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Welcome to section 4.2, Measures of Central Tendency.
In the last section, we collected groups of numbers called data. We looked at ways to organize it. Often we will summarize the data by selecting some number to represent the whole group, and call it an average.
There are three main types of averages, collectively known as measures of central tendency.
The three averages are mean, median, and mode. In this video, we'll discuss the mean.
The mean is the average you're probably most familiar with. The mean is found by adding up all of your data points and dividing by how many there are.
The formula looks like this. X bar is the symbol for the mean, and x1, x2, and so on are your individual numbers. N is how many numbers you have.
Let's look at an example. Find the mean of the following set of raw data.
As the formula suggests, we need to add up the numbers and then divide by how many there are. In this case, these are the numbers to add up, and we have 5 of them.
You can add these numbers in your head, on paper, or use a calculator. Then divide by 5 to get 4.2.
In general, we'll round off to one more decimal place than the raw data was when it was given to us. Of course, if your homework asks for a certain number of decimal places, use that.
Sometimes we use other symbols for the mean.
If the data represents the entire population, and not just a sample, we use mu for the mean instead of x bar.
Also, there is a shorthand way of writing "add up all these numbers." It's Sigma, the Greek capital S (for sum). So the formula looks like this: to get the mean, add up all the numbers and divide by how many you have.
Here's another example. This time, the data is shown in a frequency distribution, showing the ages of 42 students in an acting class.
We can't add up the individual ages of students because we don't know what they are - we just know that there are 11 students somewhere between 16 and 24.
So how do we find the mean if we can't add up all the numbers?
We'll use the midpoint of each interval to represent the whole group. How do you find the midpoint? Take the two endpoints, add them together, and divide by two.
So the midpoint of the first interval is 16 plus 24, divided by 2 - the mean of the endpoints.
That's 20, and I've gone ahead and done the rest of the midpoints and put them in this extra column here.
Now - we're treating those 11 people as all being "about" 20 years old. To add them up, we need to know how many years this whole group adds up to - that's 11 people at 20 years old each, so 11 times 20.
That's frequency times midpoint.
We do the same thing for each interval, frequency times midpoint, all the way down.
To find the mean, we need to take the total number of years in the entire class, and divide by how many students there are.
Here we have the total years, 1432, and we know there are 42 students in the class.
To find the mean, the formula for grouped data says to add up all of the "frequency times midpoint" - that's what we just did - and divide by N - ours is 42.
We get 34.09524 and so on. We'll round off to one decimal place and get 34.1.
So the mean age of students in this class is 34.1 years old.