Welcome to IaS2: What Conclusions Can We Make from Data?
Have you ever completed a physics practical, written down a page full of numbers, and wondered: "What does all of this actually mean?" If so, you are not alone! Collecting data is only the first step in science. The real superpower is knowing how to process that data, spot patterns, judge its quality, and draw rock-solid scientific conclusions.
In this chapter of your OCR GCSE (9–1) Physics B (Twenty First Century Science) course, you will learn the exact toolkit scientists use to turn raw numbers into clear evidence. These skills are tested across both of your written exam papers (Paper 1 Breadth and Paper 2 Depth), so mastering them will boost your grades across every single physics topic.
---1. Handling Numbers, SI Units, and Significant Figures
A. Standard SI Units and Prefixes
In physics, a number without a unit is meaningless. We use standard SI units (Système International) so that scientists all over the world understand our measurements.
Common base and derived SI units in physics include:
• Length / Distance: metres (\(\text{m}\))
• Mass: kilograms (\(\text{kg}\))
• Time: seconds (\(\text{s}\))
• Energy: joules (\(\text{J}\))
• Power: watts (\(\text{W}\))
• Force: newtons (\(\text{N}\))
• Pressure: pascals (\(\text{Pa}\))
Measurements can be very large or very small. We use metric prefixes to change the size of the unit:
• Mega- (\(\text{M}\)): \(\times 10^6\) or \(\times 1\,000\,000\) (e.g. \(1\text{ MJ} = 1\,000\,000\text{ J}\))
• kilo- (\(\text{k}\)): \(\times 10^3\) or \(\times 1\,000\) (e.g. \(1\text{ km} = 1\,000\text{ m}\), \(1\text{ kW} = 1\,000\text{ W}\), \(1\text{ kg} = 1\,000\text{ g}\))
• milli- (\(\text{m}\)): \(\times 10^{-3}\) or \(\div 1\,000\) (e.g. \(1\text{ mm} = 0.001\text{ m}\), \(1\text{ ms} = 0.001\text{ s}\), \(1\text{ mg} = 0.001\text{ g}\))
Memory Trick: To convert from a smaller unit to a bigger unit (e.g. \(\text{g}\) to \(\text{kg}\)), divide. To convert from a bigger unit to a smaller unit (e.g. \(\text{kJ}\) to \(\text{J}\)), multiply.
B. Standard Form and Order of Magnitude
Very large and very small numbers are written in standard form: \(A \times 10^n\), where \(A\) is a number between \(1\) and \(10\), and \(n\) is an integer (positive or negative).
Example: \(450\,000\text{ W} = 4.5 \times 10^5\text{ W}\)
Example: \(0.0032\text{ m} = 3.2 \times 10^{-3}\text{ m}\)
C. Significant Figures (\(\text{s.f.}\))
When you calculate an answer, your calculator will often give you a long string of decimals. Quoting an answer like \(3.4871923\text{ N}\) suggests an impossible level of precision!
Rule: Always quote your final calculated answer to the same number of significant figures as the least precise measurement used in your calculation.
Example: If force is \(12\text{ N}\) (2 s.f.) and area is \(2.45\text{ m}^2\) (3 s.f.), your calculated pressure must be rounded to 2 significant figures.
Key Takeaway: Check your units, convert prefixes before calculating, and round your final answer to match the least precise input data.
---2. Plotting Graphs and Spotting Mathematical Relationships
A. Rules for Graph Plotting
When presenting continuous data in physics, scatter plots and line graphs are preferred. Follow these strict examiner rules when drawing graphs:
• Axes & Scales: Use simple, linear scales (e.g. in steps of \(1, 2, 5, 10\)). Never use awkward steps like \(3\) or \(7\). Your plotted points must cover more than half of the grid in both directions.
• Labels: Label each axis clearly with the quantity and its unit (e.g. Distance (\(\text{m}\)), Time (\(\text{s}\))).
• Plotting: Plot points neatly with a sharp pencil as a small cross (\(\times\)) or an encircled dot (\(\odot\)) within \(\pm \frac{1}{2}\) of a small square.
• Line of Best Fit: Draw a single, smooth straight line (with a ruler) or a smooth curve passing evenly through or between the points. Never do dot-to-dot! Ignore any clear outliers.
B. Interpreting Linear Relationships (\(y = mx + c\))
A straight-line graph follows the general equation:
\(y = mx + c\)
• \(m\) is the gradient (slope of the line).
• \(c\) is the \(y\)-intercept (where the line crosses the vertical \(y\)-axis).
To calculate the gradient \(m\):
\(m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\)
Tip: Pick two points on your line of best fit that are far apart (drawing a large triangle) to calculate the gradient accurately.
C. Direct Proportionality vs. Inverse Proportionality
Students often mix these up, but examiners test them rigorously!
1. Direct Proportionality (\(y \propto x\)):
• Two variables are directly proportional if doubling one variable causes the other to double.
• On a graph, this shows as a straight line that passes directly through the origin \((0,0)\).
• The ratio is constant: \(\frac{y}{x} = \text{constant}\).
Common Mistake: An upward sloping straight line that does not pass through \((0,0)\) is a linear relationship, but it is not directly proportional!
2. Inverse Proportionality (\(y \propto \frac{1}{x}\)):
• As one variable increases, the other decreases such that their product remains constant: \(y \cdot x = \text{constant}\).
• On a graph of \(y\) against \(x\), this appears as a downward curve (a hyperbola).
• If you plot \(y\) against \(\frac{1}{x}\), you will get a straight line through the origin.
D. Rates of Change
The gradient of a graph often represents a physical rate of change:
• On a distance–time graph, the gradient equals speed (\(\frac{\Delta \text{distance}}{\Delta \text{time}}\)).
• On a velocity–time graph, the gradient equals acceleration (\(\frac{\Delta \text{velocity}}{\Delta \text{time}}\)).
• If the graph is curved, you can find the rate of change at any point by drawing a tangent (a straight line touching the curve at that exact point without crossing it) and calculating the gradient of that tangent.
Key Takeaway: A straight line through \((0,0)\) means direct proportionality. Gradient represents a rate of change (\(\frac{\Delta y}{\Delta x}\)).
---3. Data Quality: Accuracy, Precision, Errors, and Uncertainty
A. Accuracy vs. Precision
In everyday language, people use these words interchangeably. In physics, they mean completely different things!
• Accuracy: How close a measured value is to the true value of the quantity.
• Precision: How close repeated measurements are to each other (the spread or closeness of agreement between repeats).
The Dartboard Analogy:
• Darts clustered tightly together in the bullseye = Accurate and precise.
• Darts clustered tightly together in the top-left corner = Precise, but not accurate (consistent, but off target!).
• Darts spread out all over the board, but centered around the middle = Accurate on average, but not precise.
B. Repeatability vs. Reproducibility
• Repeatable: The original experimenter repeats the investigation using the same method and apparatus in the same lab over a short time, and gets the same results.
• Reproducible: A different person, or the same person using different equipment / methods or testing in a different lab, gets the same results.
C. Experimental Errors: Random vs. Systematic
Every measurement has some degree of uncertainty caused by errors.
1. Random Errors:
• Unpredictable variations that cause individual readings to be slightly higher or slightly lower than the true value.
• Causes: Human reaction time when using a stopwatch, minor temperature fluctuations, or slight variations in reading between scale divisions.
• How to reduce them: Take at least three repeat readings, discard any anomalous results, and calculate the mean.
2. Systematic Errors:
• Consistent errors that shift every single reading away from the true value by the same fixed amount or proportion each time.
• Causes: Incorrectly calibrated equipment, or a balance that does not read zero when empty.
• Zero Error: A specific type of systematic error where a measuring device gives a non-zero reading when the true value is zero (e.g. a voltmeter reading \(0.2\text{ V}\) when disconnected).
• How to fix them: Systematic errors cannot be eliminated by repeating readings or averaging! You must recalibrate the instrument, adjust for zero error (tare the balance), or refine the experimental technique.
D. Handling Outliers (Anomalies)
An outlier (or anomaly) is a measurement that lies distinctly outside the pattern of the rest of the data.
When you spot an outlier:
1. Identify it.
2. Investigate the possible cause (e.g. misreading a scale, timer started late).
3. Discard it before calculating the mean.
4. Repeat the measurement if possible.
E. Calculating the Mean, Range, and Uncertainty
When you have repeat readings (after removing anomalies):
1. Mean (\(\bar{x}\)):
\(\text{Mean} = \frac{\text{Sum of valid repeats}}{\text{Number of valid repeats}}\)
2. Range:
\(\text{Range} = \text{Maximum value} - \text{Minimum value}\)
3. Absolute Uncertainty:
The uncertainty represents the interval within which the true value is expected to lie. From repeat readings, uncertainty is estimated as:
\(\text{Uncertainty} = \pm \frac{\text{Range}}{2}\)
Worked Example: A student measures the time for a trolley to roll down a ramp. The repeat times are \(2.4\text{ s}\), \(3.9\text{ s}\) (anomalous), \(2.6\text{ s}\), and \(2.5\text{ s}\).
• Step 1: Discard the outlier (\(3.9\text{ s}\)).
• Step 2: Calculate the mean: \(\text{Mean} = \frac{2.4 + 2.6 + 2.5}{3} = \frac{7.5}{3} = 2.5\text{ s}\).
• Step 3: Find the range of valid repeats: \(\text{Range} = 2.6 - 2.4 = 0.2\text{ s}\).
• Step 4: Calculate uncertainty: \(\pm \frac{0.2}{2} = \pm 0.1\text{ s}\).
• Final quoted result: \(2.5 \pm 0.1\text{ s}\).
Key Takeaway: Random errors are reduced by repeats and means. Systematic errors shift all data and require instrument adjustments. Uncertainty is \(\pm \frac{\text{Range}}{2}\).
---4. Drawing Valid Conclusions: Correlation vs. Causation
A. Making Inferences
An inference is a conclusion drawn strictly from empirical data. You must be careful not to extrapolate (guess) beyond the range of data you have measured unless you have a proven scientific principle to support it.
B. Correlation vs. Causation
One of the most important concepts in science is understanding the difference between a link and a cause:
• Correlation: A relationship or pattern between two variables (e.g. as variable \(A\) increases, variable \(B\) increases). This is seen as a trend on a scatter plot.
• Causation (Causal Mechanism): Demonstrates that a change in variable \(A\) is directly causing the change in variable \(B\).
Crucial Rule: Correlation does NOT prove causation!
Just because two variables show a correlation does not mean one causes the other. A correlation could be:
1. Pure coincidence.
2. Caused by a third, unmeasured confounding variable that affects both.
Example: Ice cream sales and the number of sunburns both increase at the same time (positive correlation). Does eating ice cream cause sunburn? No! The confounding variable is hot sunny weather, which causes both.
To prove causation in physics, scientists must carry out controlled experiments where all control variables are kept constant, or demonstrate a known physical mechanism (e.g. increasing potential difference across a fixed resistor increases the electric field, forcing more electrons through per second, thereby increasing current).
Key Takeaway: A graph can show a correlation, but only controlled experiments and proven physical mechanisms can prove causation.
---5. Quick Summary & Common Exam Traps
Don't lose easy marks! Watch out for these classic exam pitfalls:
• Trap 1: Calling any upward straight line "directly proportional". Remember: it must pass through \((0,0)\).
• Trap 2: Including an anomaly when calculating the mean. Always cross out anomalies first!
• Trap 3: Confusing accuracy and precision. Precise data can still be completely inaccurate if there is a systematic zero error.
• Trap 4: Drawing "dot-to-dot" lines on graphs. Always use a single ruled line or smooth curve of best fit.
• Trap 5: Confusing repeatability with reproducibility. If the same student repeats the test in the same lesson, that tests repeatability, not reproducibility.