Graphs, Tables, and Results

Organizes data, graphs, and uncertainties to communicate experimental relationships clearly.

Conceptual context

A table preserves values, units, and uncertainties; a graph transforms that information into visual structure. Together they reveal whether two quantities are related, whether a model describes the data, and how trustworthy the results are. $x$ normally occupies the horizontal axis and $y$ the vertical axis. The trend is summarized by $m$ and $b$, while fit quality is examined through $u_y$, $u_rel$, $r$, and $R2$. Clear presentation does not merely decorate an experiment. It preserves its logic, exposes anomalies, and prevents a conclusion from appearing more precise than the evidence allows.

🟢 Essential level

A table answers which values were measured; a graph answers how they change together. $x$ is the controlled or selected quantity, and $y$ is the observed response.

If points rise, $m$ is positive; if they fall, it is negative. $b$ indicates where the model crosses the vertical axis, although it has physical meaning only when the origin belongs to the studied domain. Bars associated with $u_y$ show how much each datum may vary. $r$ compares $y_obs$ with $y_pred$ and reveals whether the model lies above or below a point. A useful graph needs a title, labeled axes, units, scale, domain, and a legend that distinguishes data, fit, and uncertainties. It should preserve every point rather than hide inconvenient observations. $R2$ summarizes in a single number, between zero and one, what fraction of the observed variability the chosen model explains, but a high value never replaces direct inspection of the $r$ nor guarantees that the model is physically correct.

🔵 Formal level

Basic formulas

A linear relation between $x$ and $y$ is represented by:

Experimental linear model

$m$ expresses response change per unit input change, while $b$ represents the extrapolated response at zero input. A graphical estimate can use $x1$, $x2$, $y1$, and $y2$:

Slope between two points

Relative precision is evaluated by comparing $u_y$ with result scale:

Relative uncertainty

Real student questions

Should I connect every point? No. Points are discrete observations; a line should represent a justified model.

Does a steeper line always have larger $m$? No, because axis scales and units matter.

Why show a table when the graph exists? The table preserves exact values, uncertainties, and rounding.

Does high $R2$ prove the law is correct? No. Bias, curvature, or spurious association may remain.

Can I extrapolate? Only with physical justification and a clear warning that the prediction lies outside the measured domain.

Final synthesis

The table preserves evidence and the graph reveals structure. A strong experimental conclusion combines $m$, $b$, uncertainties, $r$, domain, and physical coherence. Quality lies not in a beautiful line, but in a representation that shows what the model explains and where it fails.