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Shapes of Distribution

Shapes of Distribution

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Author: Ryan Backman
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Describe shapes of distributions.

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Tutorial

Shapes of Distributions

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Hi. This tutorial covers the shapes of distributions. So recall that a distribution is a way of displaying or describing how all possible data values of a variable occur or have occurred. When describing a distribution, it is important to comment on the center, spread, and shape of that distribution.

So what we're going to do today is focus on the shapes of distributions. So let's make sure we have a good working definition of what the shape means. So a shape is a way of describing how the data looks when graphed. All right, so let's compare the shapes of the dot plots of the exam scores of three different classes.

So all of these classes took a test. The test was from 0 to 100. And we got distributions for class G, H, and I. Each dot represents one student's score on that particular exam. So if we look at these three distributions, they all seem to be centered in this same area-- around 40 to 60.

They all have about the same variability, so they're all spread out in pretty similar ways. But the big distinction here is the shape. And one way to think about the shape is to almost draw a line that follows the trend of the data. So that distribution might look something like that. This distribution might look something like this. And this distribution might look something like this.

So we can see that they have three different, pretty distinct shapes. We have a big cluster in the middle here for class G, and it tails off to both sides. Here we have a big cluster on the left side, tails off to the right. And here we have a cluster on the right, tails off to the left.

OK, so let's start assigning some of these shapes and some vocabulary terms. So class G has a roughly symmetric distribution. So again, looking at class G, the top class, we're going to say that it's roughly symmetric. And what a symmetric distribution is-- a distribution where mere halves can be made with a line drawn down the middle. And then the mean and median of a symmetric distribution will be the same.

So if we, again, look at class G, if we were to split this in half around the center, it would seem like-- it's not going to be perfect. It's roughly symmetric. But it seems like we would have about two different mirror images. Whereas here, for H or I, if we split those down the middle, those certainly would not be symmetric.

So another way to think about a symmetric distribution-- so if we're dealing with a histogram like this, this would be a perfectly symmetrical distribution-- or symmetric distribution. But generally, with a symmetric distribution, a lot of times-- if you've seen a bell curve, a bell curve is symmetric. Sometimes, though, you can have distributions that are symmetric, but they don't necessarily have to look like this.

So this one would be symmetric, because these two peaks are in about the same spot on the other side of the line. So you can have lots of different symmetric distributions. The key, though, is just to make sure that you have the same thing on each side of this middle line.

If a distribution is not symmetric, sometimes it can be skewed. A skewed distribution is a non-symmetric distribution that has a tail of values-- those usually being those extreme values-- above or below the median. So if we have a tail of values to the right-hand side, those are going to pull the mean toward the tail, which then pulls that above the median.

So if we had a lower tail, that lower tail's going to pull the mean down to the left, which then will move that below the median. And we talk about two different types of skew-- a positive skew, sometimes called a right skew-- so that's what class H was. And class I had a negatively skewed distribution, or a left-skewed distribution.

So if we go back, again, to the classes, this distribution here has a positive skew. This distribution here has a negative skew. Positive because they're more-- the tail values are more towards the positive end. And this one is negative, because the tail goes toward the more negative values.

Again, this is sometimes known as a right skew. This is sometimes known as a left skew-- right versus left. OK, so then just to make sure you have a good definition of a positive skew and a negative skew-- a positively, or right-skewed distribution is a distribution that has the tail of values above or to the right of the median.

So again, positive skew, the tail is toward the right. Negatively skewed, or a left-skewed distribution, is a distribution that has a tail of values below, or to the left of the median. Again, just to reiterate this-- positive skew looks like this. Negative skew looks like this.

Again, it's not skewed toward the large cluster of data. It is always skewed toward the tail. We really need to make sure that-- it's a very common mistake, is that people think the skew is dependent on where the cluster is. It is where the tail is.

One fun way to remember which direction the skew is is you look down at your feet. So if you think about skew, if we want a right skew-- this is my right foot here. Now, if you think about the big toe almost as the cluster, and it's going to trail off toward the pinky toe. So this would be a right skew on the right foot. If we look at the left skew, again, the big toe's the cluster. It's going to tail off toward the pinky toe. So your left foot has a left skew. So just kind of a fun way to remember that.

This is typically what you'd see on a positively skewed distribution. Again, your tail is toward the positive values. Cluster is toward the more negative values. A negatively skewed distribution, or a left skew-- the negative there-- big cluster to the right, tails off to the left.

And the last type of distribution we'll look at is what's called a uniform distribution. A uniform distribution is a distribution where all values have the same frequency. So if we think about a dot plot-- just a simple dot plot-- and let's say that the values from 0 to 4 are possible. If each of those values had a frequency of 2, this is what we call a uniform distribution. If you can basically draw a straight line over the tops of all of those boxes-- a straight, horizontal line-- that's going to be a uniform distribution.

All right, so this tutorial has covered the symmetric distribution, a positively skewed distribution, a negatively skewed distribution, and a uniform distribution. So these are all important shapes that you'll see as you continue to make and look at these distributions. So that has been the tutorial on shapes of distributions Thanks for watching.

Unimodal vs. Bimodal Distribution

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Hi. This tutorial covers unimodal versus bimodal distributions. So let's start by taking a look at two distributions of scores of students from two classes on a particular exam. So we have class A and class B. We have dot plots for each of them. So just take a second to look at the distributions for class A and for class B. All right, we'll come back to those in a second.

The distribution of class A is what we call unimodal or single-peaked. The distribution of class B is bimodal. Let's define each of those, and then we'll take a look back at the distributions again. So unimodal distribution or a single-peaked distribution is a distribution with one clear peak or most frequent value. And a bimodal distribution is a distribution with two clear peaks or modes.

So we can see that for class A in the blue here, we do have just one single peak here at around 80. For class B, we can see here that we have two distinct peaks. We have a peak here at 65 and a peak here at 90.

So thinking now about that bimodal distribution is that sometimes, bimodal distributions result from the combination of two different distributions. So let's consider class B. So perhaps the distribution was made by combining two sections taught by two different teachers of the same class. So maybe due to teacher quality, one section did considerably better than the other.

So again, thinking about class B, maybe there were two separate sections of the same class. So maybe this was the distribution that came from the first section. This was the distribution that came from the second section. And maybe just due to the teacher quality, maybe students in this class did-- or, excuse me, in this section-- did a lot better than students in this section.

And then the last thing that we want to look at is what's called a multi-modal distribution. And a multi-modal distribution is a distribution with more than two peaks or modes. So if we're thinking about a multi-modal distribution, maybe you're going to get something that maybe looks like this, where this would have three distinct peaks or modes.

So this would have three modes. You could certainly have a distribution with four modes or five modes. But anything more than two, we just call that a multi-modal distribution. All right, so that has been the tutorial on unimodal versus bimodal distributions. Thanks for watching.

Terms to Know
Bimodal Distribution

A distribution where there are two distinct values or bins that contain more data than the others, usually separated by a gap.

Distribution

A display of data that shows the values the data take and how often those values occur.

Multimodal Distribution

A distribution where there are many values or bins that contain more data than other nearby bins, usually separated by gaps.

Skewed Distribution

A distribution where the majority of values are on one side of the distribution, and there are only a few values on the other.

Skewed Left (Negatively Skewed) Distribution

A distribution where the majority of values are high, and there only a few low values that form a "tail" to the left of the median.

Skewed Right (Positively Skewed) Distributions

A distribution where the majority of values are low, and there only a few high values that form a "tail" to the right of the median.

Symmetric Distribution

A distribution where the mean and median are the same. It will appear to have a "mirror line" at the median of the distribution.

Uniform Distribution

A distribution where all values are equally likely.

Unimodal/Single-Peaked Distribution

A distribution where one value or bin contains more data than the other values or bins.