Math geniuses!!! Need help with SD.

[Snoweel]

Standard Deviation?

I vaguely remember doing it in high school, however, like most of high school, it's a fuzzy blur.

Is anyone still familiar with it?

The reason is that I just received my result from my Law School Entrance Test and have less of a clue than I did before I opened the letter.

Now, the test consisted of a total of 70 questions, each of equal value. Of that I'm sure.

So I was a bit miffed when I opened the letter to find:

Your test score for the ALSET (Series B) held on 22 November 2002 is: 160

The mean score for the test was 151.88 with a standard deviation of 9.57

That's IT. That's the entirety of statistical information I received, so my question is:

How'd I really do? All I know is I did better than most.

Can anyone work out the top mark for this or which percentile I belong to, results-wise?

Thanks.

BTW, if it helps, 15 people sat the test at the institution where I took it, but I don't know if these results include others.

 

[Capellan]

http://www.robertniles.com/stats/stdev.shtml

Based on the info in the link, and the fact that you were 8.12 above the average, I'd guess you were better than 75-80% of the people who took the test.

 

[boxstop7]

It's been a while, and I'm beat, but I'll give it a whirl...

If you scored a 160, and the SD is 9.57, and the mean is 151.88, you're about one SD away from the mean. This essentially means that 68% of the people taking that test scored roughly between 142 and 160. So, with decent confidence, you can say that you scored as good or better than 68% of the people who took the test. Had you scored 169 or higher, you could say that you scored better than 95% of the people who took the test. What is more useful would be to take that information (mean and SD) and compare it to the other institutions who administered that test. See how the means and SDs compare across the board. All in all, if I were you, I'd be proud of that score.

~Box

p.s. If anyone's a better stat whiz than me (which shouldn't take much), please correct any errors I've made.

EDIT: I didn't carry the lower percentiles. Consider yourself in approximately the top 20 percent.

 

[boxstop7]

To elaborate...

If you can get info on mean and SD for other institutions who administered that same test, you can make guesses about the distribution of scores. If the SD gets bigger at some places than at your facility's, you can safely assume that there are more people scoring toward the extremes (then compare that mean to the mean from your group to see which way the distribution is skewed). If the SD is smaller at some places than at your facility's, you can usually assume the opposite. Scores would be more tightly clustered (and comparing that mean to your mean will give you a good indication of how you did comparatively). In general, the larger the SD the more variance. The more variance, the more spread out the scores are.

~Box

 

[Fast Learner]

As a non-mathematical way of saying it, think of it like this: the standard deviation indicates how far, "plus or minus," from the average (mean) almost everyone was. That is, almost everyone will fall in a range between the mean minus the standard deviation and the mean plus the standard deviation.

It gives you a sense of how far from the average most of the results were.

If, for example, the mean was 50 and the standard deviation was 2, that would mean that almost every result was 48, 49, 50, 51, or 52. Everyone was very close to the average.

If, on the other hand, the mean was 50 and the standard deviation was 20, that would mean that the scores were very spread out, with most of the people falling somewhere between 30 and 70.

In your case the numbers indicate that almost everyone who took the test scored somewhere between 142 and 161. As a result, your score was both above average (the mean) and near the top of all of the scores.

In my explanation I've said "almost everyone" a number of times. That's because when standard deviation is calculated the most extreme scores (very low and very high) are usually not included.

(All of this from my high school statistics education, over 20 years ago, but I'm pretty certain it's all correct.)

 

[nsruf]

If you assume that your test results are normal distributed, about 95% of the results will be in the range of

[mean - 1.96 * deviation, mean + 1.96 * deviation]

Scaling you result to standard normal (i.e. subtract mean and divide by deviation), yields 0.85. Looking this up in a table (distribution of standard normal) shows that the probability is about 80% to score less than you have.

 

[Snoweel]

Thanks guys! Very sweet of you.

And that's why I love this place (among other reasons).

Y'know when you're a kid and you wanna know something, you ask your dad?

Now that I'm all growed up, I come here.

Thanks again!

Love,
Snoweel

 

[dcollins]

Can anyone work out the top mark for this or which percentile I belong to, results-wise?

What percentile, as compared to others who took the test, has been discussed by others above.

However, the "top mark", by which I think you may mean "maximum possible score", cannot be determined -- you've got insufficient information to figure that out. If that's what your looking for (and/or percent of questions that you got right), then you need to go to the test adminstrators and ask for the maximum possible score.

 

[candidus_cogitens]

In my explanation I've said "almost everyone" a number of times. That's because when standard deviation is calculated the most extreme scores (very low and very high) are usually not included.

Okay, but ...
1) how do you determine which ones are "extreme"?
2) Does the range within a standard deviation include ALL remaining results, after the extremes are included?

----------

Let's talk in D&D terms, just for fun:

Let's say a PC with STR 10 is using a nonmagical greatsword. When he hits, the average damage result would be 7. Now, what is the standard deviation?

 

[nsruf]

Okay, but ...
1) how do you determine which ones are "extreme"?

This is an arbitrary decision - whatever you deem extreme.

2) Does the range within a standard deviation include ALL remaining results, after the extremes are included?

Not necessarily, no (you meant excluded, I suppose?).

Let's say a PC with STR 10 is using a nonmagical greatsword. When he hits, the average damage result would be 7. Now, what is the standard deviation?

I am too lazy to do the calculations But the example is not really relevant to the question: extreme values are excluded from the calculation if you calculate deviation based on real world data which may have some uncertainty in measurement. You do it to avoid having your result influenced by obvious errors. But the example is not concerned with observations - it contains the full stochastic model ("roll 2d6, assume fair dice"). This means I could calculate exact deviation without having to worry about errors. If I wasn't too lazy right now, that is...