Mastering NumPy in Python: The Backbone of Scientific Computing

Chapter 3: Mathematical and Statistical Operations in NumPy

🔹 1. Introduction

One of the most powerful features of NumPy is its ability to perform fast, vectorized mathematical and statistical operations on arrays. Whether you're doing basic arithmetic, aggregating data, or applying statistical analysis across large datasets, NumPy makes it both efficient and intuitive.

In this chapter, we’ll explore:

• Element-wise arithmetic
• Aggregation functions
• Universal functions (ufuncs)
• Broadcasting rules
• Real-world examples

These features form the backbone of data science, engineering simulations, image processing, and machine learning workflows.

🔹 2. Element-wise Arithmetic Operations

NumPy supports all standard arithmetic operations between arrays or between arrays and scalars.

✅ Example:

import numpy as np

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

b = np.array([4, 5, 6])

print(a + b)   # [5 7 9]

print(a - b)   # [-3 -3 -3]

print(a * b)   # [4 10 18]

print(a / b)   # [0.25 0.4  0.5 ]

print(a ** 2)  # [1 4 9]

These are vectorized operations — no need for loops!

🔹 3. Aggregate (Reduction) Functions

Aggregate functions operate over an entire array (or along an axis).

✅ Common Functions:

✅ Example:

arr = np.array([[1, 2, 3], [4, 5, 6]])

print(np.sum(arr))        # 21

print(np.mean(arr))       # 3.5

print(np.std(arr))        # 1.7078

print(np.sum(arr, axis=0))  # Column-wise sum: [5 7 9]

print(np.sum(arr, axis=1))  # Row-wise sum: [6 15]

🔹 4. Universal Functions (ufuncs)

Universal functions (ufuncs) are NumPy's way of applying element-wise operations.

✅ Examples:

x = np.array([1, 4, 9, 16])

np.sqrt(x)      # Square root

np.log(x)       # Natural logarithm

np.exp(x)       # Exponential

np.sin(x)       # Sine

np.cos(x)       # Cosine

np.abs(x)       # Absolute value

Ufuncs are much faster than looping over arrays.

🔹 5. Mathematical Constants

NumPy also provides mathematical constants:

np.pi       # 3.141592...

np.e        # 2.718281...

np.inf      # Infinity

np.nan      # Not a Number

🔹 6. Rounding and Precision

You can control the precision of floating-point results using rounding functions.

a = np.array([3.14159, 2.71828])

np.round(a, 2)    # [3.14 2.72]

np.floor(a)       # [3. 2.]

np.ceil(a)        # [4. 3.]

np.trunc(a)       # [3. 2.]

🔹 7. Broadcasting in Arithmetic

Broadcasting lets you operate on arrays of different shapes.

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

b = np.array([10, 20, 30])

print(a + b)

# [[11 22 33]

#  [14 25 36]]

✅ Broadcasting Rules:

• Dimensions are compared from right to left.
• Dimensions must either be equal or one of them is 1.

🔹 8. Applying Mathematical Functions on Axes

You can apply operations across rows or columns using the axis parameter.

arr = np.array([[1, 2, 3], [4, 5, 6]])

np.mean(arr, axis=0)  # Column-wise mean: [2.5 3.5 4.5]

np.mean(arr, axis=1)  # Row-wise mean: [2.0 5.0]

🔹 9. Cumulative and Difference Functions

NumPy supports accumulation and difference calculations:

arr = np.array([1, 2, 3, 4])

np.cumsum(arr)    # [1 3 6 10]

np.cumprod(arr)   # [1 2 6 24]

np.diff(arr)      # [1 1 1]

🔹 10. Real-World Example: Grades Analysis

grades = np.array([[87, 90, 85],

                   [70, 88, 92],

                   [95, 100, 98]])

student_avg = np.mean(grades, axis=1)

subject_avg = np.mean(grades, axis=0)

print("Student Averages:", student_avg)

print("Subject Averages:", subject_avg)

Output:

Student Averages: [87.33 83.33 97.67]

Subject Averages: [84.   92.67 91.67]

🔹 11. Summary Table