Bernoulli Distribution - Complete Guide
Introduction A Bernoulli trial or Bernoulli experiment is a random experiment with exactly two mutually exclusive …
3 min ReadAll of Raju Chaudhari's tutorials
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Binomial Distribution - Complete Guide
Introduction Binomial distribution is one of the most important discrete distributions in statistics. It models the …
5 min Read
Exponential Distribution: Complete Guide with Theory, Properties, and Examples
Introduction to Exponential Distribution The exponential distribution is a continuous probability distribution used to …
10 min Read
Gamma Distribution: Complete Guide with Theory, Properties, and Examples
Introduction to Gamma Distribution The gamma distribution is a versatile continuous probability distribution used to …
8 min Read
Geometric Distribution: Complete Guide
Introduction The geometric distribution is used to model situations where we are interested in finding the probability …
7 min Read
Hypergeometric Distribution: Complete Guide
Hypergeometric Experiment A hypergeometric experiment is an experiment which satisfies each of the following conditions: …
6 min Read
Log-Normal Distribution: Complete Guide with Theory, Properties, and Examples
Introduction to Log-Normal Distribution The log-normal distribution is a continuous probability distribution of a random …
10 min Read
Negative Binomial Distribution: Complete Guide
Introduction The negative binomial distribution is based on an experiment which satisfies the following three …
7 min Read
Normal Distribution: Complete Guide with Theory and Examples
Introduction to Normal Distribution The normal distribution, also known as the Gaussian distribution, is one of the most …
9 min Read
Poisson Distribution: Complete Guide
Poisson Distribution The Poisson distribution is one of the most important discrete probability distributions in …
9 min Read
Uniform Distribution: Complete Guide
Introduction to Uniform Distribution The uniform distribution is one of the simplest probability distributions where all …
8 min Read
Weibull Distribution: Complete Guide with Theory, Properties, and Examples
Introduction to Weibull Distribution The Weibull distribution is one of the most widely used continuous probability …
8 min Read
Hypothesis Testing & Statistical Inference - Complete Guide
Introduction to Hypothesis Testing Hypothesis testing is a statistical method for making decisions about populations …
14 min Read
Advanced Statistics - Complete Guide
Advanced statistics extends beyond introductory methods to handle complex real-world scenarios. When assumptions are …
9 min Read
Central Moments - Complete Guide
Central Moments Overview Central moments are statistical measures calculated around the mean (or any fixed point) rather …
4 min Read
Coefficient of Variation (CV) - Complete Guide
Coefficient of Variation (CV) expresses standard deviation as a percentage of the mean. It’s dimensionless, making …
1 min Read
Confidence Intervals - Complete Guide
While hypothesis testing asks “Is there a significant difference?”, confidence intervals ask …
13 min Read
Correlation and Regression Analysis - Complete Guide
Correlation measures how two variables move together; regression predicts one variable from another. These tools help …
10 min Read
Deciles - Complete Guide
Deciles Overview Deciles divide a dataset into 10 equal parts, with 9 dividing points: D₁, D₂, …, D₉. Each decile …
3 min Read
Descriptive Statistics - Complete Guide
Descriptive statistics is the foundation of data analysis. It involves methods to collect, organize, summarize, and …
16 min Read
Effect Sizes and Statistical Power - Complete Guide
Statistical significance (p-value < 0.05) tells you an effect exists; effect size tells you how big it is. Large …
10 min Read
Empirical Rule (68-95-99.7 Rule) - Complete Guide
Empirical Rule Overview The empirical rule, also known as the 68-95-99.7 rule, states that for a normal (bell-shaped) …
3 min Read
Five-Number Summary - Complete Guide
Five-Number Summary Overview The five-number summary is a set of five statistics that provides a compact description of …
3 min Read
Geometric Mean - Complete Guide
The geometric mean is the nth root of the product of n values. It’s particularly useful for data representing …
1 min Read
Harmonic Mean - Complete Guide
The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals. It’s particularly useful for …
1 min Read
Interquartile Range (IQR) - Complete Guide
The interquartile range (IQR) is the range of the middle 50% of data. It’s robust to outliers and useful for …
1 min Read
Kurtosis - Complete Guide
Kurtosis Overview Kurtosis measures the “tailedness” and peakedness of a distribution compared to a normal …
4 min Read
Mean Absolute Deviation (MAD) - Complete Guide
Mean Absolute Deviation (MAD) is the average absolute deviation from the mean. It’s more robust to outliers than …
1 min Read
Mean, Median, and Mode - Complete Guide
Mean, median, and mode are the three primary measures of central tendency. They each describe the “typical” …
4 min Read
Octiles - Complete Guide
Octiles Overview Octiles divide a dataset into 8 equal parts, with 7 dividing points: O₁, O₂, …, O₇. Each octile …
3 min Read
Outlier Detection and Analysis - Complete Guide
Outliers Overview Outliers are data points that deviate significantly from the typical pattern of a dataset. They can …
5 min Read
Percentiles - Complete Guide
Percentiles Overview Percentiles divide data into 100 equal parts. The i-th percentile is the value below which i% of …
4 min Read
Probability Distributions - Complete Guide
Probability distributions describe how values in a dataset are spread across a range. They’re fundamental to …
10 min Read
Quartile Deviation - Complete Guide
Quartile deviation (also called semi-interquartile range) is half the interquartile range. It represents the average …
1 min Read
Quartiles - Complete Guide
Quartiles Overview Quartiles are values that divide an arranged dataset into four equal parts. They consist of three …
5 min Read
Sampling and Survey Design - Complete Guide
Every statistical study begins with a fundamental question: How do we choose who to study? Sampling is the process of …
10 min Read
Skewness - Complete Guide
Skewness Overview Skewness measures the asymmetry of a distribution. It indicates whether data is symmetrically …
4 min Read
Summary Statistics - Complete Guide
Summary Statistics Overview Summary statistics provide a comprehensive overview of a dataset by summarizing key …
4 min Read
Variance and Standard Deviation - Complete Guide
Variance and standard deviation measure how spread out data is around the mean. Variance is measured in squared units, …
2 min Read
CI for Two Means (Equal Variances)
Confidence interval for difference in means when variances are unknown and equal Let $x_1, x_2, \cdots, x_{n_1}$ be a …
2 min Read
Beta Type I Distribution
Beta Type I Distribution A continuous random variable $X$ is said to have a beta type I distribution with parameters …
3 min Read
Assignment Problem
Assignment Problem The assignment problem is a special case of transportation problem. Suppose there are $n$ jobs to be …
3 min Read
Bayes Probability
Bayes’ Theorem One of the important uses of conditional probability is to find out the inverse probability. If an …
2 min Read
Bayes Probability Examples
Bayes’ Theorem One of the important uses of conditional probability is to find out the inverse probability. If an …
2 min Read
Beta Type I Distribution Examples
Definition of Beta Type I Distribution A continuous random variable $X$ is said to have a beta type I distribution with …
3 min Read
Beta Type II Distribution
Beta Type II Distribution A continuous random variable $X$ is said to have a beta type II distribution with parameters …
3 min Read
Beta Type II Distribution Examples
Beta Type II Distribution A continuous random variable $X$ is said to have a beta type II distribution with parameters …
1 min Read
Cauchy Distribution
Cauchy Distribution A continuous random variable $X$ is said to follow Cauchy distribution with parameters $\mu$ and …
8 min Read
Cauchy Distribution Examples
Cauchy Distribution A continuous random variable $X$ is said to follow Cauchy distribution with parameters $\mu$ and …
1 min Read
Chebyshev Inequality Rule
Chebyshev’s Inequality Chebyshev’s Theorem If $g(x)$ is a non-negative function and $f(x)$ be p.m.f. or …
4 min Read
Chebyshev Inequality Rule Examples
Chebyshev’s Inequality Let $X$ be a random variable with mean $\mu$ and finite variance $\sigma^2$. Then for any …
3 min Read
Chi-square Test for variance
Testing variance or standard deviation In this tutorial we will discuss a method for testing a claim made about the …
2 min Read
Chi-square Test for variance examples
Chi-square Test for the Variance In this tutorial we will discuss a method for testing a claim made about the population …
5 min Read
Column Minima Method
Column Minima Method Step 1 Select the smallest cost in the first column of the transportation table. Let it be …
1 min Read
Confidence Interval for Mean (T-Distribution)
Introduction Confidence interval can be used to estimate the population parameter with the help of an interval with some …
3 min Read
Confidence Interval for Mean examples
Confidence Interval for Mean In this tutorial we will discuss some numerical examples to understand how to construct a …
3 min Read
Confidence Interval for mean-t examples
Confidence Interval for the population mean when standard deviation is unknown This tutorial covers examples on …
4 min Read
Confidence Interval for Means (Z-Distribution)
Introduction In this article we will discuss step by step procedure to construct a confidence interval for difference …
2 min Read
Confidence Interval for Means Examples
CI for difference between two population means (variances are known) In this tutorial we will discuss some examples on …
3 min Read
Confidence interval for means t equal variances examples
Confidence interval for difference in means when variances are unknown and equal In this tutorial we will discuss step …
4 min Read
Confidence Interval for Paired Data
Confidence Interval for paired t-test Let $X_1, X_2, \cdots, X_n$ and $Y_1, Y_2, \cdots, Y_n$ be two dependent samples …
2 min Read
Confidence Interval for Paired t examples
Confidence Interval for paired t-test In this tutorial we will discuss how to determine confidence interval for the …
6 min Read
Confidence Interval for Proportion examples
Confidence Interval for Proportion In this tutorial we will discuss some numerical examples to understand how to …
3 min Read
Confidence Interval for Proportions examples
CI for difference between two population proportions In this tutorial we will discuss some examples on confidence …
6 min Read
Confidence Interval for Ratio of Variances
Confidence Interval for ratio of variances Let $X_1, X_2, \cdots , X_{n_1}$ be a random sample of size $n_1$ from …
1 min Read
Confidence Interval for Ratio of variances examples
Confidence Interval for ratio of variances In this tutorial we will discuss some examples on confidence interval for …
3 min Read
Confidence Interval for Single Mean
Introduction Confidence interval can be used to estimate the population parameter with the help of an interval with some …
2 min Read
Confidence Interval for Single Proportion
Introduction Confidence interval can be used to estimate the population parameter with the help of an interval with some …
2 min Read
Confidence Interval for Two Proportions
CI for difference between two population proportions Let $X_1$ be the observed number of individuals possessing certain …
2 min Read
Confidence Interval for Variance
Introduction Confidence interval can be used to estimate the population parameter with the help of an interval with some …
2 min Read
Confidence Interval for Variance examples
Confidence Interval for Variance In this tutorial we will discuss some numerical examples to understand how to construct …
3 min Read
Continuous Uniform Distribution
Continuous Uniform Distribution A random variable has a uniform distribution if each value of the random variable is …
4 min Read
Expected value and variance of probability distribution
Example Find the expected value and variance of $X$ for the following probability distribution. $x$ 1 2 3 4 $P(X=x)$ …
1 min Read
F test for two variances
Testing equality of two variances Many times it is desirable to compare two variances rather than comparing two means. F …
3 min Read
F test for two variances examples
F test for two variances In this tutorial we will discuss some examples on F test for comparing two variances or …
6 min Read
Hungarian Method to solve Assignment Problem
Hungarian Method to solve Assignment Problem For obtaining an optimal assignment, Hungarian method involves following …
2 min Read
Karl Pearson Correlation Coefficient
Karl Pearson’s Correlation Coefficient Let $(x_i, y_i), i=1,2, \cdots , n$ be $n$ pairs of observations then the …
4 min Read
Laplace Distribution
Laplace Distribution A continuous random variable $X$ is said to have a Laplace distribution (Double exponential …
8 min Read
Least Cost Entry Method
Least Cost Entry Method This method is also known as Matrix Minima Method. Step 1 Select the smallest cost in the cost …
1 min Read
Normal Approximation to Poisson Distribution
Normal approximation to Poisson distribution In this tutorial we will discuss some numerical examples on Poisson …
3 min Read
North West Corner Method
North-West Corner Method The North-West cornet method starts at the northwest corner cell of the transportation problem …
1 min Read
paired t-test
Paired $t$-test (Dependent Sample) Let $X_1, X_2, \cdots, X_n$ and $Y_1, Y_2, \cdots, Y_n$ be two dependent samples each …
2 min Read
paired t-test examples
Paired $t$-test (Dependent Sample) In this tutorial we will discuss some numerical examples on paired t test. Example 1 …
7 min Read
Plus Four Confidence Interval for Proportion Examples
Plus Four Confidence Interval for Proportion In this article we will discuss step by step examples to construct a plus …
4 min Read
Plus Four Confidence Interval for Proportions Examples
Plus Four CI for difference between two population proportions In this tutorial we will discuss some step by step …
2 min Read
Plus-Four Confidence Interval for Single Proportion
Introduction In this article we will discuss step by step procedure to construct a plus four confidence interval for …
2 min Read
Plus-Four Confidence Interval for Two Proportions
Plus Four CI for difference between two population proportions The plus four method produces more accurate confidence …
2 min Read
Poisson approximation to binomial distribution
Poisson approximation to binomial distribution Let $X$ be a binomially distributed random variable with number of trials …
3 min Read
Row Minima Method
Row Minima Method Step 1 Select the smallest cost in the first row of the transportation table. Let it be $c_{1j}$. …
1 min Read
Sample Size to estimate mean
Sample size to estimate mean The minimum sample size required to estimate the population mean $\mu$ is $$ …
2 min Read
Sample size to estimate proportion
Sample size to estimate proportion The minimum sample size required to estimate the population proportion is $$ n …
2 min Read
Spearman Rank Correlation Coefficient
Spearman’s Rank correlation Coefficient Let $(x_1, y_1), (x_2, y_2), \cdots , (x_n, y_n)$ be the ranks of $n$ …
2 min Read
t test p value
p value of the test p-value of the test is the probability that the test statistic under null hypothesis will take on …
3 min Read
t-test for Mean
One Sample t test for mean In this tutorial we will explain the six steps approach used in hypothesis testing to test …
2 min Read
t-test for Mean examples
One Sample $t$-Test for mean In this tutorial we will discuss some numerical examples on one sample t-test for testing …
5 min Read
t-test for two means equal variances
Two sample t test with unknown and equal variances In this tutorial we will discuss two sample t-test for testing …
2 min Read
t-test for two means equal variances examples
Two sample t test for means with unknown but equal variances In this tutorial we will discuss some numerical examples on …
6 min Read
t-test for two means unequal variances
Two sample t test for means with unknown and unequal variances In this tutorial we will discuss two sample t test for …
3 min Read
t-test for two means unequal variances examples
Two Sample $t$-Test (Independent Sample with Unequal Variances) In this tutorial we will discuss some numerical examples …
5 min Read
Testing Correlation Coefficient
Testing Correlation Coefficient Let $(x_i, y_i), i=1,2,\cdots, n$ be a random sample of $n$ pairs of observations drawn …
2 min Read
Testing Correlation Coefficient Examples
Testing Significance of Linear Relationship A test of significance for a linear relationship between the variables $x$ …
6 min Read
Testing Correlation Coefficient Part 2
Testing Correlation Coefficient Let $(X_i, Y_i), i=1,2,\cdots, n$ be a random sample of $n$ pairs of observations drawn …
2 min Read
Testing Correlation Coefficient Part 2 examples
Testing Correlation Coefficient In this tutorial we will discuss step by step solution of numerical problems on testing …
5 min Read
Testing homogeneity of Two Correlation Coefficient
Testing Homogeneity of Two Correlation Coefficient We wish to test the null hypothesis that the correlation $\rho_1$ …
2 min Read
Testing homogeneity of Two Correlation Coefficient Examples
Testing Homogeneity of Two Correlation Coefficient In this tutorial we will discuss step by step solution to some …
4 min Read
Transhipment Problem
Transhipment Problem The transhipment problem is more general than that of regular transportation problem, where direct …
1 min Read
Transportation Problem
Transportation Problem Transportation problem is a special class of linear programming problem that deals with …
2 min Read
Travelling Salesman Problem
Travelling Salesman Problem Suppose a salesman wants to visit certain number of cities, say, $n$. Let $c_{ij}$ be the …
1 min Read
UV Method for Optimal Solution
UV Method For Finding Optimal Solution of TP Step 1 First, construct a transportation table entering the origin …
2 min Read
Vogels Approximation Method
Vogel’s Approximation Method (VAM) Vogel’s approximation method is an improved version of the least cost …
1 min Read
Z Test p value
p value of the test p-value of the test is the probability that the test statistic under null hypothesis will take on …
2 min Read
Z-test for Mean
One sample Z test for mean In this tutorial we will explain the six steps approach used in hypothesis testing to test …
2 min Read
Z-test for Mean Examples
Z-test for testing population mean In this tutorial we will discuss some numerical examples on one sample z test for …
5 min Read
Z-test for Proportion
$Z$-Test for Proportion Let $X$ be the observed number of individuals possessing certain attributes (say, number of …
2 min Read
Z-test for Proportion examples
Z-Test for Proportion In this tutorial we will discuss some numerical examples on one sample Z test for testing …
4 min Read
z-test for two means
Two sample Z test for means In this tutorial we will discuss two sample Z test for testing the difference between means …
3 min Read
z-test for two means examples
Two sample Z test for means In this tutorial we will discuss some numerical examples on two sample z test for testing …
6 min Read
z-test for two proportions
Two sample proportion test Suppose we want to compare two distinct populations $A$ and $B$ with respect to possessions …
3 min Read
z-test for two proportions examples
$Z$-Test for difference of Proportions In this tutorial we will discuss some numerical examples on two sample Z test for …
6 min Read
Continuous Uniform Distribution Examples
Continuous Uniform Distribution The continuous uniform distribution is the simplest probability distribution where all …
3 min Read
Laplace Distribution Examples
Laplace Distribution A continuous random variable $X$ is said to have a Laplace distribution, if its p.d.f. is given by …
1 min Read
CI for Two Means (Unequal Variances)
Confidence interval for means (variances are unknown and unequal) Let $x_1, x_2, \cdots, x_{n_1}$ be a random sample of …
2 min Read
Sample size to test mean
Sample size to test mean The minimum sample size required to test the hypothesis about the population mean $\mu$ is $$ …
1 min Read
Confidence interval for means t unequal variances examples
CI for difference between two population means with unknown and unequal variances In this tutorial we will discuss step …
4 min Read
Normal Approximation to Binomial Distribution
Normal Approximation to Binomial Distribution Let $X$ be a binomially distributed random variable with number of trials …
2 min Read