# Determining Uncertainties from Graphs ## Error Bars - The uncertainty in a measurement can be shown on a graph as an **error bar** - This bar is drawn above and below the point (or from side to side) and shows the uncertainty in that measurement - Error bars are plotted on graphs to show the **absolute uncertainty** of values plotted - Usually, error bars will be in the vertical direction, for y-values, but can also be plotted horizontally, for x-values

Error bars represent the absolute uncertainty in a measurement, displayed as a range extending from the data point.

### Error Bar Characteristics - **Vertical error bars**: show uncertainty in the y-axis (dependent variable) measurements - **Horizontal error bars**: show uncertainty in the x-axis (independent variable) measurements - **Length**: the full length of the error bar represents $\pm$ the absolute uncertainty - **Large error bars**: indicate greater uncertainty in that particular measurement - **Small error bars**: indicate lower uncertainty and more reliable measurements ## Determining Uncertainties from Gradients - To calculate the uncertainty in a gradient, two lines of best fit should be drawn on the graph: ### The Best Line of Best Fit - This line passes as close to the points as possible - It should have roughly equal numbers of points above and below it - This gives the most representative gradient for your data ### The Worst Line of Best Fit - This is either the **steepest possible** or the **shallowest possible** line - It must still fit within all the error bars - Choose whichever gives the greatest difference from the best fit gradient - This represents the maximum reasonable deviation from your best estimate

The worst line must pass through all error bars, not just the data points themselves.

### Calculating Percentage Uncertainty in Gradient The percentage uncertainty in the gradient can be found using: $$\text{Percentage uncertainty} = \frac{\text{best gradient} - \text{worst gradient}}{\text{best gradient}} \times 100\%$$ **Example calculation:** - Best gradient = $0.25$ - Worst gradient = $0.28$ $$\text{Percentage uncertainty} = \frac{0.28 - 0.25}{0.25} \times 100\% = 12\%$$

Always use the absolute value of the difference between best and worst gradients in the numerator.

### Calculating Percentage Uncertainty in Y-Intercept The percentage uncertainty in the y-intercept can be found using: $$\text{Percentage uncertainty} = \frac{\text{best y intercept} - \text{worst y intercept}}{\text{best y intercept}} \times 100\%$$ ## Percentage Difference - The percentage difference gives an indication of how close the experimental value achieved from an experiment is to the accepted value - It is **not** a percentage uncertainty

Percentage difference = $\frac{\text{Experimental Value} - \text{Accepted Value}}{\text{Accepted Value}} \times 100\%$

### Understanding the Terms - **Experimental value**: sometimes referred to as the 'measured' value - This is the value you obtained from your experiment - **Accepted value**: sometimes referred to as the 'true' value - This may be labelled on a component such as the capacitance of a capacitor or the resistance of a resistor - Or from a databook ### Example Suppose you're measuring the acceleration due to gravity: - **Accepted value**: $g = 9.81 \text{ m s}^{-2}$ (this is the standard databook value) - **Experimental value**: from your experiment, you measure $g = 10.35 \text{ m s}^{-2}$ $$\text{Percentage difference} = \frac{10.35 - 9.81}{9.81} \times 100\% = 5.5\%$$ ### Interpreting Percentage Difference - The **smaller** the percentage difference, the more **accurate** the results of the experiment - A large percentage difference suggests systematic errors or issues with the experimental method - It provides a quantitative measure of how well your experiment replicated the accepted physics

If your percentage difference is consistently large, look for systematic errors like miscalibrated equipment or flaws in your methodology.

## Worked Example: Complete Uncertainty Analysis **Task**: Plot the following data, draw error bars, draw best and worst fit lines, and find the percentage uncertainty in the gradient. | Force / N | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | |-----------|----|----|----|----|----|----|----|----| | Extension / mm | $8.5 \pm 1$ | $11 \pm 0.5$ | $15 \pm 1$ | $15 \pm 2$ | $20 \pm 1.5$ | $19.5 \pm 2$ | $22 \pm 0.5$ | $26 \pm 1$ | ### Step 1: Draw Sensible Scales and Plot Data - Choose scales that use as much of the available space as possible - Ensure each axis is labelled with quantity and unit (e.g., "Force / N", "Extension / mm") - Plot each data point precisely to within 0.5 squares ### Step 2: Draw Error Bars - For each point, draw a vertical line extending above and below by the uncertainty value - For example, the point at Force = 10 N has Extension = $8.5 \pm 1$ mm - Draw the error bar from 7.5 mm to 9.5 mm vertically - The point at Force = 50 N has Extension = $20 \pm 1.5$ mm - Draw the error bar from 18.5 mm to 21.5 mm ### Step 3: Draw the Best Line of Best Fit - Use a ruler and draw a straight line that: - Passes as close as possible to all data points - Has roughly equal numbers of points above and below the line - Doesn't necessarily pass through the origin (unless the physics demands it) - This line represents your best estimate of the relationship ### Step 4: Draw the Worst Line of Best Fit - Draw a second line that: - Still passes through all error bars - Has the steepest or shallowest gradient possible - Represents the maximum reasonable deviation from your best fit - Choose whichever (steepest or shallowest) gives the greater difference from the best gradient ### Visualizing the Data The scatter plot below shows the experimental data points from the Force-Extension measurements:

### Visualizing Best and Worst Fit Lines The chart below illustrates both the best fit line (gradient = 0.25) and the worst fit line (gradient = 0.28) that would be drawn through the data:

The difference between these two gradients determines the percentage uncertainty in the measurement. ### Step 5: Calculate Gradients For the **best fit line**: - Choose two points far apart on your line (ideally use the full range) - Read coordinates: $(x_1, y_1) = (0, 6)$ and $(x_2, y_2) = (80, 26)$ $$\text{Best gradient} = \frac{\Delta y}{\Delta x} = \frac{26 - 6}{80 - 0} = \frac{20}{80} = 0.25$$ For the **worst fit line**: - Read coordinates: $(x_1, y_1) = (0, 4.7)$ and $(x_2, y_2) = (80, 27)$ $$\text{Worst gradient} = \frac{\Delta y}{\Delta x} = \frac{27 - 4.7}{80 - 0} = \frac{22.3}{80} = 0.28$$ ### Step 6: Calculate Percentage Uncertainty $$\text{Percentage uncertainty} = \frac{0.28 - 0.25}{0.25} \times 100\% = 12\%$$

When calculating gradients, select points from your lines (not necessarily data points) that are as far apart as possible to minimize rounding errors.

## Key Graph-Drawing Rules for Exams To gain full marks when drawing graphs: 1. **Scale**: - Ensure the scale is sensible and takes up as much paper as possible - Avoid awkward scales (e.g., multiples of 3 or 7) 2. **Axes**: - Label the axes with both quantity and unit - Format: "Quantity / Unit" (e.g., "Time / s" or "Voltage / V") 3. **Plotting**: - Precisely plot the points to within 0.5 squares - Use small, neat crosses or dots with circles 4. **Line of Best Fit**: - Leave a roughly equal number of points above and below the best fit line - Use a ruler for straight lines - Draw a smooth curve for non-linear relationships 5. **Error Bars**: - Draw the error bars accurately to the correct length - Ensure they're clearly visible and correctly positioned

A single point significantly far from the best fit line is an anomaly and should not influence your line placement.

## Summary Table: Uncertainty Calculations from Graphs | Uncertainty Type | Calculation Method | Formula | |------------------|-------------------|---------| | **Gradient** | Use best and worst fit lines | $\frac{\text{best gradient} - \text{worst gradient}}{\text{best gradient}} \times 100\%$ | | **Y-intercept** | Use best and worst fit lines | $\frac{\text{best y-intercept} - \text{worst y-intercept}}{\text{best y-intercept}} \times 100\%$ | | **Percentage difference** | Compare experimental to accepted value | $\frac{\text{Experimental} - \text{Accepted}}{\text{Accepted}} \times 100\%$ |