Summary  
This chapter explains how to implement code to calculate the mean, variance, and standard deviation of a dataset in order to quantify its central tendency and spread.

General domain of usage  
Website traffic analysis

The mean is the sum of all values divided by the number of values. It represents the **"central"** or **"typical"** value in your dataset.

Definition

**Formula:**

$$
\text{Mean} = \frac{\sum x_i}{n}
$$

**Example:**  
If your website had 100, 120, and 110 visitors over three days:

$$
\frac{100 + 120 + 110}{3} = 110
$$

**Interpretation:**  
On average, the site received 110 visitors per day.


Variance measures how far each number in the set is from the mean. It gives a sense of how **"spread out"** the data is.

**Formula:**

$$
\sigma^2 = \frac{\sum (x_i - \mu)^2}{n}
$$

**Example (using the previous data):**  

- Mean = 110;
- $$(100 − 110)^2 = 100$$;
- $$(120 − 110)^2 = 100$$; 
- $$(110 − 110)^2 = 0$$.

Sum = 200  

$$
\text{Variance} = \frac{200}{3} \approx 66.67
$$

**Interpretation:**  
The average squared distance from the mean is about 66.67.

Standard deviation is the square root of the variance. It brings the spread back to the original units of the data.

**Formula:**

$$
\sigma = \sqrt{\sigma^2}
$$

**Example:**  
If variance is 66.67:

$$
\sigma = \sqrt{66.67} \approx 8.16
$$

**Interpretation:**  
On average, each day's visitor count is about 8.16 away from the mean.

## Real-World Problem: Website Traffic Analysis

**Problem:**  
A data scientist records the number of website visitors over 5 days:  

$$
120, 150, 130, 170, 140
$$

**Step 1 — Mean:**  

$$
\frac{120 + 150 + 130 + 170 + 140}{5} = 142
$$

**Step 2 — Variance:**  

- $$(120 - 142)^2 = 484$$;
- $$(150 - 142)^2 = 64$$;
- $$(130 - 142)^2 = 144$$;
- $$(170 - 142)^2 = 784$$;
- $$(140 - 142)^2 = 4$$.

$$
\text{Variance} = \frac{484+64+144+784+4}{5} = \frac{1480}{5} = 296
$$

**Step 3 — Standard Deviation:**  

$$
\sigma = \sqrt{296} \approx 17.2
$$

**Conclusion:**  
- Mean = 142 visitors per day;
- Variance = 296;
- Standard Deviation = 17.2.

The website traffic varies by about 17.2 visitors from the average day.

What is the relationship between variance and standard deviation?

Master the mathematical foundations essential for data science. Explore core concepts in functions, calculus, linear algebra, probability, and dimensionality reduction. Build both theoretical understanding and practical coding experience to strengthen your ability to analyze data, model complex systems, and apply advanced techniques in machine learning.

Explore the foundation of mathematical functions. Learn different types of algebraic and transcendental functions, their properties, and how to implement them in Python to solve real-world problems.

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Dive into probability theory and statistics. Study conditional probability, Bayes' theorem, and statistical measures. Implement key concepts in Python, simulate distributions, and strengthen your skills through challenges and quizzes.

Understanding Central Tendency & Spread

Mean (Average)

Variance

Standard Deviation

Real-World Problem: Website Traffic Analysis