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.

Square Roots - Part I

The concept of square roots often gives students trouble. (In fact, it is the topic most searched for on my site!) However, the initial uncertainty and hesitation with this topic is quite unnecessary. Square roots sound daunting, but they're really a simple concept.

The SQUARE ROOT of a number "x" is some number "y", such that when "y"is multiplied by itself, its product is "x". Sounds confusing... but you will see that it's not. The square root sign looks like this: √25 (usually with a line over the top of the number.)  This is called a radical sign, and I will cover radicals in far more detail later.

Example:
Find the square root of 25.

So, going along with the definition I gave above, let's say x = 25. So then, we want to know y... that is, what number, when it is multiplied by itself, will equal 25. In this example, most people will be able to say immediately "5 times 5 equals 25!" And they will be right. The square root of 25 is 5, because when 5 is multiplied by itself, it gives 25.

Now, I'm sure that most of you will be saying something like "That's easy! But what about when the numbers are big... or weird... like 529?" While most calculators can tell you the square root by the touch of a single button, figuring it out by hand can take a little more guesswork. To find the square root of, say, 529 (by hand), you have to just keep trying to multiply numbers by themselves to reach it. Watch:

10 x 10 = 100..... not big enough
15 x 15 = 225..... still not big enough
20 x 20 = 400..... getting closer
25 x 25 = 625..... too big! so we've narrowed it down to between 20 and 25
22 x 22 = 484
24 x 24 = 576
23 x 23 = 529 BINGO!

It takes a bit of work, but you see how it can be done. Now, having seen this, there are a few things to note.

1) While it can be said that the SQUARE ROOT of a number "x" is some number "y" that, when multiplied by itself, gives "x", the SQUARE of a number is the result you get when you multiply a number by itself. So, the square root of 25 is 5, whereas the square of 5 is 25. It is important to understand these definitions and not to mix them up. Pay attention to what the question is asking!

2) The examples and method I described use PERFECT SQUARES. A perfect square is a number whose square root is an integer (whole number). So, the square root of 25 is 5, but the square root of 24 is.... less than 5.... but more than 4 (since the square of 4 is 16). Therefore, 25 is a perfect square, but 24 is not. We would call the square root of 24 an IRRATIONAL NUMBER.

I'll have a bit more to say about squares and square roots in following posts.  Make sure you check out my Square Roots - Part II (The Irrational Number) and Part III - Factoring Square Roots posts!

Converting Point-Slope Form to Standard Form

I previously described how to obtain the equation of a line, and how to express that in both point-slope form and standard form. While both equations describe the exact same line, sometimes you may be asked to express the line in a specific way, and you need to be able to manipulate and rearrange the provided equation to make it look like the other form. I will show an example of how this can be done.  (Please hit the Like and/or Google +1 button at the bottom if you find this helpful!)

Reminders (refer to the posts linked above for more details)

Point-slope form looks like this:
(y-y1) = m(x-x1), which is the general way of saying y=mx+b

Standard form looks like this:
Ax + By = C

Example: Express the equation y=5x-10 in standard form. State the values for A, B, and C.

Basically, what you want to do is move all the x and y terms over to one side, and move the constants (terms with no variables) over to the other. Combine and simplify where possible. That's all there is to it. "A" will be the term left over in front of x, "B" will be with y, and C will be the value not attached to a variable.

y=5x-10

10=5x-y

So:
5x-y=10
A=5, B=(-1), C=10
(remember the standard form has a "+", so a "-" in your answer implies a coefficient of (-1).

Let's try another one:

Example:Express the equation y=(4/3)x+2 in standard form. State the values for A, B, and C.

This one works the same way, but there is something else that can be done, as I will demonstrate.

y=(4/3)x+2
(-2)=(4/3)x-y
So:
(4/3)x-y=(-2)
A=(4/3), B=(-1), C=(-2)
There is nothing wrong with this answer. It is properly rearranged, and the coefficients have been stated. However, usually it is a good idea to not have fractions (ie. have nothing in the denominator). So, to do this, you work our final answer a bit further, so that all the values are in the numerators.

(4/3)x-y=(-2)
Multiply all terms by 3, to remove it from the denominator of the first term. This gives:
4x-3y=(-6)
A=4, B=(-3), C=(-6)

Again, this answer describes the exact same line as the initial answer without the extra moves, so technically, they are both right. It is just a common convention to keep things in the numerator wherever possible.  Please hit the Like or Google +1 button below if this helped you.  :)

Converting from the Standard Form to the Point-slope form is basically just the reverse. Try it for yourself with these examples!

Special Angles in Trigonometry - Part 2 (Similar Triangles)

This post will hopefully clarify how to work with special triangles when the sides are not the standard lengths (as I described in the original post for Special Angles).

Maybe this would be a good time to describe SIMILAR triangles. Similar triangles are triangles that have different side lengths, but have the same angles. Don't let the fancy name fool you... just think of them as smaller or larger versions of the same triangle. Examine this picture:

If you were to cut out all the triangles, and shrink or enlarge them, you would see that they all would fit on top of each other. This is possible because each triangle has the same angles, despite having different side lengths. When triangles are like this, they are said to be SIMILAR.

Now, to apply this to my previous post explaining the special angles in trigonometry. I explained how to derive the trig functions using the simplest triangles. However, in all likelihood, you will find triangles of different sizes, rather than these same simple triangles. You need only remember the rules of SOHCAHTOA to be able to evaulate the trig functions.

For example:

Take a right angle triangle with an angle of 30 degrees, and you know that the short side is a length of, say, 5. (Try to sketch this out... being able to draw a triangle from a description is important to learn as well!) There are a few strategies you could use here to solve the other 2 sides and last angle.

1) You know that all the angles in a triangle sum up to 180 degrees. So, 180-90-30 = 60!

2) You can apply the rules of SOHCAHTOA to determine either one of both of the unknown sides. In this case, you have the 30 degree angle known, and the short side is opposite this angle. So, you can use the Sine function to determine the length of the hypotenuse (try it! how do you solve Sin(30). Once you have the hypotenuse figured out you can then turn around and use it to to solve the Cosine function... OR you could just use the Theorem of Pythagoras, since it is a right-angle triangle.

3) Remember that you could also use either the Sine Law or Cosine Law in there as well (especially on triangles that are NOT right-angle triangles)... Sine Law works whenever you know an angle and it's opposite side, and then either of an angle or a side to complete the identity. Or the Cosine Law will work when you know 2 sides and the angle between those 2 sides.

Hopefully this quick post addresses some concerns that some of you may have had. Sorry about any confusion or lack of clarity in the first post. As always, please continue to drop me a line if you have any questions, concerns, or topics you would like me to cover!

**EDIT** - My apologies... as pointed out in the comment section, triangles as I have discussed in this post are in fact called SIMILAR, and not CONGRUENT (I have edited the post to reflect this). Similar triangles are triangles with the same angles but can have differing side lengths. Congruent triangles, on the other hand, have the same angles AND sides. This means they look either the exact same, or are a mirror image of the original. Sorry for the confusion, and thanks to the astute reader who picked out my error. :)