Statistics Methods¶

mean and np.mean¶

1. Basic Usage¶

Compute the arithmetic mean.

```python import numpy as np

def main(): a = np.array([[1, 2, 3], [4, 5, 6]])

print("a =")
print(a)
print()

print(f"{a.mean() = }")
print(f"{a.mean(axis=0) = }")
print(f"{a.mean(axis=1) = }")

if name == "main": main() ```

Output:

``` a = [[1 2 3] [4 5 6]]

a.mean() = 3.5 a.mean(axis=0) = array([2.5, 3.5, 4.5]) a.mean(axis=1) = array([2., 5.]) ```

2. Output Shape¶

```python import numpy as np

def main(): a = np.random.standard_normal((2, 3))

print(f"{a.mean().shape = }")
print(f"{a.mean(axis=0).shape = }")
print(f"{a.mean(axis=1).shape = }")

if name == "main": main() ```

Output:

a.mean().shape = () a.mean(axis=0).shape = (3,) a.mean(axis=1).shape = (2,)

3. Function Syntax¶

```python import numpy as np

def main(): a = np.random.standard_normal((2, 3))

print(f"{np.mean(a).shape = }")
print(f"{np.mean(a, axis=0).shape = }")
print(f"{np.mean(a, axis=1).shape = }")

if name == "main": main() ```

std and np.std¶

1. Basic Usage¶

Compute the standard deviation.

```python import numpy as np

def main(): a = np.array([[1, 2, 3], [4, 5, 6]])

print("a =")
print(a)
print()

print(f"{a.std() = :.4f}")
print(f"{a.std(axis=0) = }")
print(f"{a.std(axis=1) = }")

if name == "main": main() ```

2. Output Shape¶

```python import numpy as np

def main(): a = np.random.standard_normal((2, 3))

print(f"{a.std().shape = }")
print(f"{a.std(axis=0).shape = }")
print(f"{a.std(axis=1).shape = }")

if name == "main": main() ```

3. ddof Parameter¶

The ddof (delta degrees of freedom) parameter controls the divisor: N - ddof.

```python import numpy as np

def main(): a = np.array([1, 2, 3, 4, 5])

# Population std (ddof=0, default)
pop_std = a.std(ddof=0)

# Sample std (ddof=1)
sample_std = a.std(ddof=1)

print(f"Data: {a}")
print(f"Population std (ddof=0): {pop_std:.4f}")
print(f"Sample std (ddof=1): {sample_std:.4f}")

if name == "main": main() ```

var and np.var¶

1. Basic Usage¶

Compute the variance (square of standard deviation).

```python import numpy as np

def main(): a = np.array([[1, 2], [3, 1], [2, 3]])

print("a =")
print(a)
print()

print(f"{a.var() = }")
print(f"{a.var(axis=0) = }")
print(f"{a.var(axis=1) = }")

if name == "main": main() ```

2. Function Syntax¶

```python import numpy as np

def main(): a = np.array([[1, 2], [3, 1], [2, 3]])

print("a =")
print(a)
print()

print(f"{np.var(a) = }")
print(f"{np.var(a, axis=0) = }")
print(f"{np.var(a, axis=1) = }")

if name == "main": main() ```

3. std vs var¶

```python import numpy as np

def main(): a = np.array([1, 2, 3, 4, 5])

variance = a.var()
std_dev = a.std()

print(f"Variance: {variance:.4f}")
print(f"Std Dev:  {std_dev:.4f}")
print(f"sqrt(var) = std: {np.sqrt(variance):.4f}")

if name == "main": main() ```

np.median¶

1. Basic Usage¶

Compute the median (middle value).

```python import numpy as np

def main(): # Odd number of elements a = np.array([1, 3, 2, 5, 4]) print(f"a = {a}") print(f"np.median(a) = {np.median(a)}") print()

# Even number of elements (average of two middle)
b = np.array([1, 3, 2, 5, 4, 6])
print(f"b = {b}")
print(f"np.median(b) = {np.median(b)}")

if name == "main": main() ```

Output:

``` a = [1 3 2 5 4] np.median(a) = 3.0

b = [1 3 2 5 4 6] np.median(b) = 3.5 ```

2. No Method Version¶

Unlike mean/std/var, median is only a function (not a method).

```python import numpy as np

def main(): sample_data = np.array([1.5, 2.5, 4, 2, 1, 1])

# np.median() is a function only
population_median = np.median(sample_data)
print(f"{population_median = }")

# No method version
# sample_data.median()  # AttributeError

if name == "main": main() ```

3. With axis Parameter¶

```python import numpy as np

def main(): a = np.array([[1, 5, 3], [2, 4, 6]])

print("a =")
print(a)
print()

print(f"np.median(a) = {np.median(a)}")
print(f"np.median(a, axis=0) = {np.median(a, axis=0)}")
print(f"np.median(a, axis=1) = {np.median(a, axis=1)}")

if name == "main": main() ```

np.quantile¶

1. Basic Usage¶

Compute quantiles (percentiles as fractions).

```python import numpy as np

def main(): data = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])

q25 = np.quantile(data, 0.25)
q50 = np.quantile(data, 0.50)
q75 = np.quantile(data, 0.75)

print(f"Data: {data}")
print(f"Q1 (25%): {q25}")
print(f"Q2 (50%): {q50}")
print(f"Q3 (75%): {q75}")

if name == "main": main() ```

2. Five Number Summary¶

```python import numpy as np import matplotlib.pyplot as plt

def main(): data = np.array([1, 2, 0, 0, 0, 1, 3, 1, 2, 1, 2, 4, 5, -1, -2, 0, 8])

quantiles = {
    "Min": 0,
    "Q1": 0.25,
    "Median": 0.5,
    "Q3": 0.75,
    "Max": 1
}

print("Five Number Summary:")
for label, q in quantiles.items():
    quantile_value = np.quantile(data, q)
    print(f"  {label:6} : {quantile_value}")

fig, ax = plt.subplots(figsize=(2, 3))
ax.boxplot(data)
ax.set_title("Boxplot")
plt.show()

if name == "main": main() ```

3. Multiple Quantiles¶

```python import numpy as np

def main(): data = np.random.randn(1000)

# Compute multiple quantiles at once
qs = [0.1, 0.25, 0.5, 0.75, 0.9]
values = np.quantile(data, qs)

print("Quantiles:")
for q, v in zip(qs, values):
    print(f"  {q*100:5.1f}%: {v:+.3f}")

if name == "main": main() ```

scipy.stats¶

1. Skewness¶

Skewness measures asymmetry of the distribution.

```python import numpy as np from scipy import stats

def main(): x = np.random.normal(size=(1000,))

skewness = stats.skew(x)
print(f"{stats.skew(x) = :.4f}")

# Positive skew: right tail longer
# Negative skew: left tail longer
# Near 0: symmetric

if name == "main": main() ```

2. Kurtosis¶

Kurtosis measures the "tailedness" of the distribution.

```python import numpy as np from scipy import stats

def main(): x = np.random.normal(size=(1000,))

# scipy returns excess kurtosis (normal = 0)
kurt = stats.kurtosis(x)
print(f"{stats.kurtosis(x) = :.4f}")

# Positive: heavier tails than normal
# Negative: lighter tails than normal
# Near 0: similar to normal

if name == "main": main() ```

3. Complete Summary¶

```python import numpy as np from scipy import stats

def main(): np.random.seed(42) data = np.random.normal(loc=5, scale=2, size=1000)

print("Descriptive Statistics:")
print(f"  Mean:     {np.mean(data):.4f}")
print(f"  Std:      {np.std(data):.4f}")
print(f"  Var:      {np.var(data):.4f}")
print(f"  Median:   {np.median(data):.4f}")
print(f"  Skewness: {stats.skew(data):.4f}")
print(f"  Kurtosis: {stats.kurtosis(data):.4f}")

if name == "main": main() ```

Exercises¶

Exercise 1. Create a = np.array([2, 4, 4, 4, 5, 5, 7, 9]). Compute the mean, median, standard deviation, and variance. Verify that var = std**2.

import numpy as np

a = np.array([2, 4, 4, 4, 5, 5, 7, 9], dtype=float)
print(f"Mean: {np.mean(a)}")
print(f"Median: {np.median(a)}")
print(f"Std: {np.std(a):.4f}")
print(f"Var: {np.var(a):.4f}")
print(f"var == std^2: {np.allclose(np.var(a), np.std(a)**2)}")

Exercise 2. Generate 10000 samples from a normal distribution with mean 5 and std 2. Compute the sample mean and standard deviation and verify they are close to the true parameters.

import numpy as np

np.random.seed(42)
samples = np.random.normal(loc=5, scale=2, size=10000)
print(f"Sample mean: {samples.mean():.4f} (expected 5)")
print(f"Sample std:  {samples.std():.4f} (expected 2)")

Exercise 3. Compute the 25th, 50th (median), and 75th percentiles of a = np.random.randn(1000) using np.percentile. Verify that the 50th percentile matches np.median(a).

import numpy as np

a = np.random.randn(1000)
p25, p50, p75 = np.percentile(a, [25, 50, 75])
print(f"25th: {p25:.4f}, 50th: {p50:.4f}, 75th: {p75:.4f}")
print(f"Median: {np.median(a):.4f}")
print(f"50th == median: {np.allclose(p50, np.median(a))}")

Exercise 4. Explain the difference between np.std(a) and np.std(a, ddof=1). Compute both for a = np.array([2, 4, 4, 4, 5, 5, 7, 9]) and explain when to use each.

import numpy as np

a = np.array([2, 4, 4, 4, 5, 5, 7, 9])
pop_std = np.std(a)           # ddof=0 — population std
samp_std = np.std(a, ddof=1)  # ddof=1 — sample std

print(f"Population std (ddof=0): {pop_std:.4f}")   # 2.0000
print(f"Sample std (ddof=1):     {samp_std:.4f}")  # 2.1381

# ddof=0 divides by N — correct when a IS the entire population
# ddof=1 divides by N-1 (Bessel's correction) — unbiased when a
# is a sample from a larger population (the common case)

Exercise 5. Compute the z-scores of each column in a (200, 3) matrix using mean and std with axis=0. Verify that each column of the result has mean approximately 0 and std approximately 1.

import numpy as np

X = np.random.randn(200, 3) * 10 + 50
mu = X.mean(axis=0)
sigma = X.std(axis=0)
Z = (X - mu) / sigma

print(f"Column means: {np.round(Z.mean(axis=0), 10)}")  # ~[0, 0, 0]
print(f"Column stds:  {np.round(Z.std(axis=0), 10)}")   # ~[1, 1, 1]