Summary
This chapter demonstrates how to compute measures of spread—mean, population and sample variance (using ddof=1 for Bessel’s correction), and standard deviation—using NumPy functions, and how to visualize the data distribution with a histogram and overlay lines for mean ± standard deviation.

General domain of usage
Data analysis

## Define the Dataset

Here, we assign an array to the variable `data` to ensure we have a consistent dataset to work with for all calculations.

```python
import numpy as np

# Create a numpy array of daily sales
data = np.array([10, 15, 12, 18, 20, 22, 14, 17, 11, 16])
```

## Calculate Population Statistics

This function takes the array as input and returns the average value of all elements, which summarizes the central tendency of the dataset.

```python
mean_val = np.mean(data)       # Mean
variance_val = np.var(data)    # Population variance (ddof=0 by default)
std_dev_val = np.std(data)     # Population standard deviation
```

* `np.mean(data)` computes the arithmetic mean (average);
* `np.var(data)` calculates the **population variance** (divides by $$n$$);
* `np.std(data)` calculates the **population standard deviation** (square root of variance).

import numpy as np

# Create a numpy array of daily sales
data = np.array([10, 15, 12, 18, 20, 22, 14, 17, 11, 16])

mean_val = np.mean(data)       # Mean
variance_val = np.var(data)    # Population variance (ddof=0 by default)
std_dev_val = np.std(data)     # Population standard deviation

print(f"Mean: {mean_val}")
print(f"Variance (Population): {variance_val}")
print(f"Standard Deviation (Population): {std_dev_val}")

## Calculate Sample Statistics

To get **unbiased estimates** from a sample, we use `ddof=1`.
This applies **Bessel's correction**, dividing variance by $(n-1)$ instead of $n$.

```python
sample_variance_val = np.var(data, ddof=1)
sample_std_dev_val = np.std(data, ddof=1)
```

* `np.var(data, ddof=1)` - sample variance;
* `np.std(data, ddof=1)` - sample standard deviation.

import numpy as np

# Create a numpy array of daily sales
data = np.array([10, 15, 12, 18, 20, 22, 14, 17, 11, 16])

sample_variance_val = np.var(data, ddof=1)
sample_std_dev_val = np.std(data, ddof=1)

print(f"Variance (Sample): {sample_variance_val}")
print(f"Standard Deviation (Sample): {sample_std_dev_val}")

Standard deviation is the square root of variance, giving a measure of spread in the **same units as the original data**, making it easier to interpret.

Note

How do we calculate standard deviation with `numpy` library?

This course distills the essentials of Python's math module, focusing on trigonometric functions, logarithmic operations, and mathematical constants. Learners will gain hands-on experience using Python to perform these standard mathematical computations, ideal for beginners seeking practical skills with the math module.

Implementing Spread in Python

Define the Dataset

Calculate Population Statistics

Calculate Sample Statistics