1. Introduction
2. Population vs Sample
3. Types of Data
4. Levels of Measurement
5. Descriptive Statistics
6. Measures of Central Tendency
7. Measures of Dispersion
8. Percentiles and Quartiles
9. Outliers
10. Skewness
11. Kurtosis
12. Covariance
13. Correlation
14. Sampling
15. Sampling Distribution
16. Central Limit Theorem
17. Confidence Interval
18. Statistical Inference
19. Hypothesis Testing
20. P-value
21. Type I & Type II Errors
22. Statistical Tests
23. Feature Scaling
24. Standardization
25. Normalization
26. Z-Score
27. Missing Data
28. Cheat Sheet

1. Introduction

Statistics is the science of collecting, organizing, analyzing, interpreting, and presenting data. In AI and Machine Learning, statistics helps to:

Understand datasets
Discover patterns
Build predictive models
Evaluate model performance
Make data-driven decisions

2. Population vs Sample

Population	Sample
Entire dataset	Subset of population
Parameter (μ, σ)	Statistic (x̄, s)
Large	Smaller
Expensive to study	Easier to analyze

Example

Population

All students in India

Sample

1000 randomly selected students

3. Types of Data

Numerical Data

Numeric values used for calculations. Examples

Age
Height
Weight
Salary

Continuous Data

Can take any value. Example

Height = 170.35 cm

Discrete Data

Countable values. Example

Cars = 5

Categorical Data

Represents categories. Examples

Gender
Department
Blood Group
Country

4. Levels of Measurement

Level	Description	Example
Nominal	Categories only	Gender
Ordinal	Ordered categories	Rank
Interval	Equal intervals, no true zero	Temperature (°C)
Ratio	Equal intervals with true zero	Height

5. Descriptive Statistics

Descriptive statistics summarize a dataset. Common measures include:

Mean
Median
Mode
Variance
Standard Deviation
Range
Quartiles

6. Measures of Central Tendency

Mean

Average of all observations.

Formula

\bar{x}=\frac{\sum x_i}{n}

Python

import numpy as np

data = [10,20,30,40]

np.mean(data)

Median

Middle value after sorting. Python

np.median(data)

Mode

Most frequently occurring value. Python

from scipy import stats

stats.mode([1,2,2,3,3,3])

7. Measures of Dispersion

Shows how spread out the data is.

Range

Formula

Range = Maximum - Minimum

Python

max(data) - min(data)

Variance

Average squared deviation from the mean. Population Variance

\sigma^2=\frac{\sum(x-\mu)^2}{N}

Sample Variance

s^2=\frac{\sum(x-\bar{x})^2}{n-1}

Python

np.var(data)

Standard Deviation

Square root of variance. Formula

\sigma=\sqrt{\sigma^2}

Python

np.std(data)

Interpretation

High SD → More variation
Low SD → Less variation

8. Percentiles and Quartiles

Percentile

Indicates the percentage of observations below a value. Example

90th percentile

means

90% of observations lie below this value.

Python

np.percentile(data, 90)

Quartiles

Quartile	Percentage
Q1	25%
Q2	50% (Median)
Q3	75%

Interquartile Range (IQR)

IQR = Q3 - Q1

Python

np.percentile(data, [25,50,75])

9. Outliers

Outliers are observations that are significantly different from the rest.

IQR Rule

Lower Bound

Q1-1.5(IQR)

Upper Bound

Q3+1.5(IQR)

Values outside these limits are considered outliers.

10. Skewness

Measures asymmetry of data.

Type	Relationship
Symmetric	Mean = Median
Right Skew	Mean > Median
Left Skew	Mean < Median

Python

from scipy.stats import skew

skew(data)

11. Kurtosis

Measures tail heaviness.

High Kurtosis → More Outliers
Low Kurtosis → Fewer Outliers

Python

from scipy.stats import kurtosis

kurtosis(data)

12. Covariance

Measures whether two variables move together. Positive Covariance

↑ ↑

Negative Covariance

↑ ↓

Formula

Cov(X,Y)

Python

np.cov(x, y)

13. Correlation

Measures strength and direction of a relationship. Range

-1 \le r \le 1

Correlation	Meaning
+1	Perfect Positive
0	No Relationship
-1	Perfect Negative

Python

np.corrcoef(x, y)

14. Sampling

Sampling is selecting a subset from a population. Methods

Simple Random Sampling
Stratified Sampling
Cluster Sampling
Systematic Sampling

15. Sampling Distribution

Distribution of a sample statistic. Used in

Confidence Interval
Hypothesis Testing

16. Central Limit Theorem

If sample size is sufficiently large,

n \ge 30

then the sampling distribution of the mean becomes approximately normal. Why important?

Enables hypothesis testing
Basis for confidence intervals
Foundation of statistical inference

17. Confidence Interval

Range likely to contain the population parameter. 95% Confidence Interval

\bar{x}\pm z\frac{\sigma}{\sqrt n}

Interpretation

We are 95% confident that the population mean lies within this interval.

18. Statistical Inference

Drawing conclusions about a population using sample data. Includes

Confidence Interval
Hypothesis Testing

19. Hypothesis Testing

Null Hypothesis

H₀

Alternative Hypothesis

H₁

Example

H₀ : New model is not better

H₁ : New model performs better

20. P-value

Probability of obtaining the observed result assuming H₀ is true. Decision Rule

P-value	Decision
< 0.05	Reject H₀
≥ 0.05	Fail to Reject H₀

21. Type I & Type II Errors

Type I Error

False Positive Reject a true null hypothesis.

Type II Error

False Negative Fail to reject a false null hypothesis.

22. Statistical Tests

Test	Purpose
Z-Test	Large samples
T-Test	Small samples
Chi-Square Test	Categorical data
ANOVA	Compare multiple groups

23. Feature Scaling

Scaling ensures features have comparable ranges. Example

Feature	Range
Age	20–60
Salary	10,000–1,000,000

Benefits

Faster convergence
Better optimization
Improved ML accuracy

24. Standardization

Transforms data to

Mean = 0
Standard Deviation = 1

Formula

z=\frac{x-\mu}{\sigma}

Python

from sklearn.preprocessing import StandardScaler

scaler = StandardScaler()

scaled = scaler.fit_transform(X)

25. Normalization

Scales values between 0 and 1. Formula

x'=\frac{x-min}{max-min}

Python

from sklearn.preprocessing import MinMaxScaler

scaler = MinMaxScaler()

scaled = scaler.fit_transform(X)

26. Z-Score

Measures distance from the mean. Formula

z=\frac{x-\mu}{\sigma}

Interpretation

Z-score	Meaning
0	Mean
+2	Above Mean
-2	Below Mean

Python

from scipy.stats import zscore

zscore(data)

27. Missing Data

Common approaches

Remove Rows
Remove Columns
Mean Imputation
Median Imputation
Mode Imputation

Python

import pandas as pd

df.fillna(df.mean())

Statistics Libraries

import numpy as np
import pandas as pd
import scipy.stats as stats
import matplotlib.pyplot as plt
import sklearn

28. Cheat Sheet

Concept	Purpose
Mean	Average
Median	Middle Value
Mode	Most Frequent Value
Variance	Spread
Standard Deviation	Dispersion
Quartiles	Divide Dataset
IQR	Detect Outliers
Skewness	Asymmetry
Kurtosis	Tail Heaviness
Covariance	Direction of Relationship
Correlation	Strength of Relationship
CLT	Sampling Theory
Confidence Interval	Estimate Population Parameter
Hypothesis Test	Statistical Decision
P-value	Evidence Against H₀
Z-score	Distance from Mean
Standardization	Mean = 0, SD = 1
Normalization	Scale Between 0 and 1

​Table of Contents

​1. Introduction

​2. Population vs Sample

​Example

​3. Types of Data

​Numerical Data

​Continuous Data

​Discrete Data

​Categorical Data

​4. Levels of Measurement

​5. Descriptive Statistics

​6. Measures of Central Tendency

​Mean

​Formula

​Python

​Median

​Mode

​7. Measures of Dispersion

​Range

​Variance

​Standard Deviation

​8. Percentiles and Quartiles

​Percentile

​Quartiles

​Interquartile Range (IQR)

​9. Outliers

​IQR Rule

​10. Skewness

​11. Kurtosis

​12. Covariance

​13. Correlation

​14. Sampling

​15. Sampling Distribution

​16. Central Limit Theorem

​17. Confidence Interval

​18. Statistical Inference

​19. Hypothesis Testing

​Null Hypothesis

​Alternative Hypothesis

​20. P-value

​21. Type I & Type II Errors

​Type I Error

​Type II Error

​22. Statistical Tests

​23. Feature Scaling

​24. Standardization

​25. Normalization

​26. Z-Score

​27. Missing Data

​Statistics Libraries

​28. Cheat Sheet

Table of Contents

1. Introduction

2. Population vs Sample

Example

3. Types of Data

Numerical Data

Continuous Data

Discrete Data

Categorical Data

4. Levels of Measurement

5. Descriptive Statistics

6. Measures of Central Tendency

Mean

Formula

Python

Median

Mode

7. Measures of Dispersion

Range

Variance

Standard Deviation

8. Percentiles and Quartiles

Percentile

Quartiles

Interquartile Range (IQR)

9. Outliers

IQR Rule

10. Skewness

11. Kurtosis

12. Covariance

13. Correlation

14. Sampling

15. Sampling Distribution

16. Central Limit Theorem

17. Confidence Interval

18. Statistical Inference

19. Hypothesis Testing

Null Hypothesis

Alternative Hypothesis

20. P-value

21. Type I & Type II Errors

Type I Error

Type II Error

22. Statistical Tests

23. Feature Scaling

24. Standardization

25. Normalization

26. Z-Score

27. Missing Data

Statistics Libraries

28. Cheat Sheet