The description you gave of symmetric and continuous is also akin to a normal distribution curve. A normal distribution curve looks like an inverted bell therefore often called a bell curve. Using the normal distribution, one can view whether or not the data has a skew which may mean using the median or mode could be preferable.
Also, a normal distribution helps us visually see frequencies and central tendencies. From there, we can calculate the standard deviation to measure the width of the distribution.
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