Introduction to Probability and Statistics

After watching this short video, you will be able to define the terms population and sample in the context of Statistics.

This quiz will assess your ability to identify a statistical population.

After watching this video, you will understand why Statisticians often have to settle for a sample of data instead of working with the full set of population data.   

This quiz assesses your knowledge on the reasons for taking a sample.

In this video, you will learn to categorize data as qualitative or quantitative. If quantitative, you will learn to distinguish between data that is discrete and data that is continuous.

In this quiz, you will try to distinguish between different data types.

After viewing this video, you will be able to classify values as either population parameters or sample statistics. 

Assess your understanding of the terms: parameters and statistics.

This set of problems will test your knowledge of section 1.

Descriptive statistics starts with organizing and presenting data in a way that helps us to see the big picture from the small details which are provided by a set of measurements.

This video introduces the concept of relative frequency, a simple and important idea.

This short quiz assesses your ability to calculate relative frequency.

In this video, we will discuss the best practices for creating a histogram.

This quiz will test your knowledge of the best practices for constructing histograms.

This video explains the left-end-point convention for histograms and frequency distributions.

In this video, we will demonstrate the approach to evaluating an expression that uses summation notation.     

In order to properly evaluate an expression that uses summation notation, we must follow the order of operations. We perform any work that needs to be done inside parenthesis first, then we evaluate any exponents that are present, then we multiple or divide as needed, and lastly, we sum the results.

In this video, we will discuss four important properties of the mean.     

For any data set, there is one and only one mean. The mean makes use of all of the available data, and as a result, the mean is very sensitive to even slight differences between data sets. Lastly, the mean can be thought of as the center "of mass" of the data set because the sum of the deviations from the mean is always zero.

In this video, you will learn how to identify when to use each of the different measures of the center discussed: mean, median, and mode.       

When dealing with qualitative data (nonnumeric data), the mode is the best measure of the center. When dealing with data that is quantitative (numeric data), we have a choice between the mean and median. The mean is usually the preferred option; however, when the data set has a small number of extreme values on one end of the distribution, the median is usually a better option than the mean as a measure of the center.

In this video, you will learn how to identify left skewed, right skewed, and symmetric distributions.     

Left-skewed distributions will have a mean that is less than the median and a mode that is greater than the median. In a right-skewed distribution, the mean is greater than the median and the median is greater than the mode. In symmetric distributions, the mean and median are equal.

A discussion of measures of dispersion.

This quiz will assess your understanding of the uses of measures of dispersion (variation).

In this video, we will discuss the range as a measure of dispersion (or variation).

In this quiz, we assess your understanding of the role of the range as a measure of dispersion. We learned that the range is easy to calculate and easy to interpret, but it is best used to describe the dispersion of a small number of measurements. The range is not a good choice for describing the dispersion (or variation) for a large set of measurements because it only considers two values (the maximum and minimum) from any data set.       

In this video, you will learn how to derive the standard deviation from the variance.

The standard deviation is equal to the square root of the variance. The notation for the two quantities reflects this relationship. The standard deviation for the population is denoted by sigma, σ. While, the symbol for the populations variance is sigma-squared, σ². The sample standard deviation is denoted by s, and the sample variance is denoted by s². As the notation indicates, the variance is the square of the standard deviation.

In this video, you will learn how to calculate the sample standard deviation for a set of data.     

In order to calculate the sample standard deviation, we must sum all of the data values, square all of the data values and sum those squared values, and finally enter the two sums into the computational formula for the sample standard deviation.

In this video, you will be introduced to Chebyshev's theorem.

For a population of measurements with a mean of µ and standard deviation of σ, Chebyshev's theorem can be used to determine a lower bound for the percentage of data within an interval that is centered around the mean and that has the following structure: (µ - kσ, µ + kσ). The variable k represents the number of standard deviations added to (and taken away from) the mean to form the interval, and the variable k can be any real number larger than 1.

In this video, you will learn how to use Chebyshev's theorem to calculate the minimum percentage of data inside an interval that has limits that are equidistant from the mean.     

If we want to use Chebyshev's theorem to determine the minimum percentage of data inside a given interval that surrounds the mean symmetrically, we must first determine the number of standard deviations (k) used to construct the interval. Then, we plug our k into the theorem to complete the calculation. The result obtained should be reported with the qualifier, "at least" or "a minimum of,"  because the theorem gives us the minimum percentage of data that the interval contains.    

In this video, you will learn how to use Chebyshev's theorem to calculate the maximum percentage of data outside an interval that has limits that are equidistant from the mean.   

If we want to use Chebyshev's theorem to determine the maximum percentage of data outside a given interval that surrounds the mean symmetrically, we must first determine the number of standard deviations (k) used to construct the interval. Then, we plug our k into the theorem to complete the calculation. The result obtained from the theorem is not our answer. We must subtract this result from 100%. The final answer should be reported with the qualifier, "at most" or "a maximum of."

In this video, you will learn how to use Chebyshev's theorem to create an interval that captures some minimum proportion of a given data set.

In this video, you will learn a useful rule named the empirical rule.

If we can assume that a set of data (measurements) follow a bell-shaped distribution with a mean of µ and standard deviation σ, the following statements are true about the distribution of data.

The interval (µ - σ, µ + σ) contains approximately 68% of the measurements.

The interval (µ - 2σ, µ + 2σ) contains approximately 95% of the measurements.

The interval (µ - 3σ, µ + 3σ) contains approximately 99.7% of the measurements. 

In this video, you will learn how to use empirical rule to create an interval that captures some approximate proportion of a given data set.

If we know a data set follows a bell-shaped distribution, we can use empirical rule to construct intervals that contain either 68%, 95%, or 99.7% of the data set. This is done by using the following structures: (µ - σ, µ + σ), (µ - 2σ, µ + 2σ), or (µ - 3σ, µ + 3σ) respectively.

In this video, you will learn how to use empirical rule to calculate the approximate percentage of data inside a given interval.

To use empirical rule to determine the approximate percentage of data inside a given interval, we must figure out how far the interval limits are from the mean in terms of standard deviations. Once we have determined the number of standard deviations the limits are from the mean, we can use a drawing of the normal curve as a visual aid to help calculate the desired area.   

In this video, you will learn to distinguish between situations that require the use of Chebyshev's rule and situations that allow us to use the empirical rule.

In this quiz, you will distinguish between situations that require the use of Chebyshev's rule and situations that allow us to use the empirical rule.     

In this video, you will learn how to calculate and interpret z-scores as a measure of unusualness.

A z score is calculated for a given measurement by fist subtracting the mean of the data set from the given measurement, and then dividing that result by the standard deviation for the data set. Z scores that are either greater than positive 2 or less than negative 2 are typically considered to be unusual because at least 75% of all measurements belonging to any data set will have z scores between -2 and 2.

In this video, you will learn how to compare the relative standing of two measurements using z-scores.

The relative standing of measurements from different data sets can be compared by converting the measurements into z scores. This allows you to identify which of the compared measurements stands out most relative to its population.

This set of problems will assess your understanding of sections 1 and 2.

In this lecture, you will learn two different ways to express the probability of some event, A. 

This quiz will assess your understanding of the approach to calculating basic probability.

Just because an experiment has k possible outcomes doesn't mean that the probability of each of those outcomes is 1/k. This short article reminds us to be careful about assuming equally likely outcomes for an experiment.

This quiz will assess your ability to determine when it is appropriate to assume equally likely outcomes when calculating a probability.

In this video, we will demonstrate how to calculate the probability of an event from a given set of related data.

This quiz will test your ability to calculate a basic probability.

The law of large numbers says that the proportion of favorable outcomes for an event obtained from a large number of trials should be close to the expected proportion of favorable outcomes for the event. This observed proportion will move closer and closer to the expected proportion as the number of trials increases.

The law of large numbers says that the proportion of favorable outcomes for an event obtained from a large number of trials should be close to the expected proportion of favorable outcomes for the event. This observed proportion will move closer and closer to the expected proportion as the number of trials increases.

The minimum likelihood of an event is zero, and the maximum likelihood of an event is 1. If expressed as a proportion, all probability values must be between 0 and 1 inclusive.

The minimum likelihood of an event is zero, and the maximum likelihood of an event is 1. If expressed as a proportion, all probability values must be between 0 and 1 inclusive.     

This quiz will assess your understanding of what a probability value indicates.

To use the fundamental counting rule, you must complete the following steps:

1) break the process or experiment into steps or individual tasks.

2) determine the number of outcomes or possibilities for each individual step or ways to complete each individual task.

3) multiply all of the numbers obtained above. This is the number of possible outcomes for the original task (experiment or process).  

In this quiz, your ability to use the fundamental counting rule to determine the number of possible outcomes for a given experiment will be assessed.

In this video, you will learn how a combination is defined and the factorial operation will be introduced. Also, you will learn how to calculate the number of possible combinations of size r that can be drawn from a set of size n.   

A combination is a subset of items drawn from a set without replacement. The order of the items in the subset does not matter. To calculate the number of possible combinations of size r that can be drawn from a set of size n, we must know the combinations formula.

This video demonstrates how to apply the combinations formula to count combinations.

In this quiz, your ability to use the combinations formula will be assessed.     

When attempting to solve a counting problem, answer the following three questions:

·        Are we selecting a subset of r items from a set of n items?

·        Is it true that the order of the r items in the subset does not matter?

·        Are the selections made without replacement, which means repetition of items is not allowed?

 If the answer to all three of these questions is yes, you should use the combinations formula to solve the problem. If you cannot answer yes to all three of the questions, you can use fundamental counting rule to solve the problem.

In this quiz, you will demonstrate your ability to determine when a counting problem requires the use of the combinations formula.        

This short article discusses the difference between P(A ∪ B) and P(A ∩ B).

In this quiz, you will contrast the ideas of union and intersection.

In this video, you will learn to use the addition rule of probability to calculate the probability that, given a pair of events A and B, either event A or B occurs as the outcome of an experiment.

In this quiz, you will demonstrate your ability to use the addition rule of probability to calculate the probability that, given a pair of events A and B, either event A or B occurs as the outcome of an experiment. 

This article compares and contrasts mutually exclusive events and independent events.   

In this quiz, your understanding of the terms mutually exclusive and independent events will be assessed.

In this video, you will learn to use the addition rule of probability to calculate the probability that, given a pair of mutually exclusive events A and B, either event A or B occurs.

In this quiz, you will demonstrate your ability to use the addition rule of probability to calculate the probability that, given a pair of mutually exclusive events A and B, either event A or B occurs.      

In this article, we will discuss interpreting conditional probability.

In this quiz, your ability to interpret a conditional probability will be assessed.

In this video, you will learn to calculate a conditional probability without the use of a contingency table. To calculate the probability that an event (A) occurs, given that some other event (B) has occurred, we divide the probability that both of the events occur (A and B) together by the probability that the given event (B) occurs.          

In this quiz, your ability to calculate a conditional probability without the use of a contingency table will be assessed.     

In this video, you will learn to calculate a conditional probability with the help of a contingency table.        

In this quiz, you will demonstrate your ability to use the conditional rule of probability with the aid of a contingency table.

In this video, we demonstrate the use of the multiplication rule of probability for independent events.

In this quiz, you will demonstrate your ability to use the multiplication rule for independent events.

In this video, we demonstrate the use of the multiplication rule of probability for dependent events.

In this quiz, you will demonstrate your ability to use the multiplication rule for dependent events.

In this video, we demonstrate the use of complements to find a probability.

In this quiz, you will demonstrate your understanding of the approach to calculating the probability of at least one.

In this video, we find the probability of an event by first calculating its complement and then subtracting the result from one.

In this quiz, you will demonstrate your ability to calculate the probability of at least one.