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### How to Define a Random Statistical Variable

In statistics, a *random variable* is a characteristic, measurement, or count that changes randomly according to a certain set or pattern. Random variables are usually denoted with capital letters such as [more…]

### Statistics: Discrete and Continuous Random Variables

In statistics, numerical random variables represent counts and measurements. They come in two different flavors: discrete and continuous, depending on the type of outcomes that are possible: [more…]

### How to Find Right-Tail Values and Confidence Intervals Using the *t*-Table

You can use a *t*-table to find right-tail probabilities and *p*-values for hypothesis tests and to find *t**-values (critical values) for a confidence interval involving [more…]

### How to Identify the Notation for the Mean and Variance of a Discrete Random Variable

Two of the most important terms in statistics are mean and variance, and so you need to be able to identify their notations when working with discrete random variables. [more…]

### How to Identify a Random Binomial Variable

The most well-known and loved discrete random variable in statistics is the binomial. *Binomial* means *two names*and is associated with situations involving two outcomes; for example yes/no, or success/failure [more…]

### How to Find the Mean, Variance, and Standard Deviation of a Binomial Distribution

Because the binomial distribution is so commonly used, statisticians went ahead and did all the grunt work to figure out nice, easy formulas for finding its mean, variance, and standard deviation. The [more…]

### Understanding the Statistical Properties of the Normal Distribution

When you understand the properties of the normal distribution, you'll find it easier to interpret statistical data. A continuous random variable *X* has a normal distribution if its values fall into a smooth [more…]

### Using the *Z*-Distribution to Find the Standard Deviation in a Statistical Sample

One very special member of the normal distribution family is called the standard normal distribution, or *Z*-distribution. In statistics, the *Z-distribution* [more…]

### How to Change an *X*-Value to a *Z*-Value

If you have a statistical sample with a normal distribution, you can plug an *x*-value for this distribution into a special equation to find its *z*-value. The [more…]

### How to Find Probabilities for *Z* with the *Z*-Table

You can use the *Z*-table to find a full set of "less-than" probabilities for a wide range of *z*-values. To use the *Z-*table to find probabilities for a statistical sample with a standard normal [more…]

### How to Find Statistical Probabilities in a Normal Distribution

If your statistical sample has a normal distribution (*X*), then you can use the *Z*-table to find the probability that something will occur within a defined set of parameters. For example, you could look [more…]

### How to Find a Percentile for a Normal Distribution

A popular normal distribution problem involves finding percentiles for *X*. That is, you are given the percentage or statistical probability of being at or below a certain [more…]

### How to Tell a *Z*-Distribution from a *t*-Distribution

Although the normal (*Z*-) distribution and *t*-distribution are similar, they look different from each other and are used for different statistical purposes. The normal distribution is that well-known bell-shaped [more…]

### How to Indicate Possible Outcomes for a Discrete Random Variable

A discrete random variable *X* can take on a certain set of possible outcomes, and each of those outcomes has a certain statistical probability of occurring. The notation used for any specific outcome is [more…]

### How to Calculate Standard Deviation in a Statistical Data Set

By far the most common measure of variation for numerical data in statistics is the standard deviation. The *standard deviation* measures how concentrated the data are around the mean; the more concentrated [more…]

### Statistical Distributions: Binomial, Normal, and *t*-Distribution

A statistical distribution is a listing of the possible values of a variable (or intervals of values), and how often (or at what density) they occur. It can take several forms, including binomial, normal [more…]

### How Correlation, Regression, and Two-Way Tables Clarify Statistical Data

One of the most common goals of statistical research is to find links between variables. Using correlation, regression, and two-way tables, you can use data to answer questions like these: [more…]

### How *z*-Values Are Used in Statistics

If a statistical data set has a normal distribution, it is customary to standardize all the data to obtain standard scores known as *z*-values or *z*-scores. The distribution of [more…]

### Finding Standard Deviation in a Statistical Sample

Standard deviation tells you how the values are spread out in a statistical sample. For example, have you heard anyone report that a certain result was found to be “two standard deviations above the mean” [more…]

### How to Find Probabilities for a Sample Mean

In statistics, you can easily find probabilities for a sample mean if it has a normal distribution. Even if it doesn’t have a normal distribution, or the distribution is not known, you can find probabilities [more…]

### How to Calculate a Confidence Interval for a Population Mean with Unknown Standard Deviation and/or Small Sample Size

You can calculate a confidence interval (CI) for the mean, or average, of a population even if the standard deviation is unknown or the sample size is small. When a statistical characteristic that’s being [more…]

### How to Determine the Confidence Interval for a Population Proportion

You can find the confidence interval (CI) for a population proportion to show the statistical probability that a characteristic is likely to occur within the population. [more…]

### How to Interpret a Scatterplot

Scatterplots are useful for interpreting trends in statistical data. Each observation (or point) in a scatterplot has two coordinates; the first corresponds to the first piece of data in the pair [more…]

### How to Set Up a Hypothesis Test: Null versus Alternative

When you set up a hypothesis test to determine the validity of a statistical claim, you need to define both a null hypothesis and an alternative hypothesis. [more…]

### How to Find Binomial Probabilities Using a Statistical Formula

After you identify that a random variable *X* has a binomial distribution, you'll likely want to find probabilities for *X.* The good news is that you don't have to find them from scratch; you get to use established [more…]