Question

Two students count the grains of uncooked rice in a small cup. Both students repeat this measurement four times, with the following results: \(\begin{array}{ll}\text { Mike: } 256,263,262,266 & \text { Ike: } 250,242,270,278\end{array}\) The actual number of grains is 260 . Which student is more accurate? Which is more precise? Explain your answers.

Short answer

Ike is more accurate because his mean (260) exactly matches the actual number of grains (260), while Mike's mean (261.75) is 1.75 grains away from the actual number. Mike is more precise since he has a smaller range (10) compared to Ike's range (36).

Step-by-step solution

Calculate the mean for each student

To find the mean of each student's measurements, we will add up all the values and divide the sum by the number of values. For Mike, the mean is: \(\frac{256 + 263 + 262 + 266}{4} = \frac{1047}{4} = 261.75\) For Ike, the mean is: \(\frac{250 + 242 + 270 + 278}{4} = \frac{1040}{4} = 260\)

Determine accuracy of the measurements

Accuracy refers to how close a measurement is to the actual value. Since the actual number of grains is 260, we will compare each student's mean to this value to determine their accuracy. Mike's mean: 261.75, which is 1.75 grains away from the actual number Ike's mean: 260, which is 0 grains away from the actual number Considering the differences, Ike's measurements are more accurate because his mean exactly matches the actual number of grains.

Determine precision of the measurements

Precision refers to how close the measurements are to each other. To determine the precision, we will analyze the spread of the measurements for each student. Mike's measurements: 256, 263, 262, and 266 The range of Mike's measurements is 266 - 256 = 10 Ike's measurements: 250, 242, 270, and 278 The range of Ike's measurements is 278 - 242 = 36 Since Mike has a smaller range, his measurements are more precise than Ike's. In conclusion, Ike is more accurate with his measurements, and Mike is more precise.

Key concepts

Scientific Measurement

Scientific measurement is a fundamental concept that enables us to quantify the properties of matter and the changes that it undergoes. The core of scientific measurement is based on using established units and tools to collect data which can then be analyzed to draw conclusions. For instance, the counting of rice grains in a cup by both Mike and Ike is an exercise in scientific measurement.

In scientific practice, measurements must strive to be both accurate – close to the true or accepted value, like the actual number of grains – and precise – producing similar results upon repetition of the experiment. The procedure employed by Mike and Ike, where each conducted multiple counts, is an excellent example of applying repeated measurements to gauge both accuracy and precision.

Data Analysis

Data analysis in the context of scientific measurement involves scrutinizing the collected data to derive meaningful information. It is a process of inspecting, cleansing, transforming, and modeling data with the goal of discovering useful information, suggesting conclusions, and supporting decision-making.

For Mike and Ike, analyzing their data involved calculating the mean of their measurements to determine accuracy. Additional analysis, such as computing the range of their measurements, provided insight into the precision of their counting. The comparison between their results and the actual number of grains is an essential part of data analysis that helps in evaluating the level of accuracy.

Mean Calculation

The mean calculation is a method of finding the central or 'average' value in a set of numbers, which is a crucial part of data analysis. It is achieved by summing all the values in a data set and dividing by the count of those values. In the exercise, calculating the mean of the rice grain counts gave us Mike’s average count of 261.75 and Ike's average count of 260.

When assessing measurements for scientific work, the mean helps provide a single value that best represents the data set. It's important to note that while a mean is useful for comparison, it does not directly tell us about the spread or dispersion of the measurements; that's where evaluating precision comes in.

Measurement Comparison

Measurement comparison is a method used to assess the variation between different measurements and the actual value. Essentially, it helps distinguish between the concepts of accuracy and precision. In our example, comparing the mean values to the actual number of grains (260) allowed us to determine that Ike's measurements were more accurate than Mike's.

Additionally, comparing the spread of the measurements, shown by the range (difference between the highest and lowest values), highlighted that Mike's data was more precise due to a smaller range. Effective measurement comparison helps in determining reliability and validity in scientific measurements and is a necessary skill in data analysis.