Use analytical methods to manipulate experimental data in physics

If you repeat a measurement several times, calculate the mean (average) to reduce random errors. The mean is found by adding all values and dividing by the number of readings.

$\text{Mean} = \frac{\text{Sum of all readings}}{\text{Number of readings}}$

For the pendulum data above, the period for one swing is:

• At 20 cm: $T = 8.9 \div 10 = 0.89$ s
• At 40 cm: $T = 12.6 \div 10 = 1.26$ s
• At 60 cm: $T = 15.5 \div 10 = 1.55$ s
• At 80 cm: $T = 17.8 \div 10 = 1.78$ s

Sometimes your experimental result differs from the accepted (true) value. The percentage error tells you how far off you were:

$\text{Percentage error} = \frac{|\text{Experimental value} - \text{Accepted value}|}{\text{Accepted value}} \times 100\%$

If you measure the density of a stone as $2.8 \, \text{g/cm}^3$ but the accepted value is $3.0 \, \text{g/cm}^3$:

$\text{Error} = \frac{|2.8 - 3.0|}{3.0} \times 100\% = \frac{0.2}{3.0} \times 100\% = 6.7\%$

Graphs help you see the relationship between variables clearly. For most physics experiments, you will use a line graph.

Steps to Draw a Line Graph

1. Identify variables: Place the independent variable on the x-axis (horizontal) and dependent variable on the y-axis (vertical)
2. Choose scales: Use simple scales that make the graph fill most of the paper
3. Label axes: Include the quantity name and unit
4. Plot points: Mark each data pair with a clear cross (×)
5. Draw the line of best fit: If points show a linear trend, draw a straight line that passes through or near most points

Worked Example: Pendulum Graph Analysis

Using the processed pendulum data:

When you plot $T^2$ against length, you get a straight line through the origin. This shows that $T^2$ is directly proportional to length — a key relationship in pendulum physics.

The slope of this graph equals $4\pi^2/g$, where $g$ is gravitational acceleration. By measuring the slope, students can actually calculate the value of $g$ from their experiment!

From a graph, you can determine:

• Direct proportion: Line passes through origin (straight line from zero)
• Linear relationship: Straight line that does not pass through origin
• Inverse proportion: Curve that gets flatter as x increases
• Slope: Rate of change; for a straight line, slope = rise/run

In Tanzania, market vendors use simple analytical methods when weighing goods. If a scale shows 1 kg but a known 1 kg mass reads 1.05 kg, a vendor can calculate the percentage error and adjust their prices accordingly. Similarly, when buying petrol, understanding the relationship between volume and cost helps you verify you received the correct amount at the filling station.