Quantile Function¶

The CDF answers "what is the probability that \(X \le x\)?" The quantile function inverts that question: "at what value \(x\) does the cumulative probability reach \(q\)?" This inverse operation is central to statistical practice — it determines critical values for hypothesis tests, endpoints of confidence intervals, and the theoretical quantiles used in QQ plots. In scipy.stats, the quantile function is called ppf (percent point function) and its survival-function counterpart is isf (inverse survival function).

Quantile Function (ppf)¶

For a continuous random variable \(X\) with CDF \(F\), the quantile function at probability level \(q \in (0, 1)\) is:

When \(F\) is strictly increasing and continuous, this reduces to the ordinary inverse: \(F^{-1}(q)\) is the unique \(x\) satisfying \(F(x) = q\). In scipy.stats, this is the .ppf(q) method.

Example: Standard Normal Quantiles¶

The most commonly used quantiles of the standard normal distribution:

```python import scipy.stats as stats

dist = stats.norm(loc=0, scale=1)

Median¶

print(f"ppf(0.50) = {dist.ppf(0.50):.4f}") # 0.0000

95th percentile¶

print(f"ppf(0.95) = {dist.ppf(0.95):.4f}") # 1.6449

97.5th percentile (used for two-sided 95% CI)¶

print(f"ppf(0.975) = {dist.ppf(0.975):.4f}") # 1.9600 ```

The value 1.96 is the critical value that places 2.5% probability in each tail, forming the basis of the 95% confidence interval for normally distributed data.

Example: Student's t-Distribution¶

For the \(t\)-distribution, quantiles depend on the degrees of freedom. With fewer degrees of freedom, the tails are heavier and critical values are larger:

python for df in [5, 10, 30, 100]: critical = stats.t.ppf(0.975, df=df) print(f"df={df:3d}: t_0.975 = {critical:.4f}")

As \(\text{df} \to \infty\), the \(t\)-quantiles converge to the normal quantiles.

Inverse Survival Function (isf)¶

The survival function is \(S(x) = 1 - F(x) = P(X > x)\). Its inverse, the inverse survival function, returns the value \(x\) at which the upper-tail probability equals \(q\):

In scipy.stats, this is the .isf(q) method.

```python dist = stats.norm(loc=0, scale=1)

isf(0.05) = ppf(0.95)¶

print(f"isf(0.05) = {dist.isf(0.05):.4f}") # 1.6449 print(f"ppf(0.95) = {dist.ppf(0.95):.4f}") # 1.6449 ```

Relationship Between ppf, isf, cdf, and sf¶

The four functions form two inverse pairs:

Verifying the inverse relationship:

```python dist = stats.norm(loc=0, scale=1)

x = 1.5 q = dist.cdf(x) x_back = dist.ppf(q) print(f"cdf({x}) = {q:.6f}, ppf({q:.6f}) = {x_back:.6f}")

q_sf = dist.sf(x) x_back_sf = dist.isf(q_sf) print(f"sf({x}) = {q_sf:.6f}, isf({q_sf:.6f}) = {x_back_sf:.6f}") ```

For a detailed treatment of the CDF and survival function, see the CDF and Survival Function page.

Vectorized Evaluation¶

Both .ppf() and .isf() accept arrays for efficient computation of multiple quantiles at once:

```python import numpy as np

dist = stats.norm(loc=0, scale=1) probabilities = np.array([0.01, 0.05, 0.10, 0.25, 0.50, 0.75, 0.90, 0.95, 0.99]) quantiles = dist.ppf(probabilities)

for p, q in zip(probabilities, quantiles): print(f" ppf({p:.2f}) = {q:+.4f}") ```

Applications of the Quantile Function¶

The quantile function underpins several common statistical operations:

• Critical values: Hypothesis tests compare a test statistic to \(F^{-1}(1 - \alpha)\) or \(F^{-1}(\alpha/2)\).
• Confidence intervals: The .interval() method is built on two calls to .ppf(). See the Confidence Intervals page.
• QQ plots: Theoretical quantiles \(F^{-1}(p_i)\) are plotted against observed order statistics to assess distributional fit.
• Random variate generation: The inverse transform method generates samples by applying \(F^{-1}\) to uniform random numbers: \(X = F^{-1}(U)\) where \(U \sim \text{Uniform}(0,1)\).

Summary¶

The quantile function \(F^{-1}(q)\) and inverse survival function \(S^{-1}(q) = F^{-1}(1-q)\) invert the CDF and survival function respectively. In scipy.stats, .ppf() computes quantiles from lower-tail probabilities and .isf() from upper-tail probabilities. These methods provide critical values, confidence interval endpoints, and the theoretical quantiles needed for probability plots.

Exercises¶

Exercise 1. Find the critical values \(z_{0.025}\) and \(z_{0.975}\) for the standard normal distribution using .ppf(). Then find the same upper-tail critical value \(z_{0.025}\) using .isf() and verify the results match.

from scipy import stats

rv = stats.norm()
z_low = rv.ppf(0.025)
z_high = rv.ppf(0.975)
z_isf = rv.isf(0.025)

print(f"z_0.025 (ppf) = {z_low:.4f}")
print(f"z_0.975 (ppf) = {z_high:.4f}")
print(f"z_0.025 (isf) = {z_isf:.4f}")
print(f"ppf(0.975) == isf(0.025): {abs(z_high - z_isf) < 1e-10}")

Exercise 2. For a \(t\)-distribution with 10 degrees of freedom, find the critical values that enclose the middle 99% of the distribution. Compare these with the corresponding standard normal critical values.

from scipy import stats

t_rv = stats.t(df=10)
t_low = t_rv.ppf(0.005)
t_high = t_rv.ppf(0.995)

z_low = stats.norm.ppf(0.005)
z_high = stats.norm.ppf(0.995)

print(f"t(10) 99% critical values: [{t_low:.4f}, {t_high:.4f}]")
print(f"Normal 99% critical values: [{z_low:.4f}, {z_high:.4f}]")
print(f"t critical value is wider by: {t_high - z_high:.4f}")

Exercise 3. Generate 1000 samples from a uniform distribution on \([0, 1]\) and transform them to standard normal samples using stats.norm.ppf(). Verify the transformation by computing the mean and standard deviation of the resulting samples.

import numpy as np
from scipy import stats

np.random.seed(42)
u = np.random.uniform(0, 1, size=1000)
z = stats.norm.ppf(u)

print(f"Mean: {np.mean(z):.4f} (expected ~0)")
print(f"Std:  {np.std(z, ddof=1):.4f} (expected ~1)")