A student measures the density of a liquid five times and gets the results (all in gram/cm3 ) 1.8, 2.0, 2.0, 1.9, and 1.8.

What would you suggest as the best estimate and uncertainty based on these measurements?

Question

A student measures the density of a liquid five times and gets the results (all in gram/cm3 ) 1.8, 2.0, 2.0, 1.9, and 1.8.

What would you suggest as the best estimate and uncertainty based on these measurements?

BlackTom AI · Accepted Answer

The problem presents a set of density measurements for a liquid: 1.8, 2.0, 2.0, 1.9, and 1.8 g/cm^3. To decide the best estimate and uncertainty, we examine the provided options and the common ways to estimate uncertainties from repeated measurements.

Option 1: best estimate = 1.9 g/cm^3 ; uncertainty = 0.1 g/cm^3
This uses the mean of the measurements as the best estimate, which is (1.8 + 1.8 + 1.9 + 2.0 + 2.0)/5 = 9.5/5 = 1.9. The uncertainty here is 0.1, which corresponds to half of the overall range of the data (range = 2.0 − 1.8 = 0.2). This half-range approach is a common quick-and-dirty way to quote an uncertainty when you have a small sample: you state the spread of the data as ± half the range.

Option 2: best estimate = 1.9 g/cm^3 ; uncertainty = 0.2 g/cm^3
Here the uncertainty equals the full range (0.2). While this captures the total spread, it overstates the precision a bit, since a single value of ±0.2 around 1.9 would allow values from 1.7 to 2.1, but the data’s central tendency is better described with a tighter figure like ±0.1 using the half-range method or a standard error estimate. This option is not consistent with common practice for small data sets where half the range is preferred over the full range as the uncertainty.

Option 3: best estimate = 1.90 g/cm^3; uncertainty = 0.02 g/cm^3
This uses two decimal places for both the estimate and the uncertainty, implying a precision not supported by the data. The measurements lie at 1.8, 1.9, and 2.0 to one decimal place, so giving 1.90 and 0.02 would imply a precision of hundredths, which is misleading given the actual scatter and the measurement resolution. The data do not justify such a small uncertainty.

Option 4: best estimate = 1.90 g/cm^3 ; uncertainty = 0.04 g/cm^3
This reports a mean of 1.90 (same as 1.9) with an uncertainty around 0.04. The standard error of the mean for these five measurements is s/√n where s ≈ 0.1 (the sample standard deviation considering deviations −0.1, −0.1, 0, 0.1, 0.1 gives s ≈ 0.1), so the standard error is about 0.1/√5 ≈ 0.045, which rounds to 0.04. This approach reflects the precision of the mean itself rather than the spread of individual measurements and is a reasonable estimate, though some curricula prefer the half-range for uncertainty when quoting a single value with its precision.

In summary, each option uses a different convention for reporting uncertainty. The choice of method affects how confidently we present the best estimate and how wide the stated uncertainty is. The key ideas are: determine a central value from the measurements (often the mean), and quantify spread using a consistent method (half the range, standard deviation, or standard error), with the chosen method aligning to the context and instruction.

SPHY001 LAB 6 (Pre-Lab) How to Write the Conclusion of a Laboratory Report | LA010

A student measures the density of a liquid five times and gets the results (all in gram/cm3 ) 1.8, 2.0, 2.0, 1.9, and 1.8. What would you suggest as the best estimate and uncertainty based on these measurements?

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