Graphs from tables

What you will be able to do

After this lesson you can

• draw a scatter plot, a bar chart and a histogram straight from a table,
• plot a group summary, and
• use a log-log graph to find a power law.

Plotting from a table

You drew graphs in chapter 2 from arrays. A pandas table can draw itself: it has the methods plot.scatter, plot.bar, plot.hist and plot.line, which use matplotlib behind the scenes. They give back the axes, so you can use the object-style methods of chapter 2 to finish the graph. Always import matplotlib.pyplot as well, and call plt.show() at the end.

Predict first. Will the periods of the planets fall on a straight line when they are plotted against the distance?

The dots lie on a curve that gets steeper: not a straight line. The law of the planets is not linear. The inner planets are crowded in the corner, so that Mercury, Venus and Earth are hard to tell apart. This is a common problem with data that covers a wide range of sizes, and a log scale is the answer.

A power law on a log-log graph

Suppose that $T = c\,a^{p}$. If you take the logarithm of both sides,

This is a straight line when $\log T$ is plotted against $\log a$, and the slope is the power $p$. So a log-log graph turns a power law into a straight line, and the slope tells the power.

Predict first. What slope do you expect from Kepler's third law, $T^2 = a^3$?

This is a method, and not only a result about planets: whenever you suspect a power law (the area of a leaf against its length, the brightness of a lamp against the distance, the speed of a falling object against the height), plot the data on a log-log graph. If it is a straight line, the slope is the power.

Move p until the line runs through the dots. The fit in the code finds 1.5: the period grows as the distance to the power 1.5.

Bars from a table

The left graph shows each planet. The right graph shows a summary of the groups. Which one is better depends on the question. For two kinds, the summary is easy to read. For the individual planets, the left graph shows the whole picture, including the fact that Jupiter is a little bigger than Saturn and that Uranus and Neptune are about the same.

Common mistakes

• Not importing matplotlib.pyplot. On this site, the plotting tools are loaded only when the code mentions matplotlib.
• Giving the wrong names to x and y. They must be names of columns.
• Using a log scale for numbers that are zero or negative.
• Reading a slope from a log-log graph as the slope of the data. It is the power.
• Forgetting the labels and the units.

Practice

Start with two quick questions. Each New numbers gives you a fresh one, so you can practise as long as you like.

Then the labs and the problems:

1. Warm-up. Open the lab Plot from a table. You draw a log-log graph from a table, and the checker reads its scales.
2. Build. Open the problem A power law of leaves and the problem Find the power.

Remember these

• p.plot.scatter(x=..., y=...), p.plot.bar(...) and p.plot.hist(...) draw from a table. They return the axes.
• Import matplotlib.pyplot and call plt.show().
• A power law $y = c\,x^p$ is a straight line on a log-log graph, and the slope is $p$.
• Kepler's third law: $T = a^{1.5}$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. What does the slope of a straight line on a log-log graph give?

2. A graph of the period against the distance gives a straight line on log-log axes with the slope 2. What is the law?

3. How do you draw a bar chart of the mean of a column for each group?

4. Why do the inner planets look crowded on the linear graph?