When you are looking balance, such as in a statistical graph, and you are looking at symetrical balance you are looking at a graph that has a even number of people before and after the mean. This is what a standard normative deviation will look like with standardized deviations.
When you have a asymmetrical balanced graph the graph will have an uneven number of data points either before or after the mean. This will mean that the data is grouped in either a positive or negative deviation from the mean which is the middle of the data set.
When you are looking at a Standard Normative Deviation you are looking at data that has been placed into a bell curve with a mean of 0 and the deviations to the left of the mean are -1 and -2 and to the right of the mean are 1 and 2. When you are looking at a bell curve with symmetrical balance the data to the left of the mean (below the mean) falls in the same pattern (deviation) as to the right (above the mean). When you have an asymmetrical balance the data will cluster either above or below the mean caused the data to skew in one direction or the other.
For example if you have collected data on what age someone first traveled on an airplane a symmetrical graph may have a mean age of 20 with 9 data points falling below the mean to indicate they were below 20 and 9 data points falling above the mean to indicate they were above 20. However if you have 16 data points above the mean and three below the mean you have an asymmetrical graph as more people traveled on an airplane after the mean age.
Noting that this question was first posted under "Social Sciences," it's truly important to note a different theory about symmetrical vs. asymmetrical balance in regards to society's norms. This is a completely different way to analyze balance apart from the study of statistics which, of course, is also one of the later topics related to this question. Therefore, because statistics has already been dealt with, this answer will deal only with "symmetry" in regards to social interactions.
In sociology, the theory of symmetry can get quite complex (note the focus on Standard Normative Deviations above), but for now, let's keep it simple. Symmetry is a comparison between people or groups in regards to certain feelings. Most commonly, the feelings compared are important ones: "reciprocity, empathy, apology, dialog, respect, justice, and revenge."
In a society, sociological, symmetrical interactions are about similarities and equalities. Many use the phrase, "We are all the same," to describe this type of symmetrical balance. Put simply, a symmetrical relationship is a relationship between peers. Alternately, in a society, sociological, asymmetrical interactions are about one person (or group) being better than the other. In this regard, many use the phrase, "I am special, and better than you," to describe this type of asymmetrical balance. Put simply, and asymmetrical relationship is a "power relationship" where one person "rules over" or "manages" another in some way.
Creating examples for these types of relationships can get very complicated, but let me give a couple of easy ones. First, it is VERY important to understand that not all symmetrical balance is positive, and not all asymmetrical balance is negative! Cliques in high school, as negative as they are, can be considered to have a symmetrical balance. The members of the click (take Jocks or Goths, for example) create belonging and see a connection between "sameness" among the members. However, in the same high school, the relationship between the teacher and his/her students (as positive as it may be) can be considered to have an asymmetrical balance. The teacher is the leader and manager over the students and, therefore, has more power. A further example (again in the same high school) would be the relationship between the principal and his/her teachers. The principal is the leader and manager over the students and, therefore, has more power.
In conclusion, the study of the general concept of "symmetrical balance" would blow your mind! Symmetry can be applied to almost ANYTHING: math, science, nature, art, music, etc. In this answer, I dealt only with symmetrical balance and asymmetrical balance in regards to sociology.
Asymmetrical balance refers to a design that has dissimilar elements but still appears balanced, dividing a picture in half won't have the exact same elements however the elements they do have are varied and seem to balance one another out.