I think it is simple. 200 students took a test. Out of a possible 100 points
Hi Score: 87
Lo Score: 4
Mean: 60
SD: 17
So I am thinking that there must have been either lots more people below 60, or enough people close to the 4 to get that standard deviation. What more can be implied about the results from the information given?
What’s the median?
A Monte Carlo might help, randomizing all the values until they fit those parameters, and repeating lotsa times??
Some data points: the high score performed better than 94.4%. Low=4 was way below the 1st percentile, 0.05%. The data is likely skewed, which may affect this analysis. Using the median might also help, and figure out (assuming no original data) if the 4 was an extreme outlier, or there were many poor scores. If I were to curve, I would take that into consideration, boost the people who did poor but not that poor (although really, add 10 points to the 4 and they’re still doing bad).
One thing you can say for sure is that at least 75% of the scores (so, at least 150 of them out of the 200, and quite probably more than that) are within two standard deviations of the mean (so, between 26 and 94—not that that’s saying much). This is due to Chebyshev’s Theorem.
The median and mode are not currently known. I am guessing that you cannot surmise that there are more players under the mean than over, even though a half of the second standard deviation includes no high scores.
The mean and standard deviation really aren’t all that informative for distributions that aren’t particularly close to normal. The standard five number summary that works well for any data set is the minimum, first quartile, median, third quartile and maximum.
Well, there’s not really much we can say. There are lots of very different ways that the scores could be distributed that would all have the same high/low/mean/sd.
But, given that it’s test scores, I think actually it’s more likely you have fewer people below the mean of 60. For calculating the mean, each very low score (e.g. 4, -56 relative to the mean) counteracts several moderately high score (e.g. 87, +27 relative to the mean) or a bunch of just-above the mean scores (like five scores of 71). So I’d expect (again, this is just a guess) most scores in the 50’s, 60’s and 70’s (more above 60 than below), with a few in the 80’s, and a few in the abyss below 40 pulling the mean down.