Quantile-Quantile Plots

The problem, from first principles

Whether a column is normally distributed decides which statistical tests are fair, whether a linear model's errors are well behaved and whether a log transform from 04-skew-transforms worked. A histogram lets the eye judge bell-shapedness badly. A quantile-quantile plot makes the question a straight-line check: sort the data, pair each sorted value with the value a perfect normal sample would have in the same position, and plot one against the other. If the data is normal the points fall on a line, and the way they bend away from it says exactly how the distribution differs: heavy tails, skew, a flat top. This question builds the numbers behind that plot, including the inverse of the normal CDF that supplies the reference values.

From theory to code

Implement normal_ppf(p), the value below which a standard normal falls with probability p, then plotting_positions(n), the probabilities assigned to the n sorted values, then qq_points(x), the two coordinate arrays of the plot, then qq_line(theoretical, sample), the reference line through the quartiles, then qq_correlation(x), one number that summarizes how straight the plot is. The signatures and docstrings are already in the editor.

Constraints

• normal_ppf(p) returns the standard normal quantile of p as a float, accurate to at least 1e-9, for 0 < p < 1. It raises ValueError for any other p. Do not use scipy. Use math.erf and bisection (or another method you write).
• plotting_positions(n) returns the array (i - 0.5) / n for i = 1 .. n.
• qq_points(x) returns (theoretical, sample): sample is x sorted ascending and theoretical[i] = normal_ppf(plotting_positions(n)[i]), both 1D float arrays of length n.
• qq_line(theoretical, sample) returns (slope, intercept) of the line through the points at the 25th and 75th percentiles of the two arrays (np.percentile, linear interpolation): slope = (s75 - s25) / (t75 - t25) and intercept = s25 - slope * t25.
• qq_correlation(x) returns the Pearson correlation between theoretical and sample from qq_points(x), as a float.
• Inputs must not be modified.

Hints