Introduction

normal distribution vs. paranormal distribution meme

We can describe how data is spread out by talking about its skewness. This helps us to understand where the majority of the data is or is not. There are four types of skewness (or non-skewness) we will talk about here! To begin, we will talk about skewness in reference to histograms, then we will also talk about box plots!

#1: Normally distributed (or "symmetrical" or "the bell curve"):

In a normal distribution, the highest frequency is in the middle (or at least very close to the middle) and then the frequences towards the left and right decrease. Another way to think of this is that the frequencies in the "tails" of the graph get lower and lower as we go away from the center of the graph. (We will talk about this type of distribution for the majority of the course, so get ready!)

example of normally distributed histogram

In these types of distributions, you can verify that the mean and the median are the same (or are at least very close to one another). You can review how to estimate the mean and median of the histogram/distribution above if you need to!

A lot of real-life things we measure are naturally normally distributed! Things like weight, height, and age are typically normally distributed in a true random sample!

Here is another visual example of a normal distribution, showing the ages that players in the National Football League retired:

Example of a normal distribution, showing the ages that players in the National Football League retired

#2: Uniform distribution:

We say that a distribution is uniform (or uniformly distributed), the data values are perfectly spread out evenly among bars/intervals. A lot of real-life situations are not exactly uniform, but many of them we can consider roughly uniform, like the example below. In this case, the mean and median would be about equal as well, but the difference between a uniform distribution and a normal distribution is that the frequency in the "tails" of the graph, so towards the edges, are about just as high as they are in the middle of the graph.

#3: Skewed Left (or Left-Skewed):

A data set is skewed left if there are values that are pulling the left tail out further than the right. Notice in the histogram below, the left side of the histogram (the left "tail", if you will) is dragged out further than the right "tail" of the histogram. This indicates it is skewed left. Also, when we have a distribution that is skewed left, those lower values pull the mean to be lower as well, which causes the mean to be less than the median (which will be less than the mode).

left skewed histogram

#4: Skewed right (or Right-Skewed):

A data set is skewed right if there are values that are pulling the right tail out further than the right. Notice in the histogram below, the right side of the histogram (the right"tail", if you will) is dragged out further than the left "tail" of the histogram. This indicates it is skewed right. Also, when we have a distribution that is skewed left, those lower values pull the mean to be higher as well, which causes the mean to be greater than the median (which will be greater than the mode).

right skewed histogram

Comparison of Left- and Right- Skewness:

Below are two histograms showing another example of left- versus right- skewness of a histogram/data set.

comparing left and right skewness

To Summarize:

If the mean, median, and mode are all very close to one another (or equal to one another), the data is normally distributed (unless you see that it is uniformly distributed, but then there is likely no mode!)
If we have that our mean>median>mode (if the mean is greater than the median which is greater than the mode), then we are looking at a right skewed data set! (I remember this by writing those three words in that order and then seeing where the arrows are pointed! Here, they are pointed right, so it is right-skewed!)
If we have that our mean<median<mode (if the mean is less than the median which is less than the mode), then we are looking at a left skewed data set! (Using my "trick", those arrows are pointing left when we write the words in that order, so it is left-skewed!)

How does this relate to box plots?

If we have a symmetric or normal distribution, our corresponding box plot (which is plotting our 5-number summary) will also look symmetric! Also, this reinforces the idea that in this case, the mean and median are equal (or at least very close to one another).

If we have a right-skewed distribution, that right tail that is dragging out you can relate to the highest 25% or so of the data (perhaps?). So, if we think of it that way, the right "whisker" of our box plot will be longer than the left whisker! Again, that implies that the mean is greater than the median, because even though there aren't as many of those larger values, they are pulling the mean more in that direction than they pull the median.

If we have a left-skewed distribution, that left tail that is dragging out you can relate to the lowest 25% or so of the data (perhaps?). So, if we think of it that way, the left "whisker" of our box plot will be longer than the right whisker! Again, that implies that the mean is less than the median, because even though there aren't as many of those smaller values, they are pulling the mean more in that direction than they pull the median.

Skewness related to histograms and box plots

What You Will Do

Watch this video from Khan Academy talking about skewness of distributions (think of a density curve for now as the outline of the tops of our histograms):

Watch this video from The Organic Chemistry Tutor about skewness in box plots and data distributions:

Read about mean, median, and mode in our textbook:

2.6 Skewness and the Mean, Median, and Mode

What's Next

You will take a Progress Check about the material so far.