A Probability Distribution is a special kind of distribution and Joe Schmuller demonstrates how very easy it is to assign a probability to a coin toss or rolling of a die. Probabilty distributions assigns a probability to every possible outcome of an experiment. Joe reminds you to think of each possible outcome as the value of a random variable.
- [Instructor] Let's look at probability distributions. A probability distribution is a special kind of distribution. It assigns a probability to every possible outcome of an experiment. Think of each outcome as a possible value of a random variable. A probability distribution can be discrete, and so its random variable is a set of possible outcomes that you can count. It's very easy to assign a probability to each one like a coin toss or rolling a die.
And here's a simple example. Tossing a coin. The random variable is coin toss but we can refer to it as X. The possible values are head and tail. We can assign arbitrarily head equals one and tail equals zero. The possible values of the random variable X are one and zero. And here's a picture of the probability distribution for tossing a fair coin. Each possible value, one or zero, has an equal probability of occurrence, and that probability is one half.
This type of probability distribution is called a probability mass function. A probability distribution can be continuous. That means its random variable is on a continuum. The possible outcomes are not countable. The random variable can take on any value between two specified values. Here's an example of what I mean. When we measure a person's height, the ruler's precision limits the accuracy, so probability is assigned to an interval, not to an exact number.
For example, instead of the probability that a person is 69 inches tall, we'd be concerned with the probability that their height is between 68 inches and 70 inches. These kinds of distributions look like this. Possible values are on a continuum, the number of outcomes is uncountable. This type of probability distribution is called a probability density function. Note that probability density is on the Y axis. Probability density is a math concept that enables us to use area under the curve as probability.
A probability density function is often based on a complex equation. Every distribution has a mean and a variance, and a probability distribution is no exception. Calculating the mean and variance is easier for a discrete distribution than for a continuous distribution. For a continuous distribution, we'd have to get into some sophisticated mathematics and we won't do that. The mean of a discrete probability distribution is also called the expected value. To calculate the expected value, you multiply each outcome times its probability, add the products, and the result is the expected value.
Applying all this to tossing a coin, the outcomes are zero or one, each one has a probability equal to .5, so the expected value is (0)(.5) + (1)(.5) which comes out to .5. To calculate the variance of a discrete probability distribution, you subtract the expected value from each outcome and square the differences. Multiply each square difference by its corresponding probability, and the sum of the results is the variance, also labeled as V(x).
It's square root is the standard deviation. Now applying this to tossing a coin, the variance, (0-.5) squared times .5, plus (1-.5) squared times .5, comes out to .25, and the standard deviation is the square root of .25, which is .5. And now, the mean and the variance for rolling a die. Expected value, as you can see, works out to 3.5, and the variance, is 2.92.
The standard deviation is the square root of 2.92, or 1.71. In summation, we talked about probability distributions and how they can be discrete or continuous, and we showed how to calculate the mean and the variance of a discrete distribution.
He explains how to organize and present data and how to draw conclusions from data using Excel's functions, calculations, and charts, as well as the free and powerful Excel Analysis ToolPak. The objective is for the learner to fully understand and apply statistical concepts—not to just blindly use a specific statistical test for a particular type of data set. Joseph uses Excel as a teaching tool to illustrate the concepts and increase understanding, but all you need is a basic understanding of algebra to follow along.
- Understanding data types and variables
- Calculating probability
- Understanding mean, median, and mode
- Calculating variability
- Organizing and graphing distributions
- Sampling distributions
- Making estimations
- Testing hypothesis: mean testing, z- and t-testing, and more
- Analyzing variance
- Performing repeated measure testing
- Understanding correlation and regression