What is Perpendicularity?

"Perpendicular" is the term used in mathematics to describe two lines that intersect at right angles.  I recently introduced this concept in a separate post about the definition of perpendicular lines, but I thought it might be interesting to go into a little bit more detail about this rudimentary and familiar math concept.  For students who are just learning about graphing lines for the first time, this is undoubtedly sufficient, but for those more familiar with the concept, this post might provide additional insight.

I opened above by stating that we are talking about two intersecting lines that form a 90 degree angle with each other, though this definition can and should be expanded.   Though technically correct, there is more to the concept of "perpendicularity," as it's called when talking about this subject.  More correctly, this term applies to not only lines (and line segments), but planes (surfaces, not airplanes!) as well.  At first this may sound advanced, but if you think about it, it is completely obvious. If you stack two books together in a perpendicular arrangement, you essentially are viewing the intersection of two planes.  Lines are drawn on paper, but planes are like the three-dimensional versions that have depth., and you can readily find examples in real life of things that are at 90 degrees to each other.

That leads me to a second point about perpendicular lines and planes.  I have said that they intersect at 90 degrees, which is true again, but more accurately and mathematically you can say that the lines form two "congruent adjacent angles" (Wikipedia link).  This means that if you look at the perpendicular intersection in a T-shape, you see two angles next to each other (adjacent) that are the same (congruent).  And by the rules of geometry, since the angles along a straight line must add up to 180 degrees, this obviously means that the intersection must be composed of 90 degree angles.  Furthermore, if the lines extend through each other, the rules of geometry state that all angles around a point must sum up to 360 degrees, so again we have 90 degrees for each - four right angles.

Perpendicular lines

Perpendicular planes

Here's some additional information that you might find interesting.  The term "perpendicular" itself can be an adjective describing the lines, as in "the perpendicular lines are written in red ink."   Alternatively, it can be the noun, as in "the perpendicular to the ground rises to the sky."   Sorry, those examples aren't very creative!  Also, another word for perpendicular is "orthogonal," which can be used the same way to describe right angle intersections.  "Orthogonality"comes from the Greek for "straight angle" and refers to lines and planes at ninety degree angles, much like "perpendicularity."

I know these math definitions probably aren't going to make solving your graphing problems any easier, but I thought that it was good information to know!  It's a very important but basic concept that is introduced very early in math education, so hopefully I have explained it simply enough even for beginners to grasp.  Please click the +1 button below to share my post, and you can even tweet about my site if you like it!

Definition of Perpendicular Lines

A lot of traffic coming to my site lately has been specifically seeking to learn more about perpendicular lines.  At this point in the new school year, many math classes are just beginning to study graphing, and so perpendicular and parallel lines are often discussed along with other basic graphing concepts, such as slopes and intercepts.  In this post, I would like to give you a definition of perpendicular lines, maybe include a bit of review of some other graphing notes, and go over a few example questions involving them that you are almost guaranteed to encounter in your mathematics courses.  By the end of it, you should be an expert at recognizing and graphing perpendicular lines.

Let's start with the perpendicular lines definition.  It specifically is talking about the relationship between two different lines who intersect at a 90 degree angle.  Two lines that cross at 90 degrees are said to be perpendicular to each other.  A good example of this is the familiar x-y axis.  The y-axis is perfectly up and down with absolutely no slant to the side, whereas the x-axis is perfectly left to right with absolutely no deviation to the vertical.  You can easily see that they form right angles where they meet.  However, it is important to realize that the lines themselves can go in any direction, not strictly up/down and left/right.  The only critical part is that they intersect at 90 degrees.

On a side note, a corollary to this definition is that the four angles created by the intersection of two perpendicular lines are all 90 degrees.  Comparatively, the intersection of two non-perpendicular lines results in the formation of two acute angles (less than 90 degrees) and two obtuse angles (more than 90).  This also demonstrates the geometric law which states that the sum of the angles around a point equals 360 degrees.  This is a useful rule to remember when solving geometric proofs.

So then, now that you understand what this word means and how to visually recognize it on a graph, you may then wish to prove that your two lines do indeed meet the criteria.  How would you even go about that?  How can you determine if two lines are perpendicular?  Well, to do this, you need to know the mathematical equations of the lines… or, more specifically, you need to know the slopes of the two lines.  (Recall that the slope of a line is most simply expressed as "rise over run", which represents the ratio of vertical change to horizontal change.  The slope of the line, often abbreviated by "m", is easily solved by comparing two ordered pairs, and then performing the slope calculation m = y₂-y₁ / x₂-x₁.)  True perpendicular slopes will have the following relationship:

m₁ = -1/m₂

In words, this means that the slope of the first line is equal to the negative inverse slope of the second line.  Looks a bit complicated, but it's not really.  Let's take a look at an example.

Consider the lines described by the equations y = 2x - 1, and y = (-1/2)x + 2.  Are these perpendicular?

This is an example of perpendicular lines.

If you want to get some practice graphing perpendicular lines, you can go ahead and plot these curves.  They are already expressed in standard form, so it is simple to determine the slopes and y-intercepts, and you can also readily generate a table of values to plot points along the lines.  That's a great way to show that you know how to graph the lines, and in the end, you would end up with two lines crossing at 90 degrees.  But that would take an awful lot of time on a test to find a solution that can be found much more quickly and easily.  Just consider the relationship that I explained, and see if it is true in this example.  You can see that the slope of the first line is m = 2, and the slope of the second line is m = -1/2.  This precisely fits the mathematical description of perpendicular lines.  You don't even technically need to graph it out to be able to answer this!  Of course, a wise plan of attack for solving this problem would be to check this relationship first, and THEN graph it out to show that you are correct.  This comes from personal experience.  Always check your work!  ;)

With this information, you should be able to see that all you need to know about lines are their slopes to be able to say whether they are perpendicular or parallel.  Recall, parallel lines have the exact same slope.  The trick when working with this type of question is to realize the the intercept values can be 2 and 3847234, or absolutely anything else at all.  They equation of the line may look completely and extraordinarily different for each, though the only important part of them is their m values.  Keep this in mind, and don't get intimidated by complex and scary-looking equations!

Another type of question might ask you to determine the equation of a line perpendicular to a given line through a specific point.  This takes a bit more work, but it is based on the same concepts.  Let's try a question like this.

Find the perpendicular line to the line y = 2x - 1 that goes through point (4,0).

Here's the approach I would take to solve this.

1. First, recognize that you are given one of the lines' equations, so from that you can easily find its slope. 
2. Second, from the first slope, you can use the perpendicular relationship to determine the slope of the second line.  
3. Third, since you now have the slope and a point that lies on the second line, you can substitute numbers into y = mx + b to solve for b, and then rewrite in in terms of x and y to give the final equation.  

I will leave the actual work for you to try yourself, but the line in this case is the same as above, y = -1/2x + 2.

A third type of question might ask you to determine the perpendicular bisector for a given line segment.  A bisector is a line that perfectly splits another line into two equal pieces, but it can slice through at any angle.  On the other hand, a perpendicular bisector is one the does this at precisely 90 degrees.  If you can first determine what the exact midpoint of your line segment is, you can then apply the approach that I outline above to solve this question as well.

There is one last important point that I would like to make about this topic, and it is about notation.  You are not incorrect to simply state that "line AB is perpendicular to line CD" (or whatever your lines are called), but the shorthand symbol to show this is an upside-down T shape, ⊥.  The keyboard character is called the "up tack", though this term is more applicable to lattice theory, type theory, and logic.  I believe it is more appropriate to simply call it the "perpendicular sign."  So, in this case, you would simply state your answer as AB⊥CD.  That's it.  It's much simpler!

Finally, I thought I would just throw in a bit of trivia that I came across while researching this topic.  Who knows… you might be able to impress your teacher!  The word "perpendicular" originally arose in the late 14th century, and its etymology shows that it came from the Latin word "perpendiculum", which means "plumb line", and "perpendicularis", which means "vertical, as a plumb line".  A plumb line was a simple device which was composed of a small weight on the end of a string, and when holding it up, gravity pulls the weight straight down and the string represents perfectly vertical.  In relation to the ideally perfectly horizontal ground, you can see how they came up with this term.  It's not overly useful information, but you never know where extra trivia might come in handy.

So, with that information, you should now know lots about this subject, and now have no problems graphing perpendicular lines or analyzing and identifying them in either graphs or equations.  There are several different variations to the questions that you may encounter, but if you understand the basics of what it is that defines two perpendicular lines, then you should have no problems in coming up with the appropriate solutions!  Please let me know in the comments below if you would like any further explanation or examples, and don't forget to +1 my post below and follow me on Twitter!  I'm @MathConcepts.  You can even click here to tweet about my post!  Be sure to visit my follow-up post that discusses a bit more of this concept of perpendicularity.

Standard Deviation and Variance - Statistics

I've already discussed several of the ways that you can represent the center of a distribution.  I've also already presented to you the concepts of variability and spread of the data; I explained in my last post percentiles (and by extension, quartiles), and then how these points can be used together with the center and extreme values to provide you with a five-point overview of your distribution (shown in the box and whiskers graphs).  However, a far more common way of describing the spread of your data is by reporting the standard deviation.

The standard deviation provides a different view of the spread of the distribution.  Instead of reporting certain points within the data set (as with the box and whiskers plot), the standard deviation is essentially an average measure of how far away each data point is from the mean of the data set.  Quite literally, it indicates how far the data "deviates" from the mean, or, how close all the data is to the mean. Think about what this is saying, and hopefully you can see how this also appears to be a good way to describe how wide the data is spread.  For any normal, or bell-shaped distribution, a smaller standard deviation means that you will have a narrower peak, since most of the data is close to the mean.  Alternatively, a large SD would mean that the distribution is fatter and more spread out.

Unfortunately, these calculations require rather tedious formulas, if doing the math by hand.  Thankfully, calculators often have shortcuts, but I will leave that with you to explore on your own calculator.

To begin, I first need to make a comment about the notation that I am going to use. Much like what I said in the discussion of sample mean vs population mean, the standard deviation and variance statistical functions can be used when talking about either smaller samples of populations, or entire populations. Being so, they also have differing notations to designate which distribution is being analyzed:

• When talking about a population, standard deviation is denoted by "σ" (sigma). 
• When talking about a sample, standard deviation is denoted by "s". 

To calculate the standard deviation (manually), you must first find the mean, and then you find the variance of the data set, and from there you only have one final, simple operation.  Here is a general definition to mathematically describe the variance and standard deviation: The variance is equal to the average squared deviation from the mean, and the standard deviation is simply the square root of the variance. Sounds simple enough, right? Here is the formula you use to calculate the sample variance, which (logically enough) has the symbol of s²:

Notice how this is similar to how you would calculate an average.  You sum up "n" values, and then divide by the number of values.  In this case, however, there is a very important point to make.  When calculating these statistics for a SAMPLE, you use the term "n-1" instead of "n".  This is essentially a correction factor that is used, to account for the fact that not the entire population was used in the analysis.  When finding a POPULATION variance and standard deviation, it is a truer average because in that case, you do divide by the total number of units.

Since the numerator of this expression is just a large sum, we can also rewrite the formula using the shorthand "sigma" notation that I demonstrated previously in my post about means:

As I said above, if you have done all this hard work to find the variance, the standard deviation is a simple step away at this point.  All you need to do is take the square root of both sides.

or

I know... these expressions and concepts look messy and complicated.  And if you have a large data set, it can get very cumbersome to perform these mathematics by hand.  In those cases, calculators are extremely helpful, or better yet, a spreadsheet program like Excel.  However, as I've said several times, it it very beneficial for you to understand how to do things like this the long way before you learn to take the shortcuts later.  It ensures that you have a solid grasp on the math concepts at work.

In this case, it may help to not think of the formulas as they are written, but rather as what they are trying to do.  In general, you are just adding up the squared distances of each data point from the mean, and then dividing by a whole number of data points, paying attention to whether it is a sample or population.  By simply doing that, you are finding the variance first, which you can then easily use to determine the standard deviation.

But you now are probably asking "just what does the standard deviation tell me?"  Beyond the general definition I gave above, here is a bit more information that you will find useful in understanding these concepts better.  If you have calculated the standard deviation as outlined above, you have solved for s (or, for 1 times s).  One standard deviation away from the mean represents about 68% of the data set, two SD's (or 2 times s) encompass about 95%, and three SD's include almost 99%.  From this, you can now also relate the standard deviations to the general shape of your distribution (i.e. is it wide or narrow?).  I found that Robert Niles' website has quite a helpful discussion on this topic.

So, now hopefully you have a better understanding of these common statistics!  You will use these stats countless times in your mathematics studies, so I really hope that I have done a decent job explaining what is going on, if not at least just how to go about doing these analyses!  Remember, I really appreciate the social shares if you found any values in my posts!  Thanks!

Percentiles, Quartiles, and Measuring Spread - Statistics

I recently explained several math concepts relating to the measurement of the center of data.  I explained the differences and how to obtain the mean, median, and mode for a set of data. While these statistical functions are indispensable when assessing how a data set is centered, they tell you nothing about how that data is spread out, or the variability.  To get this information, percentiles and quartiles are frequently used to supplement what you have determined about the center.

To better understand why knowledge about your data's center isn't sufficient, and why we need to know about the spread of the data as well, consider this point.  If you have a data set A composed of values 36, 38, 40, 42, 44, you can easily determine that the mean (center) of this set is 40.  Now consider a data set B that has the values 0, 20, 40, 60, 80.  This set also has mean of 40.  However, even though these two data sets have the same center, they are obviously very different!  Set A has a much tighter and narrower distribution, whereas set B has a much broader range.  This is where an analysis of the spread of the data is important to better understand your distribution of data.  Having information about both center and variability of your data distribution is one of the simplest and most useful analysis there is.

To start this discussion, I will begin with where we have already been: the median.  I explained that the median is the midpoint of your ordered data set.  So, 50% of your data points are below your median, and 50% is above it.  Now, from here, let me introduce the concept of quartiles first.  Personally, when I think of quartiles, I think of quarters immediately.  If a median is at the point representing 50% of the way through your data, the quartiles represent the points that are quarters through your set, or (roughly) every 25%.  So, the first quartile (called Q1) is at the point 25% of the way along your ordered set of data points (or, it is larger than 25% of the data, but still smaller than 75%).  It is better to say this is the median of the data on the left side of your overall median.  On the other end of your data is the third quartile (Q3), representing the 75% mark of your data, where it is larger than 75% of your group but 25% is greater still.  Similarly, you would find this by finding the median of the data that is to the right of the overall median.  And for those astute enough to catch that I skipped over the second quartile, it is true that the second quartile is the same as the median (M), at the 50% mark.  So, from this, you can hopefully already see that you can get a good idea of the spread of your data with this brief analysis.  (Technically, I refer to these points as 25% and 75% marks, though I recognize that these are approximations.  It is more appropriate to think of them as medians of each half of the data, and if you do the math to say they are the x-th value out of n-values, you will find that they are around 25 or 75, but likely not exactly.  I say this here not to confuse you, but because I'm sure some of my readers will point it out to me!)

In many cases, to get a good overall view of your data, you can report the mean of your data alongside the first and third quartiles.  In addition to this, it is also good practice to report the lowest and highest (extreme) data points.

With this information, you can construct a box plot (also known as a box and whiskers plot) that conveys all of this information visually, at a glance.  Box plots are extremely helpful, especially when you can present data sets side by side for easy visual comparison.  A box plot has a mark to denote the median, inside a box that represents the range from the first to third quartile.  You can alternately think of it as being a box of the quartiles, with one edge on the first quartile, the other edge on the third quartile, and a line inside the box to denote the second quartile.  Extending past the edges are lines that end on the extreme values.  Think of the box are representing the bulk of your data (more technically, the middle 50% of your data), which is why it's thicker, and the lines on the ends represent only a thinner amount of your data, or the tails of your distribution.

Follow along below to see an example of what a box and whisker plot looks like.  This hopefully demonstrates all that I've talked about above.  Note that the center, quartiles, and extremes are all easily seen on the box plot graph, and how easy they are to compare when presented like this.  

Consider the following two data sets, and then follow along with the analysis:
Set A: 32, 5, 8, 12, 2, 6, 2, 35, 32, 15, 18, 25, 22
Set B: 33, 1, 8, 33, 32, 26, 1, 18, 1, 30, 28, 29

The first thing you have to do for this statistical analysis is arrange the data from lowest to highest.

Set A: 2, 2, 5, 6, 8, 12, 15, 18, 22, 25, 32, 35
Set B: 1, 1, 1, 8, 18, 26, 28, 29, 30, 32, 33, 33

Now, you can just count off the important points.  Here, we have a relatively small number of points in each set, so it is easy to find these important marks.  I will colour them below:

Set A: 2, 2, 5, 6, 8, 12, 15, 18, 22, 25, 32
Set B: 1, 1, 1, 8, 18, 26, 28, 29, 32, 33, 33

It is a good method to draw yourself a number line, and then use it to construct your box and whiskers right above it once you've identified your important points. Here is how you could progress through the creation of this plot, along with the final result.

As you can see, you can gain a lot of information about your distributions very quickly!  You can see that Set A has a lower center, and a narrower spread (less variability).  Compare this to Set B, which has a higher center and a much broader range.  Note that the extremes are very similar in each case, so despite having similar ranges, these distributions are quite different.  One is smaller and tighter, the other is higher but broader.  Hopefully that explains quartiles for you!

Now I will briefly extend this concept to percentiles.  Technically, quartiles are only a subset of the percentiles. They represent the 25%, 50%, and 75% (roughly) marks of your set.  Percentiles, on the other hand, can represent any position in your data.  You can talk about the 95th percentile, which indicates the point that is greater than 95% of the rest of your data, or the 7th percentile where it is greater than only 7% of your data (and through induction, smaller than 93% of your data!).  You just have to determine which of your data points represents the percentage you want, and you have it!  If you compare this to how I explained the quartiles above, you can see that it is the exact same concept applied to any point that you want!  The quartiles have a special name simply because they have historically been used the most often.

So, with that information, you hopefully can now understand the importance of measuring the variability of your data in addition to measuring its center.  You can gain a lot of useful information from this simple data analysis, and it provides a great place to start when you are performing any amount of statistical analysis on a distribution of data.

In my next post, I would like to extend my discussion of data variability of data to include one of the most well known statistical functions: the standard deviation!

Sample Mean vs. Population Mean

I did not mention this is my last post that outlined some basic statistical functions related to central tendency, but when studying stats, it is important to understand that you will be dealing with two kinds of means: a sample mean, and a population mean.  Conceptually, they both do the same kind of thing, though their meanings are slightly different.  It is a very good idea to know when to use sample mean vs population mean, and I will try to go over these concepts and their uses in this post.

Mathematically, I already explained how you determine a mean value.  It is what you have likely always known as an average value, and you can very easily find it using the mean formula.  Without actually writing out the equation, you already know that the mean is the sum of all your values, divided by the total number of your values.  This is straightforward and nothing new.  Here is where I am going to make a distinction that you need to be aware of.

In statistics, you deal with populations.  Populations are complete groups of people, of things, of measurements.  As an example, you likely know of the population of the planet Earth.  That refers to all of the people on the planet.  Or, you could have a population of bald eagles in a nesting ground, or a population of Ferrari sports car manufactured in 2011.  Populations refer to the whole group of whatever you are talking about.  However, in many cases, you don't have access to data about the entire population.  You only have access to a subset of that population... a sample of the population.  So, a sample can be considered to be a small part of the population, but is representative of that population as a whole.

A sample could also be looked at as only an estimate of the larger population.  They are frequently sufficient enough to work with, since having data for an entire population could involve a very complicated and long set of data, and the closer your sample size is to your population size, the more accurate this estimate becomes.  This is why people tend to question things that are only based upon a few observations... error is higher when sample size is smaller.  More observations means less error.

So then, with those definitions in mind, you should hopefully be able to understand what is meant by population mean and sample mean.  Literally, a population mean is the average of the entire population, whereas the sample mean is the average of a sample (which represents a larger population).  Of course, since this is mathematics, we have different ways to write the notation for these two statistics concepts.

When we are talking about a population mean, where we have data about all of the subjects or measurements of a given population, we represent that data by the Greek letter mu, which looks like a fancy lower-case u:

This is calculated by summing all of the values in the entire population, and then dividing by the total number of values in that population, which is denoted by a capital N for a population.

On the other hand, when we are dealing with a sample mean (a subset that is representative of a whole population), we denote this function by the aforementioned symbol, x-bar:

As before, we find this by summing all of the values in your set, and then dividing by the total number of values in your set, in this case, the number being denoted by a lower-case n for a sample.

As I mentioned, calculating these values means essentially doing the same thing.  However, in stats, it is wise to pay attention to the group that you are analyzing.  Making a mistake at this point could lead to much larger errors in any further statistical analysis.  Keep in mind that a sample mean is an approximation of a population mean, and that approximation becomes more accurate as the size of your sample (n values) approaches the size of your whole population (N values).