How to tell if my data distribution is symmetric? I know that if the median and mean are approximately equal then this means there is a symmetric distribution but in this particular case I'm not certain. The mean and median are quite close (only 0.487m/gall difference) which would lead me to say there is a symmetric distribution but looking at the boxplot, it looks like it's slightly positively skewed (the median is closer to Q1 than Q3 as confirmed by the values). 
(I'm using Minitab if you have any specific advice for this piece of software.)
 A: The easiest thing is to compute the sample skewness. There's a function in Minitab for that. The symmetrical distributions will have zero skewness. Zero skewness doesn't necessarily mean symmetrical, but in most practical cases it would.
As @NickCox noted, there's more than one definition of skewness. I use the one that's compatible with Excel, but you can use any other.
A: No doubt you have been told otherwise, but mean $=$ median does not imply symmetry.
There's a measure of skewness based on mean minus median (the second Pearson skewness), but it can be 0 when the distribution is not symmetric (like any of the common skewness measures).
Similarly, the relationship between mean and median doesn't necessarily imply a similar relationship between the midhinge ($(Q_1+Q_3)/2$) and median. They can suggest opposite skewness, or one may equal the median while the other doesn't. 
One way to investigate symmetry is via a symmetry plot*. 
If $Y_{(1)}, Y_{(2)}, ..., Y_{(n)}$ are the ordered observations from smallest to largest (the order statistics), and $M$ is the median, then a symmetry plot plots $Y_{(n)}-M$ vs $M-Y_{(1)}$, $Y_{(n-1)}-M$ vs $M-Y_{(2)}$ , ... and so on.
* Minitab can do those. Indeed I raise this plot as a possibility because I've seen them done in Minitab.
Here are four examples:
$\hspace{6cm} \textbf{Symmetry plots}$

(The actual distributions were (left to right, top row first) - Laplace, Gamma(shape=0.8), beta(2,2) and beta(5,2). The code is Ross Ihaka's, from here)
With heavy-tailed symmetric examples, it's often the case that the most extreme points can be very far from the line; you would pay less attention to the distance from the line of one or two points as you near the top right of the figure.
There are of course, other plots (I mentioned the symmetry plot not from a particular sense of advocacy of that particular one, but because I knew it was already implemented in Minitab). So let's explore some others.
Here's the corresponding skewplots that Nick Cox suggested in comments:
$\hspace{6cm} \textbf{Skewness plots}$

In these plots, a trend up would indicate a typically heavier right tail than left and a trend down would indicate a typically heavier left tail than right, while symmetry would be suggested by a relatively flat (though perhaps fairly noisy) plot.
Nick suggests that this plot is better (specifically "more direct"). I am inclined to agree; the interpretation of the plot seems consequently a little easier, though the information in the corresponding plots are often quite similar (after you subtract the unit slope in the first set, you get something very like the second set).
[Of course, none of these things will tell us that the distribution the data were drawn from is actually symmetric; we get an indication of how near-to-symmetric the sample is, and so to that extent we can judge if the data are reasonably consistent with being drawn from a near-symmetrical population.]
A: Center your data around zero by subtracting off the sample mean. Now split your data into two parts, the negative and the positive. Take the absolute value of the negative data points. Now do a two-sample Kolmogorov-Smirnov test by comparing the two partitions to each other. Make your conclusion based on the p-value. 
A: Put your observations sorted in increasing values in one column, then put them sorted in decreasing values in an other column.
Then compute the correlation coefficient (call it Rm) between these two columns.
Compute the chiral index: CHI=(1+Rm)/2.
CHI takes values in the interval [0..1].
CHI is null IF and ONLY IF your sample is symmetrically distributed.
No need of the third moment.
Theory:
http://petitjeanmichel.free.fr/itoweb.petitjean.skewness.html
http://petitjeanmichel.free.fr/itoweb.petitjean.html
(most papers cited in these two pages are downloadable there in pdf)
Hope it helps, even lately.
