Mastering Statistics

Types of Measurements and Distributions

If we want to do Stats, we have top find a way of measuring observations that other people will understand. No one wants to read a scientific paper about how many people are "skanky" unless they know what your exact criterion for calling people skanky is. There are 4 types of classifying observations that scientists like:

Nominal- A nominal value is a name. It has no numeric value (though it may be a number). For instance, if you have two personality types, named "introvert" and "extrovert", and we define those personality tests in such a way that every person we're studying is either one or the other. Those types are nominal values. You can tell it's nominal because you can change the name, and it wouldn't effect your study. You could call it "shy" and "talkative" or "type A" and "type B" or "pickle" and "monkey" and the stats would be exactly the same.

Ordinal- This is a measure of how an observation ranks among other observations. An ordinal measurement has no meaning on it's own, only compared to the other scores. Like in a race: you know you came in fifth, but that doesn't tell you how many miles-per-hour you ran, just that you ran faster than the fourth person and slower than the third. You can tell that a measurement is ordinal because the number looses its meaning if you take it outside of the current sample. For example, you know that someone scored third in the Miss Sideshow Freak Pageant - does that tell you anything about their "value" compared with Miss America contestants? No, because it's an ordinal measurement.

Interval- This is a measure of value on an actual "scale". It's called interval because equal intervals have equal value. Date is an interval scale, and that means that the difference between 1400AD and 1405AD (a 5 year interval) is the same as the difference between 2000AD and 2005AD. In that way, inetrval is liek a ratio (see below). Yet intervals do not have a "true zero" that represents zero value. 0 AD doesn't mean zero time, it's just another point on the scale. 1000AD is not twice as much time as 500AD.

Ratio- Like interval, a ratio has equal intervals. Unlike an interval measurement, a ratio does have a true zero that represents zero value. Height is a ratio value. 0 ft. means no distance. And thus a 10 ft. tall person is twice as tall as a 5 ft. tall person.

So here's the tests: Can you replace the labels that you call things, and still have basically the same data? If so, then it's nominal. Do the labels rank the subject in comparison to your other subjects, but not in comparison to anything else? If so, then it's ordinal. Do 5 points of difference mean the same thing on any point on the scale? If so, then it's either Interval or Ratio. If 40 on the scale has twice the value of 20, then it's ratio, and if not then it's interval.

Whatever you're trying to measure is the variable. The variable is a value that stays the same regardless of what system you use to measure it. I could rank you're height by rating you as tall or short (nominal) or by ranking you against other people (ordinal) or by actually counting the number of feet (ratio) but you're actual real height, which is the real variable, stays the same.

Variables can be continuous, which means there are infinite possibilities between any two points on the scale. Between five feet and six feet, there are an infinite number of possibilities. Or, variables can be discreet, which means there are a limited number of possibilies. When you're variable is the number of baloons, then five baloons and ten baloons, there are only a few possibilities.

Test: Which of these is an interval measurement?

A distribution is just a way of writing out a bunch of statistics. It's a list, like an inventory. There are a bunch of types of distributions, but the most common one is a "frequency" distribution. This type lists every possibility, and shows how many of each type there are. Let's say you own a store, and you need to do an inventory. This might be your distribution for cans of soup:

type of soup	number of cans
Mushroom	15
Chicken Noodle	11
Clam Chowder	4

The cans of soup are a discreet variable (also, a nominal variable).

How about you're doing an inventory on christmas trees that you have for sale:

size of tree	number of trees
0-3 ft.	17
3-6 ft.	123
7-9 ft.	87

You'll notice that because height of tree is a continuous value (also a ratio), we can't list every possible tree height, so we create arbitrary ranges.

f is a symbol that means frequency of occurrence, which means "how many times have we seen it" or "how many of these do we have". So, if you want to make a smartass distribution, instead of saying "number of cans" say "f". Another thing you can do to fancy up your table is to put a relative distribution. Relative just means, compared to the whole. In a relative distribution, instead of just counting how many times each thing occurs, we coudl what percentage of the total it is. Like this:

type of soup	f	relative f
Mushroom	15	50%
Chicken Noodle	11	36.66%
Clam Chowder	4	13.33%

Sometimes it is useful to have a number which represents what the majority of the scores are doing. When a yokel sees a basketball team and says "these guys are pretty damn tall!" that's a measure of central tendency. The measures of central tendency scientists like are mean, median and mode.

Mean is just another word for average. Add up all the scores and divide that number by n (the number of scores). If you want to find the average height in your family, add up the height of everyone in your family and divide by the number of people in your family. This is the most commonly used measure of central tendency because it is effected by every member of the population. For exactly the same reason, though, it can screw you up. Let's say most of the people in your family are aroudn 5 ft. tall, but you have a sister who is 20 ft. tall. This one very odd "outlier" would "skew" the measurement towards tallness.

Median is the middle score if the scores were put in order of value. If there are an odd number of scores, the score is the (n/2+.5)th score. If there are an even number of scores, the median is the halfway value between the n/2th and n/2+1th scores. Median scores are not quite as accurate as a median, but they are not effected by extreme scores. If you're using this method, you would line your whole family up in order of height, pick the center person, measure that person, and that would be your median.

Mode is the value that is found most frequently in the distribution. If more than one value is tied for the highest frequency then the scores are bimodal. If everyone in your family, except your 20 ft. sister, is between 5-6 ft., then that's the mode. No one uses mode, the only reason we mention it is because statisticians like saying "mean, median, mode."

Test: There are 10 people in my family. The mean number of monkeys owned by each person is 50. The mode is 0 monkeys and the median is 0 monkeys. What does that tell you about my family?

Measures of Variability

Once we've established the central tendency of a group of scores, we've got to wonder how much our scores vary from that central tendency. The average person in a group might be 5'9", but are the rest of the people approximately the same or are they all midgets and basketball players? The simplest and most obvious measure of variance is the range: the distance between the lowest and highest scores. However, this only measures the most varied members of the group.

Statisticians prefer the sum of squares. To find the sum of squares, you figure out how far everyone is from the median. The average person in your family is 5'9. One person might be 1 ft. less then that, another might be 2 inches less, another might be 6 inches more, another might be 14 ft. more. Now, take each of those differences and square them (multiply them by themselves). One thing you'll notice is that the negative numbers become positive (a negative squared is positive). Now, add everything up and you have the sum of squares. In mathematical terms:

The means sum of, means the mean of the scores and, of course, SSx is the sum of squares. Of course, this is a total of the variance, as the n gets bigger, it gets bigger. If you are measuring the sum of squares for a group of four people, it might be 8 ft. altogether, but if you are measuring the sum of squares for a hundred people there might be 90 ft. worth of variance.

So, take your sum of squares and divide by the number of scores you're measuring. Then, since we had squared everything earlier, now we square root the whole thing. You're basically measuring "what is the average of the squared difference between each member of this group and the average of the whole group." What you get is the standard deviation, symbolized by s. Here's the mathematical formula for finding s:

Test: I'm testing vitamin supplements and the average change to the lifespan (in years) of people who take it, which one do you want?

Many statistical applications use proportions. Proportions are very simple, they're just like percentages except without the hundred. 0% is equal to 0 proportion, 50% is equal to .5, 100% is equal to 1, 250% is equal to 2.5, etc. We use proportions because they are easier to work with mathematically. If .37 of a population of 500 likes Jell-O, then we can say that .37x500, or 185 people, like Jell-O. If .37 of the population likes Jell-O, then we can also say that 1 -.37 or .63 of people don't like Jell-O.

Test: Out of ten people, 2 are allergic to monkeys. Using proportions, which is true?

Probability

Statisticians are very interested in what is the probability a certain thing will happen, for reasons we will go in to later. So there is a language in statistics for defining probabilities. All probabilities range between 0 and 1. 0 probability means the thing will never happen, .5 probability means it will happen half of the time and 1 means it will always happen. For a coin, the probability of getting heads is .5. So, if you flip a coin 20 times, a statistician's best bet is that you'll get 10 heads (20x.5).

To figure out a probability, figure out all the possible outcomes (heads or tails) and find out how many of those outcomes will meet your criterion for success (in this case, only heads gives us a success) and then divide the number of good outcomes from the number of possible outcomes: 1/2 = .5. Let's look at another example, on a normal dice with the possible values of 1, 2, 3, 4, 5 and 6, what are the chances of getting an even number? There are 3 outcomes that fit our criterion, divided by 6 possible, is .5. Or what if you were asked to flip 2 coins, what is the possibility of getting 2 heads? The possible outcomes are Head-Head, Head-Tail, Tail-Tail and Tail-Head. That's one good outcome out of four possible, or .25. The same answer holds true if you were asked the chance of flipping a coin and getting a head and then flipping it again and getting another head.

In probability, there's a big difference between AND and OR. Let's say you're flipping 3 coins. I could say "you have to get them ALL heads, or it's in to the pit of doom" or my less-evil twin could say "you have to get one of them heads or it's in to the pit of doom." What I'm really saying is that you have to get "a head AND a head AND a head" but what my less-evil twin is syaing is that I have to get "a head OR a head OR a head."

When you're figuring out probability, whenever there is an AND, you multiply. The probability of getting struck by lightning is, let's say, .000005, while the probability of winning the lottery is .000002. What is the chance of both happening to you? In other words, what is the chance that you will get hit by lightning AND win the lottery? Well, AND means multiply, so multiply .000005 by .000002, and you'll get .00000000001.

On the other hand, what if the question was OR? There are two ways to deal with ORs. You have to figure out whether you're dealing with an exclusive OR or a non-exclusive OR. In an exclusive OR, when one condition happens, it means the other condition can't happen. These are the simplest because you can simply add probabilities. In a non-exclusive OR, one condition can happen, or another condition could happen, or they could both happen. These are slightly more complicated to deal with.

Here is an exclusive OR: If you pick one marble out of a bag containing 2 red, 1 blue and 1 black marble, what are the chances the marble will be either blue OR black? We know that the chance of getting a blue marble is .25 and the chance of getting a black marble is .25 so we can simply add the two and get .5.

Here is a non-exclusive OR: If you flip a coin three times, what is the chance that you'll get at least one head? To figure out the answer, we've got to find the hidden AND in that question. Flip the question around - instead of asking "what are the chances you won't get one head" ask "what are the chances you won't get any heads" - or, in other words, "what are the chances you will get all tails." - or "what are the chances you will get tails AND tails AND tails." Well, we know how to figure that you, it's .45 times .5 times .5, or 0.125. That's the chance we'll fail to get at least one head. But what is the chance we'll succeed? Well, it would be 1 minus 0.125, or 0.875.

Don't be fooled in to thinking that probability is reality. There is no such thing as a .5 chance that can be measured. Probability only exists in an imaginary world where you can do something an infinite number of times and figure out what proportion of those times a particular thing happened. Probability is a fiction, but it is an incredibly useful fiction.

Test: I have developed three automated gambling machines. You put in money, and it either gives back money or it gives you nothing. Which machine do you put your money in?

Test: You're going to flip a coin three times. Which formula will tell you the probability of getting either 3 heads in a row or 3 tails in a row?

Sampling

As I mentioned before, the strong point of statistics is that when we make educated guesses we know the chances our guesses are right. This is in part because we know how well the information we do know represents the information we don't know. If a space alien wants to know about college students and grabs one, how well will studying that one college student tell him about all college students? What are the chances that the college student he chose is so abnormal that what the alien learns about college students in general will be completely wrong? What if he lands his spaceship in the quad, opens the doors and says "anyone want to come in and get probed?" Will the people who come in to the ship accurately represent college students as a whole? What if he landed at Harvard or Bakersfeild Junior College, will those colleges give him a good representation of college students worldwide? These are the types of questions we have to worry about when we do sampling. Specifically, we have to worry about how many are selected ad how they are selected.

We sample a population because it is inconvenient to make observations about the entire population. The problem is that there's always a chance of picking a bunch of weirdoes that do not accurately represent the central tendency of the population. If you had a bag of red and black marbles and you reached in and the four you pulled out were all red, you might assume that all the marbles in the bag were red, which would be an error. Scientists are very afraid of this kind of error, they would rather admit to not knowing what the marbles in the bag are than make an assumption about that bag that turns out to be wrong. If we increase the number that we take out, it decreases the chance of us getting all red marbles. As we saw in probability, the chance of getting 10 red marbles from a bag of half red and half black marbles is .5 to the 10th power, or 0.00097652.

The more variance we have in a population, the larger a sample we need in order to be accurate. For instance: a space alien who took ten random college students might be able to make some pretty valid guesses as to what all college students are like. But let's say the space alien took 10 animals at random, would that really be a big enough sample for the alien to say something about all life on this planet? Chances are, they would all be bugs and the alien would decide that the planet has no intelligent life.

Correlation and Prediction

Are fat people really jollier than skinny people? Statisticians have nothing better to do than sit around all day wondering about silly questions like this and finding the answers. What a statistician would go is get a random sample of people, measure their weight and take an objective measure of their "joliness." Certainly, since this is people we're talking about, we would expect a lot of variation. What we want to know is: do fat people tend to be jollier, and how strong is that tendancy? What we do is run our data through a pearson's coefficient of correlation test, that test gives us a number called "r" which is the answer to our question. r is a proportion, the value of which can range from -1 to +1. Let's examine what each score would mean:

r=0: If we got an r of 0, then we woudl know that joliness and weight have absolutely nothing to do with each other. We can't tell anything about a person's Joliness by knowing their weight and visa versa. Whether they're skinny or fat, our best bet about a person's joliness is "average." If you drew a graph of weight and joliness, it would look like this:

r=1: Joliness can predict weight without any error and visa versa. If you drew a graph of weight and joliness it would look like this: Of course, one pound is probably not equal to one joliness point, but it doesn't need to be because we know the standard deviation for weight and joliness. If someone is 1 standard deviation above the average in weight, then we can be sure that they're one standard deviation above the mean in joliness. As long as we're comfortable with Z scores, we can predict any one score from any other score. If we know someone's joliness and nothing else about them, we can predict their weight and be 100% certain of being right.

r=-1: This is exactly the opposite. More weight equals less joliness and visa versa. If a person is 1 standard deviation below the mean in weight they'll be 1 standard deviation above the mean in Joliness. A correlation of -1 has the same strength of correlation as +1, meaning we can predict Joliness from Weight as exactly with a +1 as with -1. A graph of scores would look like this:

r=+.62: Joliness and Weight are somewhat related. If we know one, we can make a guess about the other. It wouldn't be a perfect guess but it would be better than nothing. Graphed, it would look soemthing like this: If we were to find a correlation between Weight and Joliness, it would probably be somewhere in this area. Most correlations are not a perfect 1 or -1. Children's age and height would probably be around .9 because we can make very good guesses about a child's height from their age but we can't make perfect guesses. Gender and mathematical ability for college students would probably be around .07: we know that there is some relationship between the two, but that relationship is very weak. We'd only be a tiny bit more successful trying to guess mathematical ability from gender than if we just guessed the average.

Still, if an r isn't zero, we can try to guess one score from another, and if all we know is the r and the person's weight, our guess on joliness is the best guess we can make at the moment. First, let's call the score we do know X and the score we want to guess, Y. We first find the Z score for the score we do know and multiply that by the r for our correlation. This result is our best guess as to how far our predicted score differs from the mean of all scores of that type. So, take the result, multiply it by the standard deviation of y and add that to the mean of the Y's. As an equation, that would look like this:

Y' means predicted Y.

Note that this whole equation works equally well for a negative r or if the z score is negative. If you use an r of 1, the equation turns to , the formula for transforming a score from one system to another. If you use an r of zero, you get , in other words, our best guess is simply the mean of all Y scores.

Test: Imagine you're at a party looking to get a little action. I've been measuring several aspects of people's behavior at the party. I've found:

Basic Statistical Tests

If there was no science and no statistics, all we would have is opinions. You might have a theory that fat people are jollier, a friend might believe that they aren't. Both of you could yell "I'm right" for hours, but none of you would have hard evidence to show you're right. Science is an objective language for proving that your theory is right. The whole hypothesis, experiment, results crap is just a standardized way of saying "I think I'm right because..." Statistics is the language of saying "I'm probably right, and here's the chance that I'm right." For the most part, if you're a scientest then other scientests don't want to hear about your discovery unless you can show there's a 95% chance that you're right. That's "significant results" in the science biz.

Statisticians have created these statistical tests that will let researchers plug in their numbers and find out if they met that 95% mark. The tests don't tell you if the research was screwed up, only if the numbers seem to be telling the story that the researcher wants them to.

The problem these tests have to deal with is random variation. What if you grab a group of people with heart problems, you put them randomly in to two groups: the control group and the leech group. The control group just sit there for an hour a day, the leech group sits there for an hour a day with leeches on them. What if all the people in the leech group get noticeably better, while the people in the non-leech group don't. Before you know it, every heart patient in the world is going in for leech treatments three times a day, then you find out that it was just the luck of the draw, that the people who got selected for the leech group, by chance, were the people who were going to get better. It had nothing to do with leeches, only individual variation. Now, you've put people around the world through a disgusting experience, endangered lives, and worst of all you've made scientists look like a bunch of bumbling dolts. That's a type I error. You know in vampire movies when crosses burn vampires? Type I errors do the same thing to scientests.

What if, on the other hand, the people you put in the leech group just happen to be the really bad patients who are doomed no matter what, and you find that the leech group is no better than the control group, even though leeches are really a life saving miracle cure. Now, you've ruled out a cure which might have saved millions of lives, but since you probably won't publish negative results like this, you're less likely to make scientists look like a bunch of idiots. We call this a type II error, and scientists don't mind these nearly as much, you're free to make as many type II errors as you want. (Of course, it's not really you making the errors, it's the statistical tests).

The difference between Type I and Type II errors is sort of like that old adage that the US Justice system is designed such that "We would rather see a 100 guilty men go free than see one innocent man put in prison." Putting an innocent man in jail would be like a type I error, letting a 100 serial killers loose would be a Type II error. When we run a statistical test, we can actually set the chances of a type I error occurring. It's usually set at .05 (95% chance you aren't making a type I error) but you can set it anywhere you want. This variable is named (alpha).

You can't take a multiple choice test where the possible answers are "A." All tests have options. Most statistical tests have 2 options, and decide between the two. One option is called the null hypothesis and the other is called the alternative hypothesis. The names are kind of stupid, but just think of it as a euphamism that scientests made up because they're so afraid of Type I errors that they don't even want to refer to them by name.

The null hypothesis (H_O) is opposite of what we're trying to prove. Most of the time, the null hypothesis will be that our independent variable does nothing and means nothing. In our leech study, the null hypothesis would be that that leeches don't do anything to help heart patients.

The alternate hypothesis (H_A) is the opposite of the null hypothesis. The alternate hypothesis in our leech case is that leeches do have some effect on the health of heart patients. When you're formulating your null and alternate hyptohesis, make sure that you have every possibility covered: the leeches either do somehting, or they don't, there's no other option.

If your H_A is that x is more than 5, then the H_O would be that X is less than or equal to five.

The statistical test chooses between the two hypothesis. If it chooses the H_A when it shouldn't have, that's a type I error. If it chooses the H_O when it shouldn't have, that's a type II error. So, as with our above adage, the H_O is that the suspect is innocent, the H_A is that the suspect is guilty.

There are some situations where we can choose between a one tailed or two tailed test. A one tailed test means we're only looking in one direction. In a one tailed test, we don't care if leeches make heart patients worse, or if they don't change them at all, we only want to know if the leeches make the patients better. Our H_O is that leeches do not make heart patients better. Our H_A is that they do. In a two tailed test, we care about any difference, positive or negative. Our H_O is that leeches make some change in health, for better or worse, in heart patients. Our H_A is that they don't make any difference. Two tailed tests are the most commonly used.

Once we've got that figured out, the alpha, the null and alternative hypothesis, whether we're using a one or two tailed test, the rest of the work is done by the statistical test which is appropriate for our particular situation. But we have to figure out all this before our test, because if we don't, then it will be us making the decision and not the statistical test, so we can't be sure that the result is impartial.

The statistical test will tell us either that it's too likely that our results mean nothing and occurred by random chance, or that it's so unlikely that our results were chance that we can assume that they mean something. We start out before the test with the inherent assumption that H_O is true, kind of like the "innocent until proven guilty" ideal. If the test says we don't have enough difference to be sure it's not random chance, then we can't accept our H_A and we're left with the H_O. H_O wasn't proven to be true, we just failed to prove anything else, just like when we let a suspect go, we haven't proven that they're innocent, we just failed to prove that they were guilty. If the test says our differences were too great to have occurred by random chance, though, then we can actually reject H_O, and if H_O isn't true, the only other possibility is that H_A is true. If we can prove beyond a reasonable doubt that the suspect is not innocent of the crime, we have to accept that he's guilty.

Test: You want to prove that punching people in the face changes their freindliness. You have a test that measures freindliness on a scale from 1 to 100, 1 being the least friendly and 100 being the most freindly. You take 10 people and sort them in to a two groups of five. One group, you just give them the test. The other group, you punch each person in the face before giving them the test. Which would be you're null and alternative hypothesis.

Test: In the above study, what would be a type I error?

T-Tests