# Midterm I report

## Summary statistics

 Problem 1 2 3 4 5 6 7 8 9 Total max poss 12 10 10 15 10 8 15 10 10 100 max achieved 12 10 10 15 10 8 15 10 10 99 min 0 0 0 0 2 0 8 3 2 43 mean 7.97 5.40 7.78 9.85 8.45 7.57 14.42 9.17 6.32 76.89 median 8 3 9 10 10 8 15 9.5 6 77 mode 8 3 9 11 10 8 15 10 6 76 std. dev 3.02 3.49 2.27 3.28 2.52 1.51 1.12 1.11 2.09 11.03 r2 with total 0.6474 0.6445 0.5078 0.7230 0.4982 0.3150 0.2981 0.1599 0.5196

## Explanation of Statistics

"Max poss" is the maximum points possible for that problem.  "Max achieved" is the highest score actually assigned to that problem.  As you can see, every problem was aced by at least one person.  Likewise, "min" is the minimum score assigned.

"Mean" is the average score on that problem.  "Median" is a slightly different summary statistic; if you line up the scores in order, this one is in the middle.  That is to say, half the scores are below the median and half are above.  The mode score is the one most frequently assigned.

"Std. dev" is the standard deviation of the scores.  It's a measure of how wide the distribution is.  In fact, it's the distance between the global maximum and the inflection point of the bell curve.  If the data is normally distributed (as test scores usually are), roughly 68% of the data lies within one standard deviation of the mean.

Robert Niles has a nice exposition on summary statistics on his website.

The last row is the correlation coefficient of each problem with the total score.  The coefficient is often denoted by r, although r2is what's commonly computed.  It denotes the amount one variable (in this case, a single problem) is related to another (in this case, the exam total).  The coefficient r does not depend on units and lies between -1 and 1.  If  r=1, the data is perfectly linearly related—an increase in one increases in the other.  If r=-1, the data is related in the other direction (an increase in one gives a decrease in the other).  If r=0, the data are completely unrelated.  The problems with the highest value of r (or r2) have the strongest correlation with the final grade, which means students who did well on those problems did well on the test overall.

I found a nice Interactive demo of the correlation coefficient online.

## So what grade am I getting?

Take your score on the midterm and subtract it from the mean total score (76.89).  Divide this number by the standard deviation (11.03).  This is your z-score on the midterm, and it measures where on the curve you lie.  I usually put the B/C line at z=-1, and the C/D line at z=-2.  This means if your score is 55 or above, you're getting a satisfactory grade so far, and if you're at 66 or above, you're getting a good grade so far.  The higher above the mean you are, the closer you are to an A.