Statistics

├── Statistics                                      
│   ├── Descriptive Statistics                      
│   │   ├── Mean, Median, Mode                      
│   │   ├── Standard Deviation                      
│   │   └── Percentiles / Quartiles                 
│   ├── Inferential Statistics                      
│   │   ├── Hypothesis Testing                      
│   │   ├── p-Value                                 
│   │   ├── Confidence Intervals                    
│   │   └── A/B Testing                             
│   ├── Correlation vs Causation                    
│   └── Sampling Methods                            
│       ├── Random Sampling                         
│       ├── Stratified Sampling                     
│       └── Bootstrap                               

Statistics exists to answer two fundamental questions about data:

1. “What do we have?” → Summarizing or describing data we already collected.
2. “What can we infer?” → Making generalizations or predictions about a larger population from a sample.

Descriptive Statistics

Goal: Summarize and describe the main features of a dataset. It doesn’t generalize or predict it only describes what is actually observed.

You surveyed 100 people and found:

• Average height = 170 cm
• Minimum = 150 cm
• Maximum = 190 cm
• 60% are male

These numbers describe your sample they do not claim anything about all people in the world.

Category	Examples	Purpose
Measures of Central Tendency	Mean, Median, Mode	Show the center or typical value
Measures of Dispersion	Range, Variance, Standard Deviation	Show how spread out data is
Graphical Tools	Histograms, Pie Charts, Box Plots	Visualize data distribution
Measures of Shape	Skewness, Kurtosis	Describe asymmetry or peakedness

Inferential Statistics

Goal: Make conclusions or predictions about a population based on a sample.

Because we can’t measure everyone, we collect a sample, compute statistics on it, and infer population-level patterns while accounting for uncertainty.

You measure the average height of 100 people in a city (170 cm) and infer that the average height of the whole city’s population is likely close to that but not exact.

You quantify that uncertainty using confidence intervals or hypothesis tests.

Category	Examples	Purpose
Estimation	Confidence Intervals, Point Estimates	Estimate population parameters
Hypothesis Testing	t-test, chi-square, ANOVA	Test if observed effects are significant
Regression & Correlation	Linear regression, correlation coefficient	Model relationships between variables
Probability Models	Normal distribution, binomial, Poisson	Model random variation

outlier

An outlier is a data point that is inconsistent with the overall pattern of the dataset.

Heights of adults (in cm):
160, 162, 165, 170, 172, 168, 450

Here, 450 cm doesn’t fit the pattern it’s likely an error or a completely different kind of observation.

So:

• Normal variation: within the expected range.
• Outlier: outside the expected range.

to Detect Outliers

There’s no single universal definition, but several statistical rules of thumb are widely used.

(a) Z-Score Method (Standard Deviation Rule)

If data is roughly normal (bell-shaped):

$$Z=\frac{x-\mu }{\sigma }$$

where

• ( x ) = data point
• ( \mu ) = mean
• ( \sigma ) = standard deviation

If |Z| > 3 → it’s far from the mean by more than 3 standard deviations → potential outlier.

(Why 3?) In a normal distribution, 99.7% of points fall within ±3σ. So anything beyond that is extremely rare.

(b) IQR Method (Boxplot Rule)

Works well for non-normal data.

1. Find Q1 (25th percentile) and Q3 (75th percentile).
2. Compute IQR = Q3 – Q1.
3. Define limits: $$\text{Lower bound}=Q1-1.5\times IQR$$ $$\text{Upper bound}=Q3+1.5\times IQR$$
4. Any point outside this range is a potential outlier.

(Why 1.5×IQR?) It empirically captures ~99% of central data for many typical distributions, while flagging extremes.

(c) Visual Methods

• Box Plot: Outliers appear as dots outside whiskers.
• Scatter Plot: Useful for multivariate outliers.
• Histogram: Can show gaps or tails where outliers live.

Five number summary

Every dataset can be summarized along two main questions:

1. Where are the values located? (center, extremes)
2. How are they spread out? (variation, range)

The five-number summary is a compact way to describe both using five key points that divide and bound your data.

It’s especially useful for non-normal, skewed, or small datasets, where mean and standard deviation can be misleading.

Position	Symbol	Name	Description
1	Min	Minimum	Smallest value in dataset
2	Q1	First Quartile (25th percentile)	Value below which 25% of data lies
3	Median (Q2)	Second Quartile (50th percentile)	Middle value (splits data in half)
4	Q3	Third Quartile (75th percentile)	Value below which 75% of data lies
5	Max	Maximum	Largest value in dataset

Let’s take a simple dataset: Data: [2, 4, 5, 7, 8, 10, 12, 13, 14, 18]

Step 1 — Order the data: (Already sorted)

Step 2 — Find the median (Q2): Middle value between 8 and 10 → (8 + 10)/2 = 9.

Step 3 — Split into halves:

• Lower half: [2, 4, 5, 7, 8]
• Upper half: [10, 12, 13, 14, 18]

Step 4 — Find Q1 (median of lower half): = 5

Step 5 — Find Q3 (median of upper half): = 13

Step 6 — Min = 2, Max = 18

Five-number summary: [2, 5, 9, 13, 18]

Value	What It Represents	What It Tells You
Min	Lower bound	Smallest observed value
Q1	Lower quartile	Start of central 50%
Median (Q2)	Center	Central location
Q3	Upper quartile	End of central 50%
Max	Upper bound	Largest observed value

The middle 50% of the data lies between Q1 and Q3. The spread of that zone is called the IQR (Interquartile Range):

$\text{IQR}=Q3-Q1$

The five-number summary directly generates a box plot, which visualizes:

Min  Q1  Median  Q3  Max
 |----|=====|====|----|

• The box spans from Q1 to Q3 (the interquartile range).
• The line inside the box marks the median.
• The whiskers extend to Min and Max (or sometimes to limits before outliers).

So, the five-number summary is the numerical backbone of the box plot.