Welcome to Experimental Analysis!

Ever felt like a detective? In Physics, collecting data is only half the job. The real "detective work" happens during analysis. This is where you take a page of messy numbers and turn them into a clear story about how the universe works. Whether you're finding the speed of sound or the resistance of a wire, these tools will help you reach a valid conclusion every time.

Note: This chapter focuses on what you do after you've collected your data. For how to set up the experiment, see "Planning and implementing experiments."

1. Understanding Your Data: The Basics

Before we do heavy calculations, we need to understand the "quality" of our numbers. Not all measurements are created equal!

Key Terms to Master

  • Accuracy: How close your measurement is to the "true" or accepted value. If the true value of \(g\) is \(9.81 \, \text{N kg}^{-1}\) and you get \(9.80\), you are very accurate!
  • Precision: How close your repeated measurements are to each other. If you measure the same length three times and get \(1.21 \, \text{m}\), \(1.21 \, \text{m}\), and \(1.22 \, \text{m}\), your results are precise.
  • Resolution: The smallest change an instrument can detect. A ruler with millimeter markings has a higher resolution than one with only centimeter markings.
  • Sensitivity: The change in output divided by the change in input. A thermometer that moves \(2 \, \text{cm}\) for every \(1^\circ\text{C}\) is more sensitive than one that moves only \(0.5 \, \text{cm}\).
  • Response Time: How long it takes for an instrument to give a stable reading after a change occurs.

Systematic vs. Random Errors

Random Errors cause readings to scatter around the mean. You can reduce their effect by taking repeats and calculating a mean.

Systematic Errors (like a zero error, where a scale doesn't start at zero) shift all your readings by the same amount. Repeats won't fix this—you have to identify the cause or subtract the error from every result!

Quick Review: Think of a target. If all your arrows hit the bullseye, you are accurate and precise. If they are all bunched together in the corner, you are precise but not accurate.

2. Processing Your Results

Once you have your data, you need to find the "best" value and see how much it varies.

Finding the Center: Mean and Median

You can use a dot-plot (a simple number line with dots for each measurement) to see your distribution.

  • Mean: Add all values and divide by the number of values.
  • Median: The middle value when they are listed in order.
  • Outliers (Anomalies): These are values that sit far away from the rest. If you spot one, check your notes—did something go wrong? If it's a clear mistake, you can exclude it from your mean calculation.

Calculating Spread and Uncertainty

In Advancing Physics, we use spread to show how much our data varies:

\( \text{Spread} = \pm \frac{\text{Range}}{2} \)

Example: If you measure \(10.2 \, \text{s}\), \(10.4 \, \text{s}\), and \(10.6 \, \text{s}\):
Range = \(10.6 - 10.2 = 0.4 \, \text{s}\)
Spread = \( \pm 0.2 \, \text{s}\)
Result = \( 10.4 \pm 0.2 \, \text{s} \)

Percentage Uncertainty is often more useful for comparing different measurements:

\( \text{Percentage Uncertainty} = \frac{\text{Uncertainty (Spread)}}{\text{Mean Value}} \times 100\% \)

3. Combining Uncertainties: The "Extreme Values" Method

Don't worry if this seems tricky! The OCR Physics B syllabus uses a simple Extreme Values approach for AS level. You don't need complex statistical formulas.

To find the uncertainty in a calculated value (like speed = distance / time), you calculate the final answer using the worst-case scenario values from your measurements.

Example: Finding the uncertainty in Speed (\(v = d/t\))

  1. Calculate the Best Value for speed using the mean distance and mean time.
  2. Calculate the Maximum Value for speed by using the largest possible distance and the smallest possible time (since dividing by a smaller number gives a bigger result).
  3. The difference between the Best Value and the Extreme Value is your uncertainty.

Key Takeaway: Always identify which measurement has the largest percentage uncertainty. This is usually the "limiting factor" of your experiment's quality.

4. Graph Skills for Success

Graphs are the "Gold Standard" of analysis. They help you see patterns that numbers alone might hide.

Plotting Rules

  • Axes: Label with the quantity and unit (e.g., \( \text{Time / s} \)). The Independent Variable (what you change) goes on the x-axis.
  • Scale: Choose a scale where the data fills at least half the page. Use "easy" numbers like 1, 2, 5, or 10 per large square.
  • Uncertainty Bars: Draw little "I-beams" through your points to show the uncertainty in the measurement.

Gradients and Intercepts

Most relationships in Physics B follow the straight-line equation: \( y = mx + c \)

  • Gradient (\(m\)): \( \frac{\Delta y}{\Delta x} \). Use a large triangle to calculate this for better accuracy.
  • y-intercept (\(c\)): Where the line crosses the vertical axis (where \( x = 0 \)).

Dealing with Uncertainty in Graphs:
To find the uncertainty in your gradient, you can draw a "Worst-fit" line. This is the steepest or shallowest line that still passes through all your uncertainty bars. The difference between the "Best-fit" gradient and the "Worst-fit" gradient is your uncertainty.

5. Presentation and Mathematical Skills

Significant Figures (sf)

Rule: Your final answer should have no more significant figures than the least precise piece of data you used. If your distance is \(1.2 \, \text{m}\) (2 sf) and your time is \(0.456 \, \text{s}\) (3 sf), your speed should be given to 2 sf.

Standard Form and Prefixes

Get comfortable with the prefixes in your Data Booklet:

  • Giga (G): \( \times 10^9 \)
  • Mega (M): \( \times 10^6 \)
  • Kilo (k): \( \times 10^3 \)
  • Milli (m): \( \times 10^{-3} \)
  • Micro (\( \mu \)): \( \times 10^{-6} \)
  • Nano (n): \( \times 10^{-9} \)
  • Pico (p): \( \times 10^{-12} \)

Memory Trick: King Henry Died By Drinking Chocolate Milk (Kilo, Hecto, Deca, Base, Deci, Centi, Milli). In Physics B, we focus mostly on the powers of 3 (kilo, milli, micro, etc.).

6. Reaching Valid Conclusions

Finally, your analysis must lead to a conclusion. This is a statement that sums up what the data shows.

Is your result valid? A result is valid if it actually measures what it was supposed to measure and is supported by the data. Check your percentage uncertainty—if it's \( 50\% \), your conclusion might not be very strong! If your line of best fit passes through the origin (0,0) and is a straight line, you can conclude the variables are directly proportional.

Quick Summary for the Exam:
1. Check for outliers and calculate the mean.
2. Find the spread (\( \pm \text{range}/2 \)).
3. Use extreme values to find the uncertainty in calculated totals.
4. Use gradients from graphs to find physical constants (like \(R\) or \(g\)).
5. Match your significant figures to the raw data.