[high school-consumer math-statistics]

[ScoztaStatic]

I am taking a class in statistics, and I understand most of it. My biggest problem is the Z-score and the "curve."
I understand raw-mean over standard deviation.
How do you apply this to a graph?
I apologize if I do not have an example to show, but they did not explain it very much anyway.
thank you :]

[Cavernio]

Re: [high school-consumer math-statistics]

This is coming way too late, but maybe someone later on will read this.

Forgive me if I'm starting too basic.

Pretend you're able to sample, say, height from every, single individual in the world. Now pretend you're plotting it. On the X axis, you have height. On the Y axis, you have frequency, the number of people at any given height. Once all the dots are in for all the heights in the world, you would theoretically get a nice, bell-shaped distribution, called the bell-curve. The highest point in the curve is the median, but also, in this theoretical example, is also the mean, because there exists the same number of people taller and shorter on either side of this median, or most popular height. 50% of the people are taller, 50% are shorter. The graph is symmetrical around the median. Now, if you calculated the standard deviation for this data, you would find that at exactly 1 standard deviation higher/lower, 34.1% of all the people measured are taller, and 34.1% are shorter. This is a property of any bell-curve. 1 standard deviation will always be 34.1%. Visually, if you drew a line at 1 standard deviation down to the x-axis from your bellcurve, and filled in the area between this line and a line drawn down to the mean, it will cover 34.1% of the area under the curve.

For a lot of different data sets that exist, people assume that if they were able to measure everyone/everything, and they plotted it, that they would get a bell-curve. This assumption about what the distribution is, is very important in statistics, and without these assumptions, statistics would not work. (As an aside to what you need to know, but something you should keep in mind, is that obviously not all data will follow this nice bell-curve shape. There are many other distributions which people assume certain data sets have. For example, if you're measuring something where the smallest number used is 0, and that there actually exists measurements close to 0, and there also exist measurements incredibly huge, like a billion, your plot will make a much different curve, and will not be at all symmetrical.)

So, what does this have to do with z scores? Going back to our height example, let's say the mean height was 123cm. (Not true, but for the sake of example). That's kinda awkward to work with though. (Ok, maybe not now, but trust me, if you learn a lot more about statistics, you need to make things easier.) Why not make it easier? Wouldn't it be easiest if we could make the mean 0? So what do we do? Well, for this nice, simple example, we'll say you could just subtract 123cm from everyone's scores, and you get a bell-curve that now has a mean of 0. You haven't actually changed the data in relation to itself at all, so all the data you have is still valid, and since you know the conversion factor, (in this case 123cm), you can easily just add 123cm to any given person's height to get back to their actual height. It would also be nice if the standard deviation were 1 too. So, we multiple each data piece by a factor (more on that later) which will change it so that the standard deviation is 1.

What we have created now is what's a special case of the bell-curve: the normal curve. The normal curve is a bell curve that is the not-too-steep/not-too-shallow of a bell-shape where the mean score is 0, and the standard deviation is 1. This normal curve is the comparison curve for every single statistical analysis you'll do where you think the population curve (if you measured every, single person/object) would make a bell shape. The normal curve is always the same, and it is used as tool and a comparison. A Z score is, well, any real number (anything from -infinity to +infinity) and represents the x axis on a graph of the normal curve. A Z score of 1 is 1 standard deviation on the normal curve. Easy numbers to work with! People have calculated specifically for this graph, the percent of people who are above/below any (reasonable) given Z value. Given this information, if you can convert a data set you're using into a normal curve, you'll be able to figure out percents for your own data set.

There's one major thing I slighted over in this example, which is the conversion of the graph we had for height to the normal curve. You usually have to add/subtract the mean value as well as multiply each score by a set factor, which I think is 1/standard deviation. (I could be wrong though, but I know it involves the standard deviation). I'm sure your textbook or whatever tells you what to do. In this example, if you think about it, if you just subtracted 123cm from everyone's scores, you wouldn't come close to a normal curve. Your standard deviation would probably be something like, oh, 20cm, which is nowhere close to 1cm.
The important thing is that if you think your data follows a bell-curve shape, you can convert it to be a normal curve. Again, since you know what you have to do to any given data point to make it fit a normal-curve, you can simply reverse the process to get back to your original piece of data. Converting data involves subtracting the mean and multiplying by something. So if you have a question which asks to convert a specific person's height to a Z score, you would take that person's height, subtract the mean and multiply by the factor, and you would get their Z score. If you want to know how many people are taller than them, go to your table that has all the Z scores listed, find the closest one, and read the percentage. Visually, once you have your Z score, just put a dot on your normal-curve using Z as your x axis value. Be careful and remember that 50% of your people will be shorter and 50% taller than the mean...your Z table might only give you numbers less than 50%, so depending on the question you may need to add 50% to your numbers.