See how to use the STDEV.S function in Excel to speed up the process of calculating a sample standard deviation.

Practice Problems for Z-Scores

Try working out these practice problems calculating z-scores for a given mean and given standard deviation.

How to Make a Histogram in 7 Steps

Learn how to construct a histogram by hand following a seven step process. Histograms are commonly used throughout statistics.

Histogram Classes - Information and Examples

The classes or bins of a histogram are what split up the horizontal line on the bottom. Learn how to determine the number and width of these classes.

Statistics Related to Father's Day

Father's Day is celebrated in the United States on the third Sunday in June. Learn more about interesting statistics that pertain to Father's Day.

What Is the Intersection of Two Sets?

In set theory the intersection of two sets is the set of elements that both sets have in common.

Formula for the Normal Distribution or Bell Curve

See the formula that generates all normal distributions, which occur throughout statistics and are commonly known as bell curves.

Is a Type I Error or a Type II Error More Serious?

We compare and contrast type I and type II errors for hypothesis tests in statistics, and determine which type of error is more serious to make.

Statistics Help and Tutorials

Statistics Help and Tutorials

Inferential Statistics

Inferential statistics is one of the major branches of the field. It involves using data from a sample to infer details about a population.

Hypothesis Tests

Hypothesis testing is one of the main topics in the area of inferential statistics.

Confidence Intervals

Confidence Intervals are a major part of inferential statistics. Learn about various aspects of this topic.

What is the F-Distribution?

Learn more about the F-distribution, which is used for inference about population variances as well as ANOVA.

What Is the Runs Test

One way to see if a sequence of data is random is to use a runs test. Learn more about this type of hypothesis test.

Example of ANOVA Calculation

One factor analysis of variance (ANOVA) is a useful statistical tool. See the worked out details of a sample ANOVA calculation.

What Is Correlation in Statistics?

One goal of statistics is to find patterns hiding in data. Correlation is a tool that does just that. Find out how this is done.

Simple Random Sampling vs. Systematic Random Sampling

Simple random samples and systematic random samples both show up in statistics. Find out the subtle difference between these sampling techniques.

How to Calculate a Confidence Interval for a Mean When We Know Sigma

See step by step instructions to calculate a two-sided confidence interval for an unknown mean, when the population standard deviation is known.

Use of the Median with the Runs Test

The median can be used with the runs test when we have quantitative data.

How to Use a Tree Diagram

Tree diagrams are helpful to compute the probabilities of several independent events. See the steps involved in making and using a tree diagram for tossing two coins.

How to Use a Tree Diagram

Tree diagrams are helpful to compute the probabilities of several independent events. See the steps involved in making and using a tree diagram for tossing two coins.

How to Use a Tree Diagram

Tree diagrams are helpful to compute the probabilities of several independent events. See the steps involved in making and using a tree diagram for tossing two coins.

How to Use a Tree Diagram for Probability

Tree diagrams help to sort the probabilities of several independent events. See a worked out example of the use of a tree diagram.

Probabilities on a Normal Distribution - Z Score Table

Although programs exist to calculate the probability that a variable lies between two z scores, using a table to compute these probabilities is a worthwhile exercise. Page 8.

Z Score Tables - Area Between Negative Z Scores

Although programs exist to calculate the probability that a variable lies between two z scores, using a table to compute these probabilities is a worthwhile exercise. Page 7.

Z Score Tables - Area to the Left of a Positive Z Score

Although programs exist to calculate the probability that a variable lies between two z scores, using a table to compute these probabilities is a worthwhile exercise. Page 2.

Z Score Table - Area to the Right of a Positive Z Score

Although programs exist to calculate the probability that a variable lies between two z scores, using a table to compute these probabilities is a worthwhile exercise. Page 3.

How to Use a Z Score Table

Although programs exist to calculate the probability that a variable lies between two z scores, using a table to compute these probabilities is a worthwhile exercise.

How to Use a Z Score Table

Although programs exist to calculate the probability that a variable lies between two z scores, using a table to compute these probabilities is a worthwhile exercise.

Calculate Probabilities - Standard Normal Distribution Table

Illustrated tutorial on computing the probability that a variable lies between two z scores, using a standard normal distribution table .

Introduction to Finding Areas With a Table

Although programs exist to find the probability that a variable lies between two z scores, computing the probability with a table is a worthwhile exercise.

Addition Rules in Probability and Statistics

Addition rules in probability provide a way to calculate the probability of the union of two events.

Groundhog Day Statistics

How many times over the years has the groundhog seen his shadow? With this compilation of Punxsutawney Phil's forecasts, you can find out.

About Statistics: Tutorials, Worksheets, and Formulas

Calculate the statistics and explain the data around you, with beginner to advanced tutorials, tools, worksheets, and formulas for students and teachers.

What Is the Plural of the Word "Data"?

Data is a word that is used quite frequently in statistics, but is frequently misused. Find out what the correct way is to use this word.

What Is a Population in Statistics?

The goal of statistics is to help in the study of a population. A population consists of every individual being studied.

Hypothesis Test for Comparing Two Proportions

See how to conduct a hypothesis test for the difference of two population proportions.

What Is Bootstrapping? - Statistics

Bootstrapping is a resampling technique in statistics that almost seems to work like magic. Find out more about this interesting topic.

Example of Normal Approximation of a Binomial Distribution

See how to use the normal approximation to a binomial distribution and how these two different distributions are linked.

Why Does Zero Factorial Equal One?

Find out why zero factorial is one, and the reason why 0! = 1 is a valid mathematical equation.

An Introduction to Hypothesis Testing

The statistical process of hypothesis testing uses intuitive ideas from probability to determine if a claim about a population is likely to be true.

What Are the Levels of Measurement in Statistics?

The four levels of measurement (nominal, ordinal, interval and ratio) help to identify what statistical techniques can be performed with our data.

What Is the Importance of the Central Limit Theorem?

Learn about what makes the Central Limit Theorem so important to the practical uses of statistics.

Confidence Interval Example for Difference of Population Proportions

See a worked out example of how to calculate a confidence interval for the difference of two population proportions.

How Do We Determine Outliers in Statistics?

The determination of outliers can sometimes seem to be quite subjective. See what objective measurements allow us to identify outliers.

What Is a Percentile in Statistics?

Percentiles, not to be confused with percents, give a way to compare elements within an ordered list of data.

How to Find Degrees of Freedom

Statistics problems require us to determine the number of degrees of freedom. See how many should be used for different situations.

Slope of Regression Line and Correlation Coefficient

See how the slope of the regression line is directly dependent on the value of the correlation coefficient r.

7 Common Graphs in Statistics

Graphs help us to display data. Find out more about seven of the most common graphs in statistics.

Type I and Type II Errors - What Is the Difference?

Type I and type II errors are part of the process of hypothesis testing. What is the difference between these types of errors?

Differences Between Explanatory and Response Variables

Learn how to distinguish between explanatory and response variables, and how these differences are important in statistics.

Confidence Interval for the Difference of Two Population Proportions

See how to construct a confidence interval for the difference of two population proportions and learn some of the theory behind this topic in statistics.

Sample Size for a Margin of Error in Statistics

See how to plan a study by determining the sample size that is necessary in order to have a particular margin of error.

When Do You Use a Binomial Distribution?

Check out what conditions are necessary before you start using binomial distribution.

Probability and Games

Many games that involve an element of chance can be analyzed using tools from probability. Games with dice or cards are particularly good examples.

Probability Help & Tutorials

Probability Help, Probability Tutorials

Probabilities for Dihybrid Crosses in Genetics

See how to calculate probabilities in genetics for a dihybrid cross.

What Is Kurtosis in Statistics?

Kurtosis considers the shape of the peaks in the distribution of data. Learn more about this statistic.

What Is the Standard Normal Distribution?

Of the infinite number of possible bell curves, we consider one special bell curve, called the standard normal distribution.

Differences Between Parameters and Statistics

Parameters and statistics are important to distinguish between. Learn how to do this, and which value goes with a population and which with a sample.

Probability and Punnett Squares in Genetics

See how to use a table called a Punnett square to help calculate probabilities in genetics.

What Is a Histogram and How Is One Used?

A common graph in statistics is a histogram. Learn more about this type of graph, including several key differences between them and bar graphs.

Empirical Rule for Mean Median and Mode

Learn about an interesting relationship between three descriptive statistics for the center of a data set: the mean, median and mode.

How to Make a Stem and Leaf Plot - Tutorial

See how to make a stem and leaf plot, a useful way to organize data while retaining all of the data values.

What Is the Probability of a Royal Flush in Poker?

A royal flush is the highest ranked hand in the game of poker. Find out how rare this hand is and what the likelihood of having it dealt.

Simple Random Samples From a Table of Random Numbers

Simple random samples are important in statistics. See how to obtain this type of sample by using a table of random digits.

Hypothesis Test for a Population Proportion

Follow these detailed instructions to conduct a hypothesis test for a single population proportioen.

Student t Distribution Table

Sometimes the standard normal distribution cannot be used. In this case, consult this table of critical values of the Student t distribution.

What Is the Formula for Expected Value?

The formula for the expected value is relatively easy to compute, and involves several multiplications and additions.

What Are the First and Third Quartiles?

Learn about the first and third quartiles - numbers that are used to split a data set into four pieces in a specified way.

Confidence Interval for a Population Proportion

See how to construct a confidence interval for a population proportion.

List of Sampling Types in Statistics

There are a number of different types of samples in statistics. Learn about each sampling techniques and how to identify them.

How to Calculate the Correlation Coefficient

The correlation coefficient provides a measurement for how well a straight-line fits a set of paired data. See how this number is calculated..

Standard Normal Distribution Table

A table of the standard normal distribution gives us the probability between any two z scores. This probability is the area under a curve.

Statistics Software

Any real-world applications of statistics requires some sort of software, such as Excel. See how to use this software.

How to Use NORM.DIST and NORM.S.DIST in Excel

See how to use Excel functions NORM.DIST and the NORM.S.DIST to do calculations with normal distributions and the standard normal distribution.

Probabilities and Liar's Dice

Liar's dice is a game that involves bidding on dice that are unseen by others. See how to calculate probabilities related to this game.

Time Series Graphs - Construction, Use and Examples

A time series graph displays paired data in which the first coordinate is time. This type of graph shows how a variable's value changes over time.

Probabilities for Rolling Three Dice

See how finding probabilities for rolling three dice is similar to calculating the probabilities associated to rolling two dice.

What is the Probability of Rolling a Yahtzee

Find out the probability of rolling five matching dice during a possible three rolls in the game of a Yahtzee.

Probabilities for Rolling Two Dice

Find out how to calculate probabilities that concern the rolling of two dice.

What Is the Difference Between Combinations and Permutations?

Find out the difference between the closely related (and easily confused) ideas of combinations and permutations.

What Value of Alpha Determines Statistical significance?

Hypothesis tests involve a level of significance, denoted by alpha. One question many students have is,

What Are the Odds of Winning the Lottery?

Find out how the stated odds of winning the lottery are calculated and how likely it is that you will win.

Glossary

This is a collection of defintions of key words and phrases from statistics and probability.

Descriptive Statistics

Descriptive statistics tell us features of a data set, such as its mean, median, mode or standard deviation.

Example of Two Sample T Test and Confidence Interval

See this worked out example of the two sample t test and two sample confidence interval.

Functions with the T-Distribution in Excel

Excel has a variety of programmed functions that involve the t-distribution.

Example of a Permutation Test

See a worked out example of a permutation test concerning the difference of two means.

Expected Value of a Binomial Distribution

See how to prove that the expected value of a binomial distribution is the product of the number of trials by the probability of success.

What Is the DIfference of Two Sets?

See how the difference of two sets can be likened to the subtraction of two numbers.

What Is the Chain Rule in Calculus

There are many formulas used to calculate derviatives. One of the most important of these formulas is called the chain rule.

What Is the Quotient Rule in Calculus

Calculus makes several appearances in mathematical statistics. One rule that is helpful in calculating derivatives is called the quotient rule. This is used to find the derivative of a quotient of two functions.

How to Calculate the Rank Correlation Coefficient

See three examples of how to calculate Spearman's rank correlation coefficient.

How to Calculate a Sample Standard Deviation

See step by step instructions on how to calculate a sample standard deviation along with a worked out example.

What Is Statistics?

This is an introduction to many of the ideas found in statistics.

What Is the Multiplication Rule for Independent Events?

Given two independent events, event A and event B, see how to use the multiplication rule to calculate the probability of the composite event

What Are the Integers

One of the most basic sets of numbers that is used in mathematics is called the set of integers. The set of integers consists of all positive and negative whole numbers, as well as the number zero.

Statistics: The Standard Deviation Formula

The standard deviation is a measure of the spread of our data.

Correlation and Causation in Statistics

There is a saying in statistics that

Correlation Coefficient

The correlation coefficient measures the strength of the relationship between two sets of data.

Z-Score Formula

Any bell curve can be transformed into the standard bell curve by using this formula.

An Introduction to the Bell Curve

Bell curve appear throughout the study of statistics. What are they? Discover what a bell curve is, and some of the surprising places that they show up in real life.

Probabilities for Defending in Game of Risk

The board game Risk involves the rolling of several dice to determine gameplay. Thus probability can be used to analyze the probability of winning a roll for the defender.

What Is the Trimean?

The trimean is a measurement of central tendency that is not as well known as the median, mode or mean. The trimean is easily calculated with the use of the median, first quartile and third quartile.

What Is the Midhinge?

The midhinge is a measure of position that incorporates information from the first and third quartiles.

Standard Deviation

The standard deviation measures how spread out a data set is.

Common Maclaurin Series

See formulas for some of the most common Maclaurin Series

Entrance and Exit Polls

Two types of political polls that ask actual, rather than potential, voters questions are entrance and exit polls. These types of polls have some advantages over other opinion polls, but there are some limitations as well.

Formulas and Tables

This is a collection of common formulas that you will use in statistics.

What Is the Range Rule to Calculate Standard Deviations

A very simple formula called the range rule only involves two arithmetic operations and provides a rough estimate for the standard deviation.

Conditional Probabiliy and Intersections

See how the formula for conditional probability can be rewritten to calculate the probability of the intersection of two events.

What Is the Interquartile Range Rule?

Use two of the numbers from a five number summary to calculate the interquartile range in order to help determine if we have a potential outlier.

Sum of Squares Formula Shortcut

The sum of squares shortcut formula allows us to find the sum of squared deviations from the mean without first calculating the mean.

Inflection Points for the Normal Distribution

See how to use some basic calculus to find the inflection points of the standard normal distribution.

Millions, Billions and Trillions

Numbers such as a billion and trillion are large, but how big are they? See all sorts of different ways to think about these large numbers.

Quantitative Data

Quantitative data consists of numerical data on which mathematical operations can be performed.

Differences in Descriptive and Inferential Statistics

Statistics has two main areas, known as descriptive statistics and inferential statistics. Although these areas use the same procedures, there are a number of differences between them.

What Are De Morgan's Laws?

The complement of a union and the complement of an intersection can be rewritten using De Morgan's Laws.

How to Interpret a Scatterplot

A scatterplot is a type of statistical graph that looks like a jumble of points. Learn what questions you should ask about a scatterplot.

Mathematical Statistics

Mathematical statistics looks at the subject of statistics from a mathematical standpoint. Areas of mathematics such as calculus, probability, linear algebra and real analysis are used to formally prove theorems in statistics.

Calculus

Mathematical statistics requires one to be well versed in the basics. Here are a collection of Calculus topics that are helpful in many areas of mathematics.

How to Prove De Morgan's Laws

See how to prove a result known from set theory that is called De Morgan's law.

Definition of Bimodal in Statistics

Some data sets have two values that tie for the highest frequency.

Binomial Table for n=7, n=8 and n=9

This binomial distribution table for n = 7, 8 and 9 and various probabilities.

Binomial Table for n=10 and n = 11

See a binomial distribution table for n = 10 and 11 and a variety of probabilities.

Binomial Table for n=2, n=3, n=4, n=5 and n=6

The binomial distribution table displays the probabilities for n =2 to 6.

What Is a Convenience Sample?

Convenience samples are easy to obtain, but are quite problematic in statistics.

What Is Paired Data in Statistics?

See why it sometimes makes sense to consider data points as a pair of values.

What Is the Interquartile Range?

The interquartile range (IQR) is the difference between the first and third quartiles. See example calculations for the IQR.

What Is a Scatterplot?

A scatterplot is a type of graph that is useful for working with paired data.

How to Use the NORM.INV Function in Excel

Calculate critical values by using the NORM.INV function in Excel.

Examples of Z-score Calculations

See how to use the formula for Z-scores in these example calculations.

Differences Between Population and Sample Standard Deviations

Learn about the differences between the sample and population standard deviations.

What Is the Probability You Just Inhaled a Part of Lincoln's Last Breath?

Learn how to determine the probability that you just inhaled a molecule from Lincoln's last breath.

Probability and Chuck-a-Luck

Learn more about calculating probabilities for various types of rolls in the game of Chuck-a-Luck.

- By Category

An index of categories in the

Expected Value for Chuck-a-Luck

See how to calculate the expected value for a game called chuck-a-luck.

What Is Skewness in Statistics?

Distributions that are not symmetric are said to be skewed. See how to measure this asymmetry with a statistic called skewness.

What Is a Real Number

The real number system is used throughout mathematics and statistics. Learn about the important and useful properties of these types of numbers.

What Is the Five Number Summary?

The five number summary includes the descriptive statistics of minimum, first quartile, median, third quartile to give us a full picture of our data.

How Are Odds Related to Probability?

Probability and odds both describe the likelihood of an event, but they do so in slightly different ways. Learn what the connection is.

What Is the Factorial (!) in Mathematics and Statistics?

Use the factorial (denoted by !) to compactly write the product of several consecutive positive whole numbers.

What Is a Simple Random Sample

Learn more about simple random samples, the most important type of sample in statistics.

What Is Ecological Correlation?

Learn more about ecological correlation, a relationship observed for a group of individuals.

How to Plot a Least Squares Regression Line in Excel

See how to use Microsoft Excel to plot a least squares regression line on a scatterplot.

What Are Residuals?

Learn about residuals in statistics and how to use these quantities to discern trends in data sets.

What Are the Conditions for Linear Regression

Learn what the conditions must be met in order to perform linear regression and how to check that these conditions are met.

Classroom Activities

Classroom activities, statistics activities, statistics fun, statistics games

Benford's Law Activity

Benford's law is a surprising result in statistics. See how to explore this concept in a classroom activity.

What Is Set Theory

Set theory is fundamental in mathematics. Learn more about this topic and see how it is used in the study of probability.

Christmas Statistics for 2015

Celebrate Christmas with these Christmas related statistics.

What Is a Systematic Sample?

Learn more about how the sampling technique known as systematic sampling can be used to select individuals from a population.

Baseball Statistics for Beginners

Learn more about statistics in baseball and the commonly abbreviations that are used to refer to them.

Student's t Distribution Formula

Student's t distribution distribution is used throughout statistics, here is the formula for all t distributions.

What Is a Standard Deck of Cards and Its Features?

Learn about the details concerning a standard deck of cards, which is a widely used sample in basic probability.

What Is a Weighted Average?

Weighted averages are used in a number of different ways. Learn more about this topic.

Degrees of Freedom for Chi-Square Test for Homogeneity

See how to determine the number of degrees of freedom in a chi square test for homogeneity.

Probability Mass Function Example

See worked out examples of mean and variance calculations for a discrete random variable.

Standard Error for Linear Regression Example

See a worked out example of the calculation of standard error for linear regression.

How to Calculate Expected Counts in Chi-Square Test for Independence

See how to calculate expected counts for a two-way table of categorical variables for use with a chi-square test.

Difference Between Null and Alternative Hypotheses

Learn the differences between the null and alternative hypotheses and how to distinguish between them.

What Is a Converse Error

Learn more about a converse error, which is a logical fallacy resulting from an incorrect understanding of conditional statements.

Worksheet on Combination and Permutations

Practice working with combinations and permutations by using these problems.

Sampling With or Without Replacement

Learn about the differences between replacing and not replacing the objects or individuals when we sample.

The Meaning of Mutually Exclusive in Statistics

Mutually exclusive events have no outcomes in common and have empty intersection. See examples of mutually exclusive events.

Thanksgiving Statistics

This assortment of statistics give details about various aspects of Thanksgiving.

BINOM.DIST in Excel

See how to use the BINOM.DIST function in Excel to speed up calculations with the binomial distribution.

What Are the Simple Conditions for Statistical Inference?

Statistical inference is typically introduced in a simple setting. Learn what the conditions are for this.

Confidence Interval Activity with Candy

See how to use M&M candies in a classroom activity with confidence intervals.

How to Use the Z.TEST Function in Excel

See how to use the Z.TEST function in Excel to conduct statistical tests of significance.

Examples of Hypothesis Tests with Z.TEST in Excel

See several examples of how to perform hypothesis tests in Excel with the Z.TEST function.

What Is Multistage Sampling?

Multistage sampling provides a way to produce a random sample that is different from a simple random sample.

Chi-Square Test for Independence

The Chi-square hypothesis test for independence determines if two categorical variables are independent.

Veterans Day Statistics

Celebrate Veterans Day by seeing some statistics related to veterans.

Chi-Square Activity with Candy

See how you can use M&Ms in a statistics classroom activity. Use a chi-square test to answer the question,

How to Use the AVERAGE Function in Excel

See how to use the AVERAGE function in Excel, as well as its variations.

How to Use the Definition of Derivative

Calculate the derivative of a function by using the definition of a derivative.

What Is the Slope-Intercept Form of a Line?

Learn more about the slope intercept form of the equation of a line.

What Is the Rank Correlation Coefficient?

Spearman's rank correlation coefficient can be used to analyze the agreement between two sets of rankings of the same list.

What Is a Prime Number?

Prime numbers show up throughout mathematics. Learn about prime numbers and how they connect to the topic of probability.

What Is the Difference Between the Variance and Standard Deviation?

The variance and standard deviation are two closely related statistics. See the reason for why the standard deviation is the square root of the variance.

ANOVA (Analysis of Variance) - Statistics Definition

Analysis of Variance or ANOVA is a statistical technique used to compare several populations. Rather than compare parameters two at a time, ANOVA allows us to consider all of the parameters at once.

Linear Regression Example

See a worked out example of a linear regression problem.

The DEVSQ and AVEDEV Functions in Excel

Learn how to use both the DEVSQ and the AVEDEV functions in Excel, both of which are concerned with deviation from the mean.

What Does "If and Only If" Mean?

Learn about the phrase

KURT Function for Kurtosis in Excel

See how to calculate kurtosis in Excel with the KURT function.

Different Methods for Inference About the Mean

Learn which methods are appropriate to construct confidence intervals for a population mean.

What Is the Gamma Function?

The gamma function generalizes the factorial for numbers other than nonnegative whole numbers. Learn more about the details of this function.

Table of Probabilities for the Birthday Problem

This is a table of probabilities for the birthday problem, to see the probability that a pair of people share a birthday in a group of size 1 up to 100.

Standard Normal Distribution Problems

See how to use the standard normal distribution in statistics problems regarding proportions of data sets.

Calculations with the Gamma Function

See sample calculations of how the formula for the gamma function is used to calculate the gamma function for certain values of z.

Worksheet on Combinations and Permutations

Combinations and permutations are two concepts that are easily confused. Check out these problems dealing with these topics.

When Is the Standard Deviation Equal to Zero?

Find out the necessary and sufficient conditions for the sample standard deviation to be equal to zero.

What Are the Differences Between Histograms and Bar Graphs?

Bar graphs and histograms are two types of graphs that appear very similar. Learn what the differences are between these two types of graphs.

Asymptotes in Statistics

The graphs of many functions possess asymptotes. Learn more about this topic and these relate to the study of statistics.

Symmetry in Statistics

Learn what the three types of symmetry are and how they are used in statistics.

What Is Robustness in Statistics?

Certain models and procedures in statistics are described as robust. Learn what this term and the term robustness mean in the field of statistics.

What Is the Range of a Function?

The range of a function is the set of numbers that form the outputs of a function. Find out more about this topic and how it applies to statistics.

What Is the Power of a Statistical Test?

One feature of tests of significance is called power. Find out what the power of a statistical test is.

What Is the Domain of a Function?

Learn more about the domain of a function.

Books, Software & Resources

Statistics books, statistics software, statistics resources

Probability Distributions in Excel

See a listing of probability distributions that are available in Excel.

What Are the Maximum and Minimum

The minimum and maximum are two descriptive statistics that tell us the smallest and largest values in our data set, and are used for other calculations.

What Is an Event in Probability?

Many sets in probability have names. Learn what an event is, and the meaning of an event being a subset of the sample space.

What Are the Contrapositive, Converse and Inverse?

See how the converse, contrapositive and inverse are obtained from a conditional statement by changing the order of its statements and using negations.

What Are the Differences Between Tallies and Counts?

In statistics a tally system is sometimes used to organize data. See how this is done and how a tally differs from a count.

What Is a Two-Way Table

When dealing with two categorical variables, a two-way table is a helpful way to display this data. Find out what is a two-way table.

Proof that the Harmonic Series Diverges

The harmonic series is counterintuitive in that it diverges although its terms seem to decrease quickly enough for the series to diverge. See a proof of why this series diverges.

What Is the P-Series Convergence Test?

There are a number of convergence tests when working with infinite series. A simple one of these tests is used for a p-series.

Example of a Bimodal Data Set

Some data has more than one mode. See this example of a bimodal data set.

Finding the Average: Mean, Median, Mode, and Midrange

The average or center of a data set is an important area of descriptive statistics. The mean, median, mode and midrange are measure the center of the data.

Common Mistakes in Statistics

Be careful not to make these mistakes when you're doing statistics.

How to use the chi-square statistic formula

The chi-square statistic is relatively easy to compute by hand for experiments with a small number of observed and expected counts.

The Formula for Chi-Square Statistic

The formula for the chi-square statistic involves a combination of several basic mathematical operations.

The Number e

The mathematical constant e is found throughout mathematics, including probability and statistics. This number has many important properties.

What Is Benfords Law?

Benford's Law is a surprising result regarding the first or leading digits of quantitative data.

Four Confidence Interval Mistakes in Statistics

Prevent these four mistakes from happening when you are analyzing confidence intervals.

How to Calculate Backgammon Probabilities

See how to calculate probabilities for particular rolls in backgammon.

What Is a Conditional Statement?

Conditional statements, also know as if-then statements, are commonly used in the proof of theorems. Learn more details about this topic in logic.

How to Use the CORREL Function in Excel

See how to use the CORREL function in Excel to calculate the correlation coefficient.

How to Calculate a P-Value with a Chi-Square Table

See how to calculate a p-value with a chi-square table. This skill is useful for hypothesis tests of independence and goodness of fit tests.

What Are the Pythagorean Means?

Learn more about the three Pythagorean means and how what we call

Statistics Formulas

This is a collection of common formulas that you will use in statistics.

Table for Student's T-Statistic for Two Tails

Use a table of values for Student's T-Statistic. This table is organized for two-tailed tests and two-sided confidence intervals.

AVERAGE, GEOMEAN and HARMEAN Functions

The Pythagorean means of arithmetic mean, geometric mean and harmonic mean can all be calculated in Excel using built-in functions.

How to Find Critical Values with a Chi-Square Table

See how to use a chi-square table to look up critical values for either confidence intervals or hypothesis tests.

Why Is Division by Zero Impossible?

Learn the reason why you can't divide by zero, and what this implies in the field of statistics.

How to Generate Random Data Along a Chi-Square Distribution

See how to use Excel to generate random data that is distributed along a chi-square distribution.

How to Generate Random Data Along a T-Distribution

See how to use Excel to generate random data that is distributed along a t-distribution.

How to Generate Normally Distributed Random Data

See how to use Excel to generate random data that is distributed normally.

Example of Confidence Interval for Variance

See a worked out example of how to calculate a confidence interval for a population variance.

How to Construct a Confidence Interval for the Population Variance

Learn the steps and the theory behind the construction of a confidence interval for a population variance.

Chi Square Table

See and use a chi-squared table. The number of degrees of freedom range from 1 to 80.

What Are Sten Scores?

Sten scores are an adaptation of a normal distribution to a ten point scale. Sten is an abbreviation of

Applications of Statistics

You can find statistics just about anywhere. See how different areas of statistics apply to real world problems.

Stanine Score Example

See a worked out example of how to calculate stanine scores for a set of data.

What Is a Stanine?

Stanine scores are a way to use a normal distribution with a nine point scale.

Top Four Mathematical Constants

Some numbers are more important than others. Learn more about the top four mathematical constants.

What Is the Cauchy Distribution

The Cauchy distribution is a pathological example of a probability distribution as it does not have a mean or a variance.

Functions Used in Mathematical Statistics

Mathematical statistics involves the use of several functions that are often confused. Sort out the differences between the definitions of these functions.

Probability Density Function Examples

See several examples of problems that use a probability density function.

Continuous in Probability and Statistics

Data can be classified in a number of different ways. One way to draw a distinction is to ask, is the data discrete or is it continuous. Continuous ideas pop up throughout probability and statistics.

What Is the Barber Paradox

Russell's barber paradox shows what kinds of problems can arise when we are not careful with our definitions. The traditional definition of a set was turned on end by Russell's paradox.

What Is the Prisoner's Dilemma

The prisoner's dilemma is an example of a game where the rational choice is to choose an outcome that is not the most beneficial. It has been studied as an example of a non-zero sum game and has wide applications to other areas.

Statistics Major Courses

There are a variety of different mathematics and statistics courses required for an undergradute degree in the area. What will these courses involve?

How to Derive the Formula for Combinations

See how to derived the formula for combinations based upon some basic counting properties.

Leap Day Statistics

From leap day birthdays and mean solar years, a variety of statistical concepts are related to leap day and leap year.

How to Generate Pascal's Triangle

Pascal's triangle is a triangular array of numbers that has a variety of applications in mathematics. See how to generate this triangle.

What Is Pascal's Triangle?

Pascal's triangle is an interesting arrangement of numbers with connections to many areas of mathematics.

What Are Taylor Series?

Taylor series are an important tool in mathematics. These generalize the concept of a Maclaurin series.

A List of Convergence Tests

See a list of convergence tests for infinite series and learn when to use each of these tests.

The LIPET Strategy for Integration by Parts

LIPET is an acronym that is helpful for when we want to integrate by parts. Each letter in the acronym indicates a particular type of function.

What Is a Plus Four Confidence Interval?

Plus four confidences intervals are a variation on methods used to estimate the value of an unknown population proportion.

Example of Confidence Interval for Difference of Two Means

See a worked out example of how to find a confidence interval for the difference of two means.

What Is an Unbiased Estimator?

An unbiased estimator is a statistic with expected value that matches its corresponding population parameter.

What Are Inner and Outer Fences?

Inner and outer fences are constructed using the interquartile range and can be used to detect the presence of outliers in a set of data.

What Is a Quantile-Quantile Plot?

Quantile-Quantile plots are helpful ways to visually represent how well a proposed probability distribution matches a set of data.

Absolutely and Conditionally Convergent Series

There are two types of convergence. Find out the difference between absolutely and conditionally convergent series.

What Is the St. Petersburg Paradox?

The St. Petersburg paradox is an illustration where simply relying upon the expected value to analyze a situtation will lead us to a conclusion that no rational person would make. Would you play the St. Petersburg lottery?

What Is a Sigma-Field?

A sigma-field is an idea from set theory that is used in some theory behind probability

Moment Generating Function for Exponential Distribution

See how to calculate the moment generating function for a random variable with a exponential distribution.

How to Calculate the Mean and Variance of an Exponential Distribution

See how to use calculus to find the mean and variance of an exponential distribution.

How to Calculate the Median of an Exponential Distribution

See how to use calculus to calculate the median of an exponential distribution.

Mean and Variance of a Uniform Distribution

See how to calculate the mean and variance of a uniform distribution

What Is the Skewness of an Exponential Distribution

Skewness is one important parameter that tells us something about the shape of a probability distribution. See how to calculate this for an exponential distribution.

Probability of a Small Straight in Yahtzee in One Roll

The dice game Yahtzee requires players to roll a small and large straight. Find out how to calculate the probability of rolling a small straight in a single roll.

Probabilities for Risk with a Total of Four Dice

Risk is a board game that involves the rolling of multiple dice. See what the probabilities are for attacking and defending with two dice each.

What is the Symmetric Difference

In set theory the symmetric difference is a construction that is not as well known as the union or intersection. Find out what the symmetric difference is.

Valentines Day Statistics

This is a collection of Valentine's Day statistics.

Probability of a Large Straight in Yahtzee in One Roll

The dice game Yahtzee requires players to roll a small and large straight. Find out how to calculate the probability of rolling a large straight in a single roll.

What Is Simpson's Paradox With Data?

Simpson's paradox, also known as the simpson-yule effect, is an unexpected result that occurs when data is grouped together. A correlation or relationship for all of several groups can be reversed when the data is aggregated.

What Is the Difference Between an Experiment and Observational Study

Learn the subtle difference between experiments and observational studies, both of which are widely used in statistics.

Fourth of July Statistics

Find out more interesting statistics pertaining to U.S. Independence Day for July 4th.

Worksheet for Chebyshev's Inequality - Solutions

Practice using Chebyshev's inequality with this collection of problems. Check your work with the solutions.

Worksheet for Chebyshev's Inequality

Practice using Chebyshev's inequality with this collection of problems.

What Is a Poisson Process?

A Poisson process involves a certain type of discrete random variable. Learn what conditions must be met for this.

How to Calculate the Variance of a Poisson Distribution

See how to use the moment generating function of the Poisson distribution to calculate its variance.

How to Find the Mean of the Poisson Distribution

See how to work with a Poisson distribution and prove what its mean is.

What Is a Type I Improper Integral?

A type I improper integral involves the presence of at least one infinite quantity in the limits of integration. See how to evaluate this type of integral.

How to Make a Histogram in Excel

Histograms are important graphs in the study of statistics. See how to use Excel to make a histogram.

Transformations and Linear Regression

Many times paired data are not linear. See how to use a transformation to work with nonlinear data.

What Is a Probability Mass Function?

Probability mass functions are found in mathematical statistics when working with discrete random variables. See how these functions are used.

What Is the Geometric Distribution?

One type of distribution for a discrete random variable is the geometric distribution. This is used when we count the number of trials necessary to have a single success.

Probabilities for Monopoly

Monopoly is a board game that involves the rolling of a pair of dice. We can use this probabilistic feature of the game to calculate the probabilities of landing on certain spaces during the first two turns of the game.

Probability of Going to Jail in Monopoly

The board game Monopoly features a space known as jail. What is the probability that on your turn you will end up going to jail?

How to Calculate Powerball Odds

See how to calculate the probabilities associated with winning any prize in the Powerball lottery.

A Catalog of Integration Techniques

Many problems in mathematical statistics require us to integrate. Learn more about the various kinds of integration techniques that are available to use.

Worksheet on Markov's Inequality - Solutions

Check your work at applying Markov's inequality can be applied to real world problems.

What Is Markov's Inequality

Markov's inequality is a result from probability, that like Chebyshev's inequality, is a general result applying to a wide variety of distributions.

Uniform in Probability

One common assumption in basic probability problems is that the sample space being used is uniform. This describes what is meant by a uniform probability distribution.

What Is the Difference between the Associative and Commutative Properties?

Associativity and commutativity are two important properties in mathematics. Learn more about these and their differences.

What Is a Matrix?

A matrix is simply a rectangular array of numbers. Learn more about this topic from mathematics.

How to Use RAND and RANDBETWEEN Functions in Excel

Both of the functions RAND and RANDBETWEEN are useful for modeling random phenomena. Learn more about the similarities and difference of these functions.

What Is a Universal Set?

The universal set that is being used in a problem or for an application is the set that we can form other sets from. Learn more about this special set.

Example of Use of Bayes' Theorem

See a worked out example of a probability problem that employs the use of Bayes' Theorem.

How to Use the COMBIN, FACT and PERMUT Functions in Excel

See how to use the COMBIN, FACT and PERMUT functions in Excel to easily calculate the number of combinations, factorials and permutations.

What Is a Relative Frequency Histogram?

A refinement of the idea of a histogram is a relative frequency histogram. See what the features are of this type of statistical graph.

What Are Probability Axioms?

Axioms are statements that we accept as true. The foundations of probability are based upon three axioms. Other theorems in probability can be logically deduced from these three statements.

Example of How to Use the Complement Rule

See a worked out example of how to use the complement rule to solve problems in probability.

How to Prove the Complement Rule in Probability

See how to prove the complement rule in probability, a result that relates the probability of an event to the probability of its complement.

Bunco Probabilities

See how the dice game of Bunco can be analyzed using probability, and learn how likely it is to score a Bunco.

What Is the Birthday Problem?

The birthday problem, also known as the birthday paradox, asks what the probability is that there is at least one shared birthday in a group of people. Find out how to solve this problem in probability.

Basic Arithmetic Functions in Excel

See how to use Excel to perform some calculations involving basic arithmetic.

What Is an Equivalence Relation?

An equivalence relation is an idea from mathematics that generalized the concept of equality. There are three properties that must be checked to show that a relation is an equivalence relation.

Blog

How To Calculate the Margin of Error

Typically poll results are accompanied by a margin of error. How does is this margin of error calculated? It's really just a statement of the sample size and level of confidence being used.

What Is a Degree of Freedom

The number of degrees of freedom is a measure of how many values can vary in a statistical calculation.

What Is the Difference between Alpha and P-Values

P values and alpha are both probabilities that are used in tests of significance. Learn how to differentiate between these two numbers.

An Example of a Hypothesis Test

It is one thing to read about the method hypothesis testing, it is another to see an example worked out. Read an entired worked out example.

Qualitative Data Definition

Qualitative data is distinguished by traits or attributes that are not numeric.

How to Conduct a Hypothesis Test in Statistics

Hypothesis testing is one of the main topics of inferential statistics. Find out how to tell the likelihood of an observed event occurring randomly.

March Madness Bracket Statistics From 1985 to 2012

The NCAA Division I Basketball Championship has a wealth of statistics. Find out how often do #7 seeds beat #10 seeds, as well as other statistical aspects of March Madness.

Parametric and Nonparametric Methods in Statistics

One type of division in statistics is between parametric and nonparametric methods. Find out what the differences are between these.

Frequencies or Relative Frequencies for Histograms

In the construction of a histogram we have an option of using frequencies or relative frequencies for the heights of the bars

Probability of Union of Three or More Sets

The addition rules in probability are typically stated for two sets. These formulas can be generalized for three or more sets.

What Is Statistical Sampling? Populations and Censuses

Statistics approaches large problems by sampling, a technique in which only some of the population are studied. Data about the sample allow us to reach conclusions about the population.

Moments - Definition of Statistics Terms

Moments of a data set are calculated b

What Is a Bar Graph and How to Make One

A bar graph is used when you have qualitative data. Find out how to construct a bar graph from a set of data.

Chebyshev's Inequality (Statistics)

Chebyshev's inequality is a mathematical theorem that states that at least a certain amount of data must fall within a stated number of standard deviations from the mean.

Range in Statistics Formula and Description

The range is a descriptive statistic that gives a very crude indication of how spread out a set of data is.

Calculating a Confidence Interval for a Mean

Read through step by step instructions on how to calculate a two-sided confidence interval for a mean, when the population standard deviation is unknown.

What Is a Probability Distribution?

A probability distribution is a function or rule that assigns probabilities to a random variable. This can be done in a list, a table or even a graph.

What Is the Expected Value

The expected value is a measure of the center of a probability distribution.

Why Say Fail to Reject in Statistics?

One misconception that arises in statistical testing is that some are not careful with their language.

Description of a Pie Chart and a Graph

To graph qualitative data, one helpful way to depict it is to make a pie chart.

What Is Normal Approximation to Binomial Distribution

Binomial probabilities can be difficult to calculate. To help with this, we can approximate some binomial distributions with a bell curve.

Levels of Measurement Worksheet - Solutions

Check your work on identifying levels of measurement with this worksheet.

What Is a Table of Random Digits in Statistics?

A table of random digits can be very helpful in the practice of statistics. See how such a table can be used.

Interpreting Statistics and Political Polls

One of the most common applications of statistics is in the area of political polling. Find out how statistics is used and what questions you should ask when you read the results of an opinion poll.

Roulette - Calculation of Expected Value

The game of roulette is a good example of the application of expected value. We can analyze what the winnings will be if we continually bet on red.

Example of a Sample Variance Calculation

See an example of how the sample variance is calculated.

What Is the Mode and How Is it Used?

The mode is one measure of central tendency. See how this statistic is applied.

What Is the Median in Statistics? - An Overview

The median is one of the three main ways to find the average or center of a data set.

How to Succeed in Statistics and Math Classes

Doing well in a college statistics course can be challenging. Here are some tips to help you do better in class.

Calculate Z-Scores Worksheet - Solutions

Check your work with solutions to problems calculating the z score for a given variable with mean and standard deviation.

Hypothesis Test Example of Calculating Probability

See an example of a hypothesis test, complete with the calculation of the probability of type I and type II errors.

Common Hypothesis Test Mistakes

One area of statistical inference involves hypothesis tests. There are many subtle mistakes that are made in this process. Find out what they are, and what the correct way is to avoid these pitfalls.

What Is the Empty Set

The empty set is a feature of set theory that can seem somewhat strange. It is an example of where nothing can become something.

Example of Bootstrapping

Bootstrapping is a statistical technique that is very computational in nature. This article shows how one such bootstrap resampling would be done to compute a confidence interval about a mean.

Statistics and Probability in the Common Core

Find out more about the details of where the Common Core intersects with statistics and probability.

What Is the Fibonacci Sequence?

The Fibonacci sequence is a famous sequence of numbers that has wide-ranging applications to the real world.

What Is a Random Walk?

A random walk is a concept from probability that has several real world applications. Learn more about what a random walk is.

Probability and Hockey

Probability informed decisions are all around us, even in sports. See how we can use this to decide whether to pull a hockey goalie.

How Much Data Is in the Universe?

Data surround us, most of the time without us realizing it. How much data is there in the known universe?

What Is the Derivative of the Inverse Tangent Function?

The derivative of the inverse tangent function can be found by using the technique of implicit differentiation.

What Is the Gambler's Fallacy

The gambler's fallacy is a mistaken notion about random processes that ignores the independence of certain events. Learn more about this.

A Collection of Integration Formulas

The process of integration involves a collection of rules or formulas that can be used and combined to calculate a wide variety of integrals.

What Is Integration?

Integration is one of the basic operations of calculus. Mastery of the integral is important in areas of mathematical statistics.

What Is the Power Rule In Calculus

Derivatives are used in a number of places in statistics. From just a few rules we can find the derivative of many different kinds of functions. One of these rules is the power rule. The power rule is particularly helpful to find the derivative of a polynomial.

What Is the Product Rule in Calculus

The product rule is a result from calculus that is used to find the derivative of a product of two functions. It is an example of a theorem that does not immediately match with our intuition.

What Is Implicit Differentiation?

Implicit differentiation is a calculus technique that can find the derivative of a relation in which we do not have a function defined explicitly in terms of an independent variable.

What Is the Law of Large Numbers?

The law of large numbers is a result from probability that states in the long run the empirical probability of a random event approaches its theoretical probability.

Probability a Pair of Numbers is Relatively Prime

An interesting intersection between number theory and probability occurs when we ask,

Proof that there are infinitely many prime numbers

There are infinitely many positive integers, and there are infinitely many prime numbers as well. See how to use logic and deductive reasoning to prove that this is true.

What Is the Union

The union is a basic operation in set theory that provides a way to group the elements from two sets into one new set.

Discrete in Probability

Discrete topics in mathematics are concerned with items can be counted with the natural numbers and separated from one another. These ideas make many appearances throughout probability and statistics.

Birthday Problem Activity

This classroom activity uses the probability illustration known as the birthday problem.

The Use of the Sign Test for the Population Median

The sign test is an easy to use nonparametric method for hypothesis tests about the population median. See how to perform one of these tests.

What Is the Probability of Randomly Selecting a Prime Number

If we choose a positive integer at random from the numbers 1 through n, what is the probability of selecting a prime number?

How to Construct a Confidence Interval About the Population Median

The sign test is an important nonparametric tool that can be used to construct a confidence interval about a population median. See how to use this method.

What Final Exam Score Do You Need for a Particular Course Grade?

One type of problem asks for what a missing data value would need to be to obtain a particular mean. This type of thinking is very helpful in knowing what score is needed on an exam for an overall course grade.

Synonyms for Hypothesis Tests

Forming the null and alternative hypotheses involves requires reading information carefully. It is helpful to know various synonyms and key phrases in order to form these hypotheses.

What Is a Two-Tailed Test?

Hypothesis tests are classified by the parts of the underlying distribution that they use. Two-tailed tests are named such because they employ both extremes of a distribution.