Calculate: Sum of Squares

September 29, 2008 by  
Filed under Calc Sum of Squares, How To Calculate

Like range, variance and standard deviation, Sum of Squares (SS for short) is a measure of dispersion. The more inconsistent the scores are (less homogeneous) the larger the dispersion. The more homogenous the scores (alike), the smaller the dispersion.

Using the formula above, let’s go through it, step by step Assume this is the distribution at issue:

X
12
6
5
4
5
10
3

First, each number is squared, and put into another column:

X       X2
12     144
  6       36
  5       25
  4       16
  5       25
10     100
  3         9

 

Second, we sum each column. The sum of the first column is 45. This is called the sum of X.

The sum of the second column is the sum of X-squared. Remember, we squred the scores and then added them up. The sum of the squared-X’s is 355.

Third, we square the sum of X (45 times itself = 2025) and divide it by N.

Since N = 7, we divide 2025 by 7 (which equals 289.29).

Fourth, we recall the sum of the X2 and subtract 240.67 from it. So 355 minus 289.29 = 65.71. The Sum of Squares is 65.71.

 

NOW YOU CHOOSE:
Day 3: Dispersion
A Bit More About Dispersion
Even More About Dispersion
Range
MAD
Sum of Squares
Variance
Standard Deviation
How To Calculate
Range
MAD
Sum of Squares
Variance
Standard Deviation
Formulas For Dispersion
Practice Problems
More Practice Problems
Basic Facts About Dispersion
Vocabulary
Quiz 3
Summary

 

Calculate: z-Scores

September 29, 2008 by  
Filed under How To Calculate, z-scores

A z-score tells you how many standard deviations your score is away from the mean. If z is positive, your score is above the mean. If z is negative, your score is below the mean. If your score is at the mean, z = 0.
Take your score (X) and subtract the mean from it. X with a bar over it is the symbol for the mean, and it’s pronounced bar-x (like the ranch on an old cowboy movie).
Divide the result of your subtraction by the standard deviation.
Takes it. It’s easier than mixing muffins, and there’s no preheating the oven. 🙂
NOW YOU CHOOSE:
Day 4: z-Score
A Bit More About z-Scores
Even More About z-Scores
How To Calculate z-Scores
Practice Problems
Basic Facts About z-Scores
Vocabulary
Formulas For z-Scores
Quiz 4
Summary

Baseball Stats

September 24, 2008 by  
Filed under Featured Articles

 

In the 1950s, it wasn’t unusual for children to be quite conversant about baseball statistics. Part of that ability was tied to the popularity of the game, but part of it must be attributed to bubble gum.

It was a pretty good deal, compared to the chewable cigarettes, for example. You got a big piece of (not terribly flavorful) gum. And with the gum, you got a baseball card. It was a single card for a penny, or a five pack (sometimes 6) for a nickel.

The cards were comparable to playing cards in size and shape. The front of the card had a picture of the player, his name, team affiliation, and the position he played. The back of the card listed the player’s stats: height, weight, bats (left or right), throws (left or right), some major achievements, where he was born, and his birthday.

Then the good stuff. There were numbers for games, AB, runs, hits, 2B, 3B, HR, RBI and B ave. There also were some numbers about fielding (PO, A, and E). For pitchers, there were stats for wins, earned run average (ERA) and strikeouts.

Here’s what the abbreviations mean:

            AB = at bat. It’s the number of times up to bat; but not counting getting hit by a ball, getting to base on balls, and other unusual events.

            2B = a two-base hit, also called a double.

            3B = a three-base hit, also called a triple.

            HR = home runs (a four base hit).

            RBI = runs batted in (number of other runners to cross the plate because of the player’s batting.

            B Ave = batting average (hits divided by at bats).

            PO = put outs (tagging out opposing runners)

            A = assists (helping other fielders)

            E = errors (mistakes)

In the case of baseball cards, statistics can be profitable too. Check your attic. If you have a 1951 Mickey Mantle rooky card (made by Bowman) or the 1952 Mickey Mantle card (made by Topps), I’ll give you a dollar for it. That’s just the kind of guy I am.

The card’s worth a lot of money but I’ll give you a dollar. As I said, that’s just the kind of guy I am.

Not Your Average American City

September 24, 2008 by  
Filed under Featured Articles

 

As you can see, this is not your average American city. For one thing, this is Hong Kong. Minimally, to be the average American city, it has to be in America. 🙂

But Hong Kong can serve as an example of how to best approach the problem of description.

Every city can be described by numbers. There are certain items that could be counted: number of lights shining, height of buildings, number of people living in it, etc. And each of these variables (as opposed to constants) could be used to describe where you live. So, clearly, the first step is to decide what to measure. Are we looking for average population, average rainfall, or average income?

Let’s stick with population for now, and see where it leads. After all, how hard could this be? Let’s just take the population of the US and divide it by the number of cities in the country. That will give us the average city size.

But what exactly is a city? If we count only people who live within the actual boundaries of the city limits, aren’t we underestimating its size? For example, Hong Kong Island has a population around 1.3 million people, but the greater Hong Kong area has a population of nearly 7 million people. In the US, the greater Chicago would be comparable in population, depending on how big you make your “greater area.”

A related issue is that defining a city turns out to be a bit tricky. For example, Maza, North Dakota claims to be a city, and yet boasts a population of five. Another contender, Marineland in Florida, has a population of 7. Just think, if a family of five moves in, they could double your population.

And yet, Framingham, MA, which has a population of about 67,000, claims it is not a city. It says it is the largest town in the US.  Apparently, being a city may depend on your type of local government, on if it has formed itself into a corporation, or just how you feel about it.

Calculating something as simple as an average can have its complications. Numbers look so clear cut and stable. But even descriptive statistics depend a lot on our definitions. So when we see a number, we have to remember that assumptions went into it. And those assumptions are critically important. We can’t separate our numbers from our assumptions.

 

Statistics Should Be Fun

September 24, 2008 by  
Filed under Featured Articles

Statistics should be fun. Well, maybe not as fun as a Ferris wheel but still…

Actually, a Ferris wheel might be a good analogy. It can be big, seem overwhelming, move rapidly and still be significantly enjoyable. So is statistics.

Statistics is a large field of study. It encompasses everything from political polls to rating scales to actuarial tables to business, education and scientific research. Many people use statistics as part of their job.

It also can seem overwhelming. Most people don’t spend a lot of time with formulas, numbers and data tables. Learning the vocabulary alone can be daunting. Then there’s the calculations. Fortunately, we have computers to do the heavy crunching, but getting used to working with numbers can take some doing.

Many classes in statistics move quickly. Even textbooks often only state a formula or theorem once. That’s one of the advantages of this site: we can take our time with the material. You can proceed at your own pace.

And, although you probably won’t become a researcher or professional statistician, I think you will gain an appreciation for how hard it is to get answers. If research was easy, it would already have been done. So it takes skill, creativity and luck to get some partial answers to the most difficult problems we face. But I think you’ll be pleased with your progress and success. And maybe the process will even be enjoyable. We’ll sure try to make it so.