From our sample data:

• Force ranges from 0.00 N to 2.50 N
• Extension ranges from 0.00 cm to 12.23 cm

To allow for error bars and some margin, we'll set our axes to:

• x-axis (Force): 0 to 3.0 N
• y-axis (Extension): 0 to 13.0 cm

Guidelines for choosing appropriate scales:

• The plotted graph should occupy at least 75% of the available graph space.
• Choose scales that yield convenient, easily readable intervals (e.g., 0.5 N per division for force).
• Each axis should have 5-10 major divisions.
• The scale should be linear unless the data clearly indicates a logarithmic relationship.

For our example:

• x-axis: 0.5 N per division (assuming standard graph paper)
• y-axis: 2.0 cm of extension per division

For our example:

• Force measurement uncertainty: ± 0.05 N (given)
• Extension measurement uncertainty: ± 0.1 cm (given)

These values will be used directly as the length of the error bars above and below each data point.

For more complex scenarios where multiple measurements are involved, we would use error propagation formulas as shown in the theory section.

Key elements for proper axes:

• Draw x-axis (horizontal) and y-axis (vertical) with a ruler.
• Mark regular intervals on each axis according to your chosen scale.
• Label each axis with the quantity name followed by its unit in parentheses. For example: "Applied Force (N)" and "Extension (cm)".
• Number the axes clearly, placing the numbers below the x-axis and to the left of the y-axis.

For best practices:

• Use a sharp pencil to plot small, distinct points.
• A small cross (×) or dot with a circle around it (⊙) works well to mark data points.
• Be precise in positioning each point according to its coordinates.
• For our sample data, plot each (Force, Extension) pair: (0.00, 0.00), (0.50, 2.45), etc.

For each data point:

1. Draw a vertical line extending ± 0.1 cm (the uncertainty in extension) above and below each data point.
2. Draw a horizontal line extending ± 0.05 N (the uncertainty in force) to the left and right of each data point.
3. Add small caps at the ends of each error bar to improve readability.

The resulting error bars form a "cross" around each data point, showing the uncertainty in both variables.

Guidelines for drawing the best-fit line:

• The line should pass as close as possible to all data points, considering their error bars.
• There should be roughly an equal number of data points above and below the line.
• For a visual method: place a transparent ruler on the graph and adjust it until you find the position where the line best represents all data points.
• For our spring example, we expect a linear relationship according to Hooke's Law: F = kx, where k is the spring constant.

Mathematical method for linear fit:

For data points \((x_i, y_i)\) with a linear relationship \(y = mx + c\), the slope \(m\) and intercept \(c\) can be calculated using:

\[ m = \frac{n\sum x_i y_i - \sum x_i \sum y_i}{n\sum x_i^2 - (\sum x_i)^2} \]

\[ c = \frac{\sum y_i - m\sum x_i}{n} \]

Where \(n\) is the number of data points.

The slope represents the relationship between the variables. For our spring example, the slope gives the spring constant k in N/cm.

To calculate the uncertainty in the slope:

\[ \Delta m = m \cdot \sqrt{\sum \left(\frac{y_i - (mx_i + c)}{y_i}\right)^2 \cdot \frac{1}{n-2}} \]

Or, using a simplified approach for a linear fit through the origin:

\[ \Delta m = \sqrt{\frac{\sum(y_i - mx_i)^2}{(n-1)\sum x_i^2}} \]

For our sample data, this calculation would yield the spring constant k = 4.89 N/cm with an uncertainty of approximately ± 0.02 N/cm.

Essential elements:

• Title: "Relationship Between Force and Extension for a Spring"
• Include your name, date, and any experiment identification information.
• If multiple data series are plotted, add a legend explaining each symbol or color used.
• You may also include the equation of the best-fit line: F = (4.89 ± 0.02)x N/cm