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# Statistical Analysis
**Learning Objectives:**
- Descriptive statistics (mean, median, variance, std deviation)
- Histograms and probability distributions
- Correlation and regression
- Uncertainty, error bars, confidence intervals
---
## Engineering Models
Why care? - By analyzing data engineers can use statistical tools to create a mathematical model to help us predict something. You've probably used excel for this before, we will do it with python.
- Curve fitting
You've probably used excel for this before, we will do it with python.
## Statistics Review
Let's take a second to remind ourselves of some statistical terms and how we define it mathematically
| | Formula |
| ------------------------ | ---------------------------------------------------------------------------------------------------------------- |
| Arithmetic Mean | $$\bar{y} = \frac{\sum y_i}{n}$$ |
| Standard Deviation | $$s_y = \sqrt{\frac{S_t}{n - 1}}, \quad S_t = \sum (y_i - \bar{y})^2$$ |
| Variance | $$s_y^2 = \frac{\sum (y_i - \bar{y})^2}{n - 1} = \frac{\left(\sum y_i^2 - \frac{(\sum y_i)^2}{n}\right)}{n- 1}$$ |
| Coefficient of Variation | $$c.v. = \frac{s_y}{\bar{y}} \times 100\%$$ |
## Statistics function in python
Both Numpy and Pandas come with some useful statistical tools that we can use to analyze our data. We can use these tools when working with data, it’s important to understand the **central tendency** and **spread** of your dataset. NumPy provides several built-in functions to quickly compute common statistical metrics such as **mean**, **median**, **standard deviation**, and **variance**. These are fundamental tools for analyzing measurement consistency, uncertainty, and identifying trends in data.
```python
import numpy as np
mean = np.mean([1, 2, 3, 4, 5])
median = np.median([1, 2, 3, 4, 5])
std = np.std([1, 2, 3, 4, 5])
variance = np.var([1, 2, 3, 4, 5])
```
Pandas also includes several built-in statistical tools that make it easy to analyze entire datasets directly from a DataFrame. When working with pandas we can use methods such as `.mean()`, `.std()`, `.var()`, and especially `.describe()` to generate quick summaries of your data. These tools are convenient when working with experimental or simulation data that contain multiple variables, allowing you to assess trends, variability, and potential outliers all at once.
## Problem: Use pd.describe() to report on a dataseries
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Great, so we
## Statistical Distributions
Normal distributions
- Design thinking -> Motorola starting Six sigma organization based on the probability of a product to fail. Adopted world wide.
## Problem: Spectroscopy
Let's
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