometer.

The values in Table 1 indicate that the hydrometer and pycnometer were the least precise of the measuring devices. Not surprisingly, the values in Table 2 indicate that the hydrometer was not the most accurate measuring device. The refractometer and densitometer appeared to be similarly accurate. The pycnometer, on the other hand, was the most accurate.

During measurement of specific gravity, any deviation from the developed procedure typically resulted in a dramatic increase in the %RSD. This resulted in a loss of precision for that measurement. Similarly, any deviation from the procedure gave rise to a loss of accuracy in the measurement. It was found that even a 0.1°C deviation increased the %RSD for that measurement. Therefore, it is very important to note that all measurements were obtained at exactly 20.0°C after 20 min of attenuation of the sample. It should be noted that while the temperature chosen for the measurements in this study was 20.0°C, any temperature could be chosen as long as the pure water sample is also measured at that temperature.

Parallax played a very large role in the error found with each measurement of the hydrometer and refractometer. With practice to avoid parallax, the %RSD and percent error decreased to a minimum.

Conclusions

A summary of the four instruments used to measure specific gravity is provided in Table 3. Based on this information, our recommendation of the best instrument for use in measuring the density of a wort sample in the brewery is the densitometer. Its benefits in terms of the length of the procedure, ease of use, and precision of the measurement clearly make this the most appropriate choice.

However, each of these instruments has a place in the brewery. For example, an inexpensive hydrometer may suffice for measurements when accuracy and precision are not critical. In cases where the accuracy of the measurement is essential and the time to obtain the result is not critical, the pycnometer becomes the better choice.