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The never-repeating digits of `\pi` can be approximated by `22/7 = 3.142857`

to within 0.04%. These pages artistically and mathematically explore rational approximations to `\pi`. This 22/7 ratio is celebrated each year on July 22nd. If you like hand waving or back-of-envelope mathematics, this day is for you: `\pi` approximation day!

Want more math + art? Discover the Accidental Similarity Number. Find humor in my poster of the first 2,000 4s of `\pi`.

Curiously, the 22/7 rational approximation of `\pi` is more accurate (to within 0.04%) than using the first three digits `3.14`

, which are accurate to 0.05%.

It seems that `\pi` Approximation Day is 20% more accurate (verify on Wolfram Alpha)! And therefore definitely worth celebrating. $$ \frac{(\pi-3.14)-(22/7-\pi)}{\pi-3.14} = 0.206 $$

The poster shows the accuracy of 10,000 rational approximations of `\pi` for each `m/n` and `m=1...10000`. Read about the details of the method.

These posters show warped circles, which embody the 22/7 approximation of `\pi`, using a retro 1970's color scheme. Read about the details of the method.

In collaboration with the Phil Poronnik and Kim Bell-Anderson at the University of Sydney, I'm delighted to share with you our 8-part video series project about thinking about drawing data and communicating science.

We've created 8 videos, each focusing on a different essential idea in data visualization: encoding, shapes, color, uncertainty, design, drawing missing or unobserved data, labels and process.

The videos were designed as teaching materials. Each video comes with a slide deck and exercises.

What are you trying to say

Of significance?

—Steve Ziliak

We've written about P values before and warned readers about common misconceptions about them, which are so rife that the American Statistical Association itself has a long statement about them.

This month is our first of a two-part article about P values. Here we look at 'P value hacking' and 'data dredging', which are questionable practices that invalidate the correct interpretation of P values.

We also illustrate how P values can lead us astray by asking "What is the smallest P value we can expect if the null hypothesis is true but we have done many tests, either explicitly or implicitly?"

Incidentally, this is our first column in which the standfirst is a haiku.

Altman, N. & Krzywinski, M. (2017) Points of Significance: P values and the search for significance. *Nature Methods* **14**:3–4.

Krzywinski, M. & Altman, N. (2013) Points of significance: Significance, P values and t–tests. Nature Methods 10:1041–1042.

*Appeal to intuition when designing with value judgments in mind.*

Figure clarity and concision are improved when the selection of shapes and colors is grounded in the Gestalt principles, which describe how we visually perceive and organize information.

The Gestalt principles are value free. For example, they tell us how we group objects but do not speak to any meaning that we might intuitively infer from visual characteristics.

This month, we discuss how appealing to such intuitions—related to shapes, colors and spatial orientation— can help us add information to a figure as well as anticipate and encourage useful interpretations.

Krzywinski, M. (2016) Points of View: Intuitive Design. Nature Methods 13:895.

*Constraining the magnitude of parameters of a model can control its complexity.*

This month we continue our discussion about model selection and evaluation and address how to choose a model that avoids both overfitting and underfitting.

Ideally, we want to avoid having either an underfitted model, which is usually a poor fit to the training data, or an overfitted model, which is a good fit to the training data but not to new data.

Regularization is a process that penalizes the magnitude of model parameters. This is done by not only minimizing the SSE, `\mathrm{SSE} = \sum_i (y_i - \hat{y}_i)^2 `, as is done normally in a fit, but adding to this minimized quantity the sum of the mode's squared parameters, `\mathrm{SSE} + \lambda \sum_i \hat{\beta}^2_i`.

Lever, J., Krzywinski, M. & Altman, N. (2016) Points of Significance: Regularization. *Nature Methods* **13**:803-804.

Lever, J., Krzywinski, M. & Altman, N. (2016) Points of Significance: Model Selection and Overfitting. *Nature Methods* **13**:703-704.

Lever, J., Krzywinski, M. & Altman, N. (2016) Points of Significance: Classifier evaluation. *Nature Methods* **13**:603-604.

Lever, J., Krzywinski, M. & Altman, N. (2016) Points of Significance: Logistic regression. *Nature Methods* **13**:541-542.

*With four parameters I can fit an elephant and with five I can make him wiggle his trunk. —John von Neumann.*

By increasing the complexity of a model, it is easy to make it fit to data perfectly. Does this mean that the model is perfectly suitable? No.

When a model has a relatively large number of parameters, it is likely to be influenced by the noise in the data, which varies across observations, as much as any underlying trend, which remains the same. Such a model is overfitted—it matches training data well but does not generalize to new observations.

We discuss the use of training, validation and testing data sets and how they can be used, with methods such as cross-validation, to avoid overfitting.

Lever, J., Krzywinski, M. & Altman, N. (2016) Points of Significance: Model Selection and Overfitting. *Nature Methods* **13**:703-704.

Lever, J., Krzywinski, M. & Altman, N. (2016) Points of Significance: Classifier evaluation. *Nature Methods* **13**:603-604.

Lever, J., Krzywinski, M. & Altman, N. (2016) Points of Significance: Logistic regression. *Nature Methods* **13**:541-542.

*It is important to understand both what a classification metric expresses and what it hides.*

We examine various metrics use to assess the performance of a classifier. We show that a single metric is insufficient to capture performance—for any metric, a variety of scenarios yield the same value.

We also discuss ROC and AUC curves and how their interpretation changes based on class balance.

Lever, J., Krzywinski, M. & Altman, N. (2016) Points of Significance: Classifier evaluation. *Nature Methods* **13**:603-604.

Lever, J., Krzywinski, M. & Altman, N. (2016) Points of Significance: Logistic regression. *Nature Methods* **13**:541-542.