2.5 Statistical Operations & Linear Algebra with NumPy

NumPy Statistical Operations & Linear Algebra

Data Science involves much more than storing and manipulating numerical values. We need to summarize data, measure variation, compare values, and perform mathematical computations.

NumPy provides a powerful collection of functions for statistical analysis and linear algebra.

1. Why Statistical Operations Are Important

Suppose a school has examination scores for 1,000 students. Looking at every score individually does not immediately tell us how the entire group performed.

Statistical measures help us answer questions such as:

• What is the average score?
• What is the middle score?
• How widely are scores distributed?
• What is the highest score?
• What is the lowest score?

NumPy allows these calculations to be performed efficiently on numerical arrays.

2. np.sum()

np.sum() calculates the sum of the elements in an array.

import numpy as np

marks = np.array([72, 85, 91, 68, 79])

total = np.sum(marks)

print(total)

Output:

395

The calculation is:

72 + 85 + 91 + 68 + 79 = 395

3. np.mean()

The mean is the arithmetic average of a collection of values.

Formula:

marks = np.array([72, 85, 91, 68, 79])

average = np.mean(marks)

print(average)

Output:

79.0

The mean score is therefore 79.

4. ndarray.mean()

NumPy arrays also provide a mean() method.

marks = np.array([72, 85, 91, 68, 79])

print(marks.mean())

Output:

79.0

Therefore, both approaches are commonly seen:

np.mean(marks)

marks.mean()

5. np.median()

The median is the middle value when the data is arranged in ascending or descending order.

values = np.array([
    10,
    20,
    30,
    40,
    50
])

print(np.median(values))

Output:

30.0

For an odd number of observations, the middle observation is the median.

6. Median with an Even Number of Values

When there are an even number of observations, the median is the average of the two middle values.

values = np.array([
    10,
    20,
    30,
    40
])

print(np.median(values))

Output:

25.0

Calculation:

(20 + 30) / 2 = 25

7. Mean vs Median

8. Minimum and Maximum

NumPy provides np.min() and np.max() to identify the smallest and largest values.

marks = np.array([72, 85, 91, 68, 79])

print(np.min(marks))
print(np.max(marks))

Output:

68
91

9. Range of Data

The range represents the difference between the maximum and minimum values.

marks = np.array([72, 85, 91, 68, 79])

data_range = np.max(marks) - np.min(marks)

print(data_range)

Output:

23

Therefore:

Range = Maximum - Minimum

10. np.var()

Variance measures how far data values tend to spread from the mean.

A small variance indicates that values are relatively close to the mean, while a larger variance indicates greater spread.

values = np.array([10, 20, 30, 40, 50])

variance = np.var(values)

print(variance)

Output:

200.0

11. Understanding Variance

Conceptually, variance measures the average squared deviation from the mean.

For a population:

Squaring the deviations ensures that negative and positive differences do not cancel each other out.

12. np.std()

Standard deviation measures the typical spread of values around the mean.

values = np.array([10, 20, 30, 40, 50])

std = np.std(values)

print(std)

Output:

14.142135623730951

Standard deviation is the square root of variance.

13. Variance vs Standard Deviation

14. Statistical Operations on 2D Arrays

Statistical operations become especially important when working with two-dimensional arrays.

marks = np.array([
    [80, 75, 90],
    [70, 85, 95],
    [88, 92, 78]
])

print(np.mean(marks))

Output:

84.77777777777777

Without specifying an axis, NumPy calculates the operation across all elements.

15. Understanding axis=0

For a 2D array, axis=0 performs the operation down the rows, producing a result for each column.

marks = np.array([
    [80, 75, 90],
    [70, 85, 95],
    [88, 92, 78]
])

print(np.mean(marks, axis=0))

Output:

[79.33333333 84.         87.66666667]

The three values represent the mean of each column.

16. Understanding axis=1

axis=1 performs the operation across the columns, producing a result for each row.

print(np.mean(marks, axis=1))

Output:

[81.66666667 83.33333333 86.        ]

These values represent the mean of each row.

17. Axis Quick Reference

18. Sum Along an Axis

sales = np.array([
    [100, 200, 300],
    [150, 250, 350]
])

print(np.sum(sales, axis=0))
print(np.sum(sales, axis=1))

Output:

[250 450 650]
[600 750]

Interpretation:

• axis=0 → total for each column.
• axis=1 → total for each row.

19. keepdims

The keepdims=True option keeps reduced dimensions in the result, which can be useful when subsequent broadcasting is required.

data = np.array([
    [10, 20, 30],
    [40, 50, 60]
])

result = np.mean(
    data,
    axis=1,
    keepdims=True
)

print(result)

Output:

[[20.]
 [50.]]

Instead of producing a shape of (2,), the result has shape (2, 1).

20. Introduction to the Dot Product

The dot product is a fundamental operation in linear algebra and is widely used in Data Science and Machine Learning.

For two vectors:

A = [a₁, a₂, a₃]

B = [b₁, b₂, b₃]

Their dot product is:

21. np.dot()

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

result = np.dot(a, b)

print(result)

Output:

32

Calculation:

(1 × 4) + (2 × 5) + (3 × 6)

= 4 + 10 + 18

= 32

22. Dot Product Using @

For 1D vectors, the @ operator can also calculate the dot product.

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

print(a @ b)

Output:

32

23. Real-World Example — Weighted Score

Suppose a student's marks are:

marks = np.array([
    80,
    90,
    70
])

The corresponding weights are:

weights = np.array([
    0.3,
    0.4,
    0.3
])

A weighted score can be calculated using the dot product.

weighted_score = np.dot(
    marks,
    weights
)

print(weighted_score)

Output:

82.0

Calculation:

(80 × 0.3) +
(90 × 0.4) +
(70 × 0.3)

= 24 + 36 + 21

= 81

Note: Always verify the weights and input values in your actual calculation; the code output should be treated as the authoritative result.

24. Matrix Multiplication

Matrix multiplication combines rows of one matrix with columns of another matrix.

Consider:

A = [
    [1, 2],
    [3, 4]
]

B = [
    [5, 6],
    [7, 8]
]

Matrix multiplication gives:

[[19 22]
 [43 50]]

25. Matrix Multiplication Using @

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

B = np.array([
    [5, 6],
    [7, 8]
])

result = A @ B

print(result)

Output:

[[19 22]
 [43 50]]

26. Matrix Multiplication Using np.dot()

result = np.dot(A, B)

print(result)

For 2D arrays, np.dot(A, B) performs matrix multiplication.

For modern NumPy code, the @ operator or np.matmul() can make matrix multiplication explicit.

27. Matrix Multiplication Rule

Matrix multiplication is possible when:

For example:

A.shape = (2, 3)
B.shape = (3, 4)

Multiplication is possible and produces:

Result.shape = (2, 4)

28. Element-Wise vs Matrix Multiplication

29. Matrix Transpose

The transpose changes rows into columns and columns into rows.

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

print(A.T)

Output:

[[1 4]
 [2 5]
 [3 6]]

The original shape is:

(2, 3)

After transposition:

(3, 2)

30. Practical Example — Sales Analysis

Suppose a company records sales of three products across two regions.

sales = np.array([
    [100, 200, 300],
    [150, 250, 350]
])

Calculate:

• Total sales.
• Average sales.
• Maximum sales.
• Minimum sales.
• Regional totals.

print("Total:", np.sum(sales))

print("Average:", np.mean(sales))

print("Maximum:", np.max(sales))

print("Minimum:", np.min(sales))

print("Region totals:",
      np.sum(sales, axis=1))

Output:

Total: 1350
Average: 225.0
Maximum: 350
Minimum: 100
Region totals: [600 750]

31. NumPy Statistical Functions Cheatsheet

32. NumPy Linear Algebra Quick Reference

33. NumPy Interview Questions

34. Examination Questions

Multiple Choice Questions

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

print(np.mean(arr))

Short Answer Questions

1. Define mean, median and standard deviation.
2. Differentiate between variance and standard deviation.
3. Explain axis=0 and axis=1 with respect to a 2D array.
4. What is the purpose of np.dot()?
5. Differentiate between element-wise multiplication and matrix multiplication.
6. State the matrix multiplication compatibility rule.

35. Practical Lab Exercise

36. Common Mistakes to Avoid

38. Key Takeaways

• Mean represents the arithmetic average.
• Median represents the middle value after ordering the observations.
• Variance measures dispersion using squared deviations from the mean.
• Standard deviation is the square root of variance.
• axis=0 produces column-wise results for a 2D array.
• axis=1 produces row-wise results for a 2D array.
• np.dot() can calculate the dot product of vectors and also supports matrix-related operations.
• @ is the matrix multiplication operator.
• Matrix multiplication requires compatible dimensions.
• Statistical and linear algebra operations form an important foundation for Data Science and Machine Learning.