Recall the following information from Unit 5.

Continuous vs. Discrete

The difference between the two is whether or not they are obtained by measuring or counting.

Continuous Data

Definition:  Continuous data is data that is obtained by measuring.  You can measure data in different ways, including time and distance. Because there is always a data point that can exist between two data points and there is a possibility of infinite data points, measured data is continuous and organized into intervals.   

Example

Discrete Data

Definition:  Discrete data is data that is obtained by counting.  This type of data was what you focused on in the first half of this course.  Discrete data points, unlike continuous ones, do not have points between points. There are a finite number of possibilities.  

Examples

Calculating Probabilities

We have spent the first half of the course learning about how to calculate probabilities for discrete distributions, by counting the total number of possible ways to arrive at a specific outcome and divide by the total possible ways to arrive at all outcomes.  As discrete variables are measured by counting, discrete probabilities are calculated by counting.

In a discrete probability histogram, each of the outcomes would have its own unique probability, and you can read the probabilities of each outcome off of the graph.

The focus of this unit will be on calculating probabilities for continuous variables.  The probabilities for continuous variables are calculated by summarizing data that shows up on an interval. 

Definition: Similar to a discrete probability histogram, a continuous probability density graph shows us the percentage or probability of different amounts occurring except the histogram is shown on intervals.  We can find the probabilities of specific intervals by finding the area of the graph between those intervals.  What makes the probability density graph very useful is the fact that the area of the bars add to 1.

This is most easily seen in a graph of uniformly distributed data:

Now, often, you will see a probability density graph with intervals that are not a width of 1. See the following graphing on the distribution of teacher ages at an Ontario High School.  

Age is a continuous variable, and for this example, teacher's ages would be the time since they were born. In other words, if a teacher was 35, they would be in the 35-40 interval. If a teacher was 40, they would be in the 40-45 interval.

Now, the area would not work here to give the probability of intervals other than the ones given. The total area under the curve is  or 5 times what it should be.  

This is because the interval size is 5. What we can do is take each of the interval boundaries and call 25 = 0, 30 = 1, 35 = 2 and so on....  

This would be the same as calling 5 years equal to 1 unit.  

This would be illustrated in the following graph, by taking each age, subtracting 25 and then dividing by 5:

Now we can calculate probabilities involving the teachers by using the area of the bars because the total area is: 

You will calculate probabilities for this graph in the quiz in this activity.

A common question for discrete probabilities is whether or not the question includes the number given. For example, if you wanted to find the probability of less than 2 heads flipped on 3 coins, you would have to ask whether or not 2 heads is included.  Asking "What is the probability of 2 or less heads on the flip of 3 coins?" is different from asking "What is the probability of less than 2 heads on the flip of 3 coins?" For the first, we could write:  . For the second we could write  which is the same as .

For continuous probabilities, since the probability of any specific value is 0%, the probability that a number is less than a certain number is the same as the probability that a number is less than or equal to that certain number. In other words,  is the same as . The area of the bars over these intervals are the same.

Challenges with Continuous Variables

Unlike discrete variables like the ones we saw in the first half of this course, continuous variables do not have theoretical probabilities associated with them. Often we can only base our probabilities on past data and create a histogram based on a sample of the data. The length of time it takes you to get to school, for example, can be modelled after collecting a sample of the times it takes you to get to school. There will be some variability(definition:The extent to which numbers are different.) in the data, requiring a measure of the mean and the standard deviation, which we will see in the remainder of the unit. There is also a need to assume that the data will behave in a certain way, by being normally distributed around the mean.  

In Activity 4, you will see some data for Hurricane wind speeds. Now, since it is impossible to calculate the theoretical probability of a Hurricane having a wind speed of greater than a certain amount, the best we can do is to make a mathematical model based on the past data that we have. This may include only a sample of the data, which will need to represent the population. It may also require us to use data that may not be as accurate, because it was calculated by different sources.