Friday, February 26, 2010

Which Measure of Central Tendency to Use? Mode, Mean, or Median?

A concept which may need a bit more explanation is: which average is appropriate for a given question? What is the best measure of central tendency? When would you use a median instead of a mean, or perhaps use a mode instead?

For any data set, you can perform the analysis to come up with a value for each average. However, here are a few basic guidelines to help you choose the most appropriate form of central tendency to describe your data.  If this is helpful, it would be great if you could please hit the +1 button to share it!

1. For a normal, random distribution of data (evenly distributed), the mean is preferred.
2. For a skewed data set, a median is more appropriate than a mean. The skewed data set (ie. extreme data points) will cause the mean value to be much more extreme than the median, and therefore less central.
3. The mode can be used for non-numerical data. Eg. hair colour in a classroom.

Here are a few examples of where each would be appropriate:

1) students' heights in a classroom
2) temperature over a length of time

1) income of a group of people
2) test scores for a group of students

1) finding the most common hair colour in a room
2) finding the most common car in a parking lot

Hopefully these guidelines will help you to determine which is the most appropriate measure of central tendency to report for your data set.

Again, if this was helpful, please share and hit the +1 for me! :)

Thursday, February 25, 2010

Perfect Squares

This post should have been put up when I posted about square roots (here, here, and here), because it is the exact opposite of a square root!

Where a square root of a number "x" is some number "y" that, when multiplied by itself, gives "x", a (perfect) square of a number is the result of multiplying a number by itself. That is to say, the square of "y", by multiplying "y" by "y", is "x". You can also talk about "squaring" a number, which is to find what the square is. (It can be both a noun and a verb.)

For example: the square of 4 is "4 x 4" = 16. Also, if you square 4, you get 16.

If you want to think of a visual representation of it, "what is the square of 5" is essentially the same as asking "what is the area of a square with a length of 5?" (Of course, all sides have equal lengths in a square.) As you can probably figure out already, to find the area of a square (or, in general, a rectangle) you multiply the length by the width. So here, it is obviously 5 x 5 and the area, or the square of 5, is 25.

This concept of perfect squares can also be extended to polynomials. For example, let's look at the following case:

(x+1)*(x+1) is the square of (x+1). It can also be written as (x+1)^2. You can do the visual trick i just described above if you want, using x+1 as the side length.

If you multiply (FOIL) these binomials, you get (x^2 + 2x + 1). As it is equal to our original expression, you can also say that this product is a perfect square, just as you can say that 16 or 25 is a perfect square. To find out what the square root of this is expression is, it is the same as asking what the square root of (x+1)*(x+1). Over time you will see patterns and be able to quickly notice that the square root of (x^2 + 2x + 1) is (x+1).

This shouldn't be too difficult of a concept to understand. I will do a few short examples, but post comments if you require clarification:

Find the square of 12:
12 x 12 = 144

Find the square of 25:
25 x 25 = 625

Find the square of (x-1):
(x-1)*(x-1) = (x-1)^2 = (x^2 - 2x + 1)

Find the square of (2x+3):
(2x+3)*(2x+3) = (2x+3)^2 = (4x^2 + 12x + 9)

It can also be noticed, and should be kept in mind, that squares will always be positive. Try it to see: plus x plus = plus... negative x negative = plus. A plus times a negative is NOT a square! Squares multiply the SAME number (sign and all!).

See my previous post about "completing the square" for some more stuff relevant to this post.

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