Introduction to Statistics - Mode

The third concept of introductory statistics that I will explain here is the mode. (Refer to my previous posts about mean and median.)

The mode is also a measure of average (central tendency), but again, different from how you have likely thought of an average before.

The mode is simply the value in a data set that is represented the most times.

Again, let's refer to my ongoing test grade example:

Example:
The grades received for a test in a math class, composed of 13 students, were:
65, 98, 92, 43, 76, 64, 69, 72, 75, 85, 96, 90, 90

When you rearrange them from lowest to highest, you can quickly identify which value appears the most times:
43, 64, 65, 69, 72, 75, 76, 85, 90, 90, 92, 96, 98

So, for this data set of math scores, the mode was 90... that is, more students scored 90 than any other single grade achieved.

Hopefully this series of posts has helped to explain the calculation of median, mean, and mode. As you will see in your studies, each has its own applications and are useful in their own ways. It is important to realize though that they are all related as forms of average, and they all describe the centeredness of a data set. See my post here for tips on how to choose which of these measures of central tendency to use.

As always, leave a comment if you need clarification or more information on the topics I've posted. As well, as I have done with this series of posts, I will try to address any requests students may have as well in future posts. :)

Introduction to Statistics - Median

The next introductory statistical concept I will discuss is the median. It is similar to the mean in that it is a form of an average, or a measure of central tendency, though most likely not in the terms that you have thought of an average before.

While the mean is the sum of a group of values divided by the number of values, the median is the point at which half of the data points of the set are below it, and half of the data points are above it. In other words, the median is the midpoint of the data set, with 50% of the data points on either side of it. As you will see, while this number CAN be equal to the mean, it does not have to be.

Let's continue with the example given in the post about means.

Example:
The grades received for a test in a math class, composed of 12 students, were:
65, 98, 92, 43, 76, 64, 69, 72, 75, 85, 96, 90
What is the median of this set of data?

The first thing to do is to rearrange the data points from lowest to highest.
43, 64, 65, 69, 72, 75, 76, 85, 90, 92, 96, 98

To determine the median, you simply have to pick out the MIDDLE value of this data set. For data sets with an odd number of values, this is easy. This data set, however, has 12 values, so the median is actually represented by the AVERAGE of the center TWO values. In this case, the middle 2 values are the 6th and 7th values, 75 and 76. Therefore, the median of this data set is the average of 75 and 76, which is 75.5

Let us pretend that one student was sick on test day, and when he took it later on, he scored a 90 on the exam. If we now factor this in, we have a data set of 13 values (an odd number), and so as you can see, the middle point is the 7th point, which is 76.

Also, convince yourself that that addition of this student's score increases the mean to 78%.

The final concept to discuss is the mode, which I will explain next. See my post here for tips on how to choose which of these measures of central tendency to use.

Introduction to Statistics - Mean

As I have received several requests to do so, I am going to put up a couple of posts that explain some basic statistical concepts. The first few will be the mean, median, and mode, the three most common measures of central tendency. Several students find these concepts difficult to grasp at first, but you will see that they are really quite simple. This post will explain the mean.

The mean may sound like a bad thing, but the mean is actually just another word for a concept that you have UNDOUBTEDLY used SEVERAL times up to this point... the AVERAGE! That's right, the mean is what you have always known as the average. (In fact, the average is not the most precise word to use to describe this function... mean is the correct name.) It is remarkable how many times this connection is not immediately presented to the students, and so they feel they are struggling with a new concept, and one that is probably not being taught well! I will briefly go over the calculation of the mean, but please leave a comment if you wish to have any additional details about it.

The arithmetic mean is the sum of a group of values, divided by the number of values used to determine that sum. A familiar example would go something like this:

Example:
The grades received for a test in a math class, composed of 12 students, were:
65, 98, 92, 43, 76, 64, 69, 72, 75, 85, 96, 90
What is the mean grade received on this test?

To compute the mean of this set of data, first determine the sum of the grades... this is 925.
Next, you determine the number of values, which was stated as 12.
Divide the sum by the number of values to give.... 77.083333
So the mean grade received on this test was 77%.

It's that simple. The sum of the values, divided by the number of values. :) Let me know if you want more. Otherwise, I will now move on to median and mode. See my post here for tips on how to choose which of these measures of central tendency to use.