Three part question
Two students each measured the density of a quartz sample three times:
Student A Student B
1. 3.20g/mL 2.82g/mL
2. 2.58g/mL 2.48g/mL
3. 2.10g/mL 2.59g/mL
mean 2.63g/mL 2.63 g/mL
a. which student measured density with the greatest precision? explain your answer.
b. which student measured density with the greatest accuracy? explain your answer.
c. are the errors for these students random or systematic? explain
a. To determine which student measured density with the greatest precision, we need to compare the variability of their measurements. Precision refers to the degree to which repeated measurements give similar results. One way to measure precision is by calculating the standard deviation of the measurements. The lower the standard deviation, the higher the precision.
For Student A:
Measurements: 3.20g/mL, 2.58g/mL, 2.10g/mL
Mean: 2.63g/mL
To calculate the standard deviation, we can use the following formula:
Standard deviation = √[ (∑(xi - mean)^2) / (N-1) ]
where xi represents each measurement, mean is the average, and N is the total number of measurements.
Standard deviation for Student A:
√[ ( (3.20 - 2.63)^2 + (2.58 - 2.63)^2 + (2.10 - 2.63)^2 ) / (3-1) ] ≈ 0.55g/mL
For Student B:
Measurements: 2.82g/mL, 2.48g/mL, 2.59g/mL
Mean: 2.63g/mL
Standard deviation for Student B:
√[ ( (2.82 - 2.63)^2 + (2.48 - 2.63)^2 + (2.59 - 2.63)^2 ) / (3-1) ] ≈ 0.16g/mL
Comparing the standard deviations, we can see that Student B has a lower standard deviation (0.16g/mL) compared to Student A (0.55g/mL). This indicates that Student B measured density with greater precision.
b. To determine which student measured density with the greatest accuracy, we need to compare their measurements to the actual or accepted value. Accuracy refers to the closeness of a measured value to the true value.
Since we don't have the actual value of the quartz sample's density, we can't directly determine accuracy. However, we can compare the means of both students' measurements. The one whose mean is closer to the actual value is considered more accurate.
Both students have a mean density of 2.63g/mL. Without knowing the actual value, we cannot determine which student measured density with greater accuracy.
c. To determine if the errors for these students are random or systematic, we need to assess the consistency of the differences between their measurements and the mean.
For Student A, the differences between their measurements and the mean are:
3.20g/mL - 2.63g/mL = 0.57g/mL
2.58g/mL - 2.63g/mL = -0.05g/mL
2.10g/mL - 2.63g/mL = -0.53g/mL
For Student B, the differences between their measurements and the mean are:
2.82g/mL - 2.63g/mL = 0.19g/mL
2.48g/mL - 2.63g/mL = -0.15g/mL
2.59g/mL - 2.63g/mL = -0.04g/mL
Looking at the differences, we can see that there is variability in both students' measurements, with positives and negatives. This indicates random errors, meaning the measurements are affected by various factors that cause slight inconsistencies.
Therefore, the errors for both students are random, rather than systematic.
a. To determine the student who measured density with the greatest precision, we need to calculate the standard deviation for each student's measurements. The standard deviation measures the spread or variability of the data points around the mean.
For Student A:
- Calculate the differences between each measurement and the mean: (3.20 - 2.63), (2.58 - 2.63), (2.10 - 2.63)
- Square each difference: (0.57)^2, (-0.05)^2, (-0.53)^2
- Calculate the average of the squared differences: [(0.57)^2 + (-0.05)^2 + (-0.53)^2]/3 = 0.3447
- Take the square root of the average: sqrt(0.3447) ≈ 0.588
For Student B:
- Calculate the differences between each measurement and the mean: (2.82 - 2.63), (2.48 - 2.63), (2.59 - 2.63)
- Square each difference: (0.19)^2, (-0.15)^2, (-0.04)^2
- Calculate the average of the squared differences: [(0.19)^2 + (-0.15)^2 + (-0.04)^2]/3 = 0.0467
- Take the square root of the average: sqrt(0.0467) ≈ 0.216
Comparing the standard deviations, we can see that Student A has a larger standard deviation (0.588) compared to Student B's standard deviation (0.216). Therefore, Student B measured density with greater precision since their measurements were less spread out from the mean.
b. To determine the student who measured density with the greatest accuracy, we need to assess how close their mean value is to the true or accepted value. However, we do not have the true or accepted value for density, so we cannot determine accuracy in this case.
c. To assess whether the errors for these students are random or systematic, we need to look at the pattern of the differences between each measurement and the mean for each student.
For Student A, the differences are: +0.57, -0.05, -0.53. There is no clear pattern, indicating random errors.
For Student B, the differences are: +0.19, -0.15, -0.04. Again, there is no clear pattern, suggesting random errors.
Therefore, the errors for both students appear to be random rather than systematic, as there is no consistent bias in their measurements.