Normal Approximation to Binomial Distribution
Normal Approximation to Binomial Distribution Let $X$ be a binomially distributed random variable with number of trials …
2 min ReadAll of Raju Chaudhari's tutorials
Confidence interval for means t 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 …
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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 …
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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
Bernoulli Distribution
Introduction In the theory of probability and statistics, a Bernoulli trial or Bernoulli Experiment is a random …
3 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 …
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Beta Type II Distribution
Beta Type II Distribution A continuous random variable $X$ is said to have a beta type II distribution with parameters …
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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
Binomial Distribution
Introduction Binomial distribution is one of the most important discrete distribution in statistics. In this tutorial we …
7 min Read
Bowleys Coefficient of skewness for grouped data
Bowley’s Coefficient of Skewness for grouped data Skewness is a measure of symmetry. The meaning of skewness is …
6 min Read
Bowleys Coefficient of skewness for ungrouped data
Bowley’s Coefficient of Skewness for Ungrouped data Skewness is a measure of symmetry. The meaning of skewness is …
5 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
Central moments for grouped data
Central moments for grouped data Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution. The mean of $X$ is …
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Central moments for ungrouped data
Central moments for ungrouped data Let $x_1, x_2,\cdots, x_n$ be $n$ observations. The mean of $X$ is denoted by …
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 …
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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
Coefficient of variation for grouped data
Coefficient of variation for grouped data Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution. Formula …
2 min Read
Coefficient of variation for ungrouped data
Coefficient of variation for ungrouped data Coefficient of variation is an absolute measure of variation and is used for …
2 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
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 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
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 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
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 t
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 …
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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
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 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
CI for difference between two population proportions Let $X_1$ be the observed number of individuals possessing certain …
2 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 Variance
Introduction Confidence interval can be used to estimate the population parameter with the help of an interval with some …
1 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
Covariance between X and Y
Covariance between X and Y Let $(x_i, y_i), i=1,2, \cdots , n$ be $n$ pairs of observations. Covariance measures the …
2 min Read
Deciles for grouped data
Deciles for grouped data Deciles are the values of arranged data which divide whole data into ten equal parts. They are …
5 min Read
Deciles for ungrouped data
Deciles for ungrouped data Deciles are the values of arranged data which divide whole data into ten equal parts. They …
4 min Read
Discrete Uniform Distribution Guide
Discrete Uniform Distribution A discrete random variable has a discrete uniform distribution if each value of the random …
4 min Read
Empirical Rule for grouped data
Empirical rule for grouped data Empirical rule is the general rule of thumb that applies to bell shaped (symmetrical) …
4 min Read
Empirical Rule for ungrouped data
Empirical Rule for ungrouped data Empirical rule is the general rule of thumb that applies to bell shaped (symmetrical) …
2 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
Exponential Distribution
Exponential distribution Exponential Distribution Definition A continuous random variable $X$ is said to have an …
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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
Five number summary for grouped data
Five number summary for grouped data A five number summary is a quick and easy way to determine the the center, the …
5 min Read
Five number summary for ungrouped data
Five number summary for ungrouped data A five number summary is a quick and easy way to determine the the center, the …
2 min Read
Gamma Distribution
Gamma Distribution Gamma distribution is used to model a continuous random variable which takes positive values. Gamma …
5 min Read
Gamma Distribution Examples
Definition of Gamma Distribution A continuous random variable $X$ is said to have an gamma distribution with parameters …
0 min Read
Geometric Distribution
Introduction Geometric distribution is used to model the situation where we are interested in finding the probability of …
9 min Read
Geometric mean for ungrouped data
Geometric mean for ungrouped data Let $x_i, i=1,2, \cdots , n$ be $n$ positive observations then the geometric mean of …
0 min Read
Harmonic Mean for grouped data
Harmonic mean for grouped data Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution. Then the harmonic …
1 min Read
Harmonic Mean for ungrouped data
Harmonic mean for ungrouped data Let $x_i, i=1,2, \cdots , n$ be $n$ positive observations then the harmonic mean of $X$ …
1 min Read
Hungarian Method to solve Assignment Problem
Hungarian Method to solve Assignment Problem For obtaining an optimal assignment, Hungarian method involves following …
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Hypergeometric Distribution
Hypergeometric Experiment A hypergeometric experiment is an experiment which satisfies each of the following conditions: …
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Inter Quartile Range for grouped data
Inter Quartile Range for grouped data Inter quartile range is given by $IQR = Q_3-Q_1$ where, $Q_1$ is the first …
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Inter Quartile Range for ungrouped data
Inter Quartile Range for ungrouped data Inter quartile range is given by $IQR = Q_3-Q_1$ where $Q_1$ is the first …
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Karl Pearson coefficient of skewness for grouped data
Karl Pearson coefficient of skewness for grouped data Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency …
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Karl Pearson coefficient of skewness for ungrouped data
Karl Pearson coefficient of skewness for ungrouped data Let $x_i, i=1,2, \cdots , n$ be $n$ observations. Formula The …
3 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
Kelly Coefficient of skewness for grouped data
Kelly’s coefficient of skewness for grouped data For a symmetric distribution, the first decile namely $D_1$ and …
5 min Read
Kelly Coefficient of skewness for ungrouped data
Kelly’s Coefficient of Skewness for ungrouped data For a symmetric distribution, the first decile namely $D_1$ and …
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
Mean Absolute Deviation for grouped data
Mean absolute deviation for grouped data Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution. The mean of …
2 min Read
Mean Absolute Deviation for ungrouped data
Mean absolute deviation for ungrouped data Let $x_i, i=1,2, \cdots , n$ be $n$ observations. The mean of $X$ is denoted …
1 min Read
Mean median and mode for grouped data
Mean, median and mode for grouped data Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution. Formula …
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Mean median and mode for ungrouped data
Mean, median and mode for ungrouped data Let $x_i, i=1,2, \cdots , n$ be $n$ observations. Formula The mean of $X$ is …
2 min Read
Moment coefficient of kurtosis for grouped data
Moment coefficient of kurtosis for grouped data Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution. The …
3 min Read
Moment coefficient of kurtosis for ungrouped data
Moment coefficient of kurtosis for ungrouped data Let $x_1, x_2,\cdots, x_n$ be $n$ observations. The mean of $X$ is …
1 min Read
Moment coefficient of Skewness for grouped data
Moment coefficient of skewness for grouped data Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution. The …
3 min Read
Moment coefficient of skewness for ungrouped data
Moment coefficient of skewness for ungrouped data Let $x_1, x_2,\cdots, x_n$ be $n$ observations. The mean of $X$ is …
1 min Read
Negative Binomial Distribution
Introduction A negative binomial distribution is based on an experiment which satisfies the following three conditions: …
8 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
Normal Distribution
Normal Distribution Normal distribution is one of the most fundamental distribution in Statistics. It is also known as …
10 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 …
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Octiles for grouped data
Octiles for grouped data Octiles are the values of arranged data which divide whole data into eight equal parts. They …
4 min Read
Octiles for ungrouped data
Octiles for ungrouped data Octiles are the values of arranged data which divide whole data into eight equal parts. They …
1 min Read
Outliers for ungrouped data
Outliers for ungrouped data An outlier is any observation that is $1.5(IQR)$ away from the first quartile ($Q_1$) or …
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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 …
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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
Percentiles for grouped data
Percentiles for grouped data Percentiles are the values which divide whole distriution into hundred equal parts. They …
4 min Read
Percentiles for ungrouped data
Percentiles for ungrouped data Percentiles are the values of arranged data which divide whole data into hundred equal …
4 min Read
Plus Four Confidence Interval for 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 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
Plus Four CI for difference between two population proportions The plus four method produces more accurate confidence …
2 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
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
Poisson Distribution
Poisson Distribution Poisson distribution helps to describe the probability of occurrence of a number of events in some …
6 min Read
Quartile Deviation for ungrouped data
Quartile Deviation for ungrouped data Quartile Deviation is given by $QD = \dfrac{Q_3-Q_1}{2}$ Coefficient of Quartile …
3 min Read
Quartiles for grouped data
Quartile for grouped data Quartiles are the values which divide whole distriution into four equal parts. They are 3 in …
5 min Read
Quartiles for ungrouped data
Quartiles for ungrouped data Quartiles are the values of arranged data which divide whole data into four equal parts. …
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
Simple linear regression from raw data
Simple linear regression from raw data Let $(x_i, y_i), i=1,2, \cdots , n$ be $n$ pairs of observations. The simple …
3 min Read
Simple linear regression from sum and sumsquares
Simple linear regression from sum and sum of squares Let $(x_i, y_i), i=1,2, \cdots , n$ be $n$ pairs of observations. …
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Simple linear regression from summarize data
Simple linear regression from summarize data Let $(x_i, y_i), i=1,2, \cdots , n$ be $n$ pairs of observations. The …
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
Summary Statistics for grouped data
Summary statistic for grouped data Summary statistic summarize and provide information about the sample data. It …
5 min Read
Summary Statistics for ungrouped data
Summary statistic for ungrouped data Summary statistic summarize and provide information about the sample data. It …
4 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
Variance and standard deviation for grouped data
Variance and standard deviation for grouped data Let $(x_i,f_i), i=1,2, \cdots , n$ be the observed frequency …
2 min Read
Variance and standard deviation for ungrouped data
Variance and standard deviation for ungrouped data Let $x_i, i=1,2, \cdots , n$ be $n$ observations. Formula Sample …
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
Weibull Distribution
Weibull Distribution A continuous random variable $X$ is said to have a Weibull distribution with three parameters …
4 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
Bernoulli Distribution Example
Bernoulli Distribution In this tutorial, we will discuss example on Bernoulli’s distribution. The discrete random …
1 min Read
Binomial Distribution Examples
Binomial Distribution In this tutorial, we will provide you step by step solution to some numerical examples on Binomial …
4 min Read
Continuous Uniform Distribution Examples
Continuous Uniform Distribution The continuous uniform distribution is the simplest probability distribution where all …
3 min Read
Discrete Uniform Distribution Examples
Discrete uniform distribution In this tutorial we will discuss some examples on discrete uniform distribution and learn …
3 min Read
Exponential Distribution Examples
Exponential Distribution In this tutorial, we will provide you step by step solution to some numerical examples on …
3 min Read
Geometric Distribution Examples
Geometric Distribution In this tutorial, we will provide you step by step solution to some numerical examples on …
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
Log-normal Distribution
Definition of Log-normal distribution The continuous random variable $X$ has a log-normal distribution if the random …
3 min Read
Log-normal Distribution Examples
Log-normal distribution Normal distribution is not suitable when the data are highly skewed or data contains outliers. …
2 min Read
Negative Binomial Distribution Examples
Negative Binomial Distribution In this tutorial, we will provide you step by step solution to some numerical examples on …
4 min Read
Poisson Distribution Example
Poisson distribution In this tutorial, we will provide you step by step solution to some numerical examples on Poisson …
3 min Read
Confidence Interval for means t 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
Hypergeometric Distribution Examples
Hypergeometric Experiment In this tutorial, we will provide you step by step solution to some numerical examples on …
4 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
Normal Distribution Examples
Normal Distribution A continuous random variable $X$ is said to have a normal distribution if its probability density …
3 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
Weibull Distribution Examples - Step by Step Guide
Weibull Distribution In this tutorial we will discuss about the Weibull distribution and examples. Weibull distribution …
3 min Read
Geometric mean for grouped data
Geometric mean for grouped data Let $(x_i,f_i), i=1,2, \cdots , n$ be the given frequency distribution then the …
1 min Read
Quartile Deviation for grouped data
Quartile Deviation for grouped data Quartile deviation is half the difference between the thrid quartile $Q_3$ and the …
5 min Read