Problem 148 - Introductory Chemistry Atoms First

Question

A student measures the mass of an object three times and reports the numeric average of her measurements. If her three measurements are \(212 \mathrm{~g}\), \(260 \mathrm{~g}\), and \(233 \mathrm{~g}\) and the actual mass is \(235 \mathrm{~g}\), which of the following statements is true: (a) The student is accurate but not precise. (b) The student is precise but not accurate. (c) The student is both accurate and precise. (d) It is impossible to tell whether the student is accurate and \(/\) or precise without knowing how she determined the mass.

Step-by-Step Solution

Answer

The student is accurate but not precise.

Step 1: Understand Accuracy and Precision

Accuracy refers to how closely a measurement or set of measurements matches the true value. If the measured values are very close to the true value, the measurement is considered accurate. Precision, on the other hand, refers to how closely the measurements are to each other irrespective of their closeness to the true value. If the measurements are consistent and close to each other, the measurement is considered precise.

Step 2: Calculate the Numeric Average of the Measurements

To determine the accuracy and precision, let's first calculate the numeric average of the three measurements (\({212g, 260g, 233g}\)). The numeric average (or mean) can be calculated as follows: \[ Mean = \frac{Sum \: of \: the \: measurements}{Number \: of \: measurements} \]

Step 3: Calculate the Mean of the Measurements

Now, let's calculate the mean of the given measurements: \( Mean = \frac{212g + 260g + 233g}{3} = \frac{705g}{3} = 235g \) The calculated mean is \( 235g \), which is equal to the actual mass of the object.

Step 4: Compare the Mean with the Actual Mass

Since the mean of the measurements is equal to the actual mass, we can say that the student's measurements are accurate. However, we must analyze the precision. Given measurements are \({212g, 260g, 233g}\), notice that the measurements have a range of \(48g\) (considering the difference between the highest and the lowest values). This indicates a significant variation in the measurements.

Step 5: Determine the Accuracy and Precision

Based on our analysis: 1. The measurements are accurate since the mean of the measurements is equal to the actual mass. 2. The measurements are not precise since there is significant variation between the measurements. Therefore, the correct statement is: (a) The student is accurate but not precise.

Key Concepts

Mean Calculation

Mean calculation is a fundamental concept in statistics and measurement that helps to find the central value of a data set. It is also known as the

• average,
• arithmetic mean,

and is calculated by summing all the values in a given set and then dividing the sum by the number of values.
Let's consider the example where the measurements taken were:- 212 g- 260 g- 233 g
To calculate the mean of these measurements, you add them together:\[ 212 + 260 + 233 = 705 \]
Then divide the sum by the number of measurements, which in this instance is 3:\[ \text{Mean} = \frac{705}{3} = 235 \text{ g} \]
This calculated mean of 235 g tells you the average value of your measurements.
In relation to accuracy and precision, the mean can reveal how close the data is to the actual or true value.

Measurement Analysis

Measurement analysis involves examining both the mean and the range of values obtained during measurements to evaluate and understand their accuracy and precision.
Analyzing measurements means looking at how data aligns with the known standard (true value) and how scattered or cohesive the data points are.
In this exercise, the actual mass of the object is known to be 235 g. After calculating the mean, - We found that the mean is exactly 235 g, which suggests that the measurements are accurate.
However, to fully assess the overall quality of the measurements, precision must be considered as well.
In this context, the individual measurement values (212 g, 260 g, and 233 g) show some inconsistency when compared to each other, despite their average aligning with the true mass.

Precision Definition

Precision refers to the consistency or repeatability of measurement results.
It is all about how close repeated measurements are to each other.- When measurements are precise, there will be a minimal range and low variability.- They do not necessarily need to match the true value closely.
Considering our example:- The measurements were 212 g, 260 g, and 233 g.
The deviation between these measurements is large:- Difference between highest and lowest value: \[ 260 - 212 = 48 \text{ g} \]
This large range indicates a lack of precision, since a precise set of measurements would be closely grouped together with small differences.

• Thus, data with high precision may not always be accurate, but it shows consistency.
• In our scenario, despite the average being correct, the individual readings demonstrate low precision.

Accuracy Definition

Accuracy is a measure of how close a measurement or a set of measurements is to the actual or true value.
It reflects the correctness of measurements. - Unlike precision, accuracy addresses the agreement with a true standard.
In our example with a known actual mass of 235 g, the calculated mean of the measured values was also 235 g.

• This direct match signifies accuracy, meaning that the measurements on average align perfectly with the true value.
• Being accurate implies that the measurement is close to the target value.

However, accuracy as a standalone does not indicate closeness among individual measurements, which is where precision comes into the picture.
Achieving both high accuracy and high precision is ideal for reliable and valid measurement results.