Data Evaluation and Drawing Conclusions

In contrast, random errors are unpredictable fluctuations in measurements, such as changes in room temperature or human reaction time (typically $\approx 0.2\text{ s}$ when using a stopwatch). Reducing random errors improves precision, defined as the closeness of agreement between repeated measurements. Precise results cluster tightly together with a small range. We test precision through repeatability (same person, same equipment) and reproducibility (different person or different equipment).

Worked Example: Evaluating Reliability Two groups measure the acceleration due to gravity ($g$). The true value is $9.81\text{ m/s}^2$.

• Group A Results: $9.8, 9.6, 9.7$ (Mean $= 9.70\text{ m/s}^2$, Range $= 0.2$)
• Group B Results: $9.1, 10.5, 9.8$ (Mean $= 9.80\text{ m/s}^2$, Range $= 1.4$)

Evaluation:

• Arguments for Group A: Their data is more reliable because it is more precise. The range ($0.2$) is much smaller than Group B ($1.4$), suggesting lower random error and higher repeatability.
• Arguments for Group B: Their mean ($9.80$) is closer to the true value ($9.81$) than Group A's mean ($9.70$). Therefore, Group B’s result is more accuracy|accurate.
• Summary Judgement: Although Group B is more accurate, Group A's data is more trustworthy because the high precision suggests a well-controlled method. Group B should repeat their test with a larger sample size to reduce the impact of the obvious random errors.

Evaluating Experimental Strategies

To improve an experiment, you must identify weaknesses and suggest justified improvements:

• Increasing the Range: To better identify a trend or see if a relationship is truly linear, you should increase the range of data points. For example, in an electricity experiment, use a variable resistor to take readings across a wider range of voltages and currents.
• Timing: Use light gates or data loggers to eliminate human reaction time. Alternatively, measuring 10 oscillations instead of one reduces the percentage uncertainty.
• Thermal Control: Use insulation, a lid, or a water bath to keep thermal control variables constant, ensuring validity.
• Resolution: Upgrading to a vernier caliper ($0.1\text{ mm}$) or a micrometer ($0.01\text{ mm}$) instead of a ruler ($1\text{ mm}$) allows you to detect smaller changes.

Interpreting Data and Drawing Conclusions

A conclusion must be supported by a breakdown of data patterns using specific evidence.

Worked Example: Drawing Conclusions from Data A student measures the extension of a spring under different loads:

Force (N)	Extension (cm)
2.0	4.0
4.0	8.0
6.0	15.5 (Anomaly)
8.0	16.0

1. Identify Anomalies: $15.5\text{ cm}$ is an anomaly because it differs by more than $10\%$ from the expected trend. It should be excluded from calculations.
2. Determine Proportionality: Looking at the first two points, as the Force doubles ($2.0\text{ N}$ to $4.0\text{ N}$), the Extension also doubles ($4.0\text{ cm}$ to $8.0\text{ cm}$). This provides evidence for a directly proportional relationship ($y \propto x$).
3. Draw Conclusion: The data shows a linear relationship up to $8.0\text{ N}$, as the extension increases by $4.0\text{ cm}$ for every $2.0\text{ N}$ of force (excluding the outlier).

Scientific Confidence and Hypotheses

A hypothesis is a tentative explanation. We build confidence through evidence.

• Sample Size: A larger sample size (e.g., testing 100 components instead of 5) increases confidence because it reduces the impact of random errors on the mean, making the results more representative.
• Refutation: If new data contradicts a prediction, the hypothesis is refuted or modified.
• Peer Review: To reach the level of a scientific theory, findings must undergo peer review—scrutiny by independent experts—to ensure the method was valid and results are reproducible.

Worked Example: Evaluating Confidence Hypothesis: "Resistance increases with temperature."

• Evidence: At $20^\circ\text{C}$, $R = 10\Omega$. At $40^\circ\text{C}$, $R = 20\Omega$. Confidence increases as this supports the trend.
• New Evidence: A second lab (different equipment) gets the same results. Confidence increases further because the results are reproducible.