Symmetric distribution

Symmetry is a mirror test. Fold the graph at its center and a symmetric distribution has its two halves land on each other, which means a value a given distance above the center is about as common as one the same distance below. For real data the honest word is approximately symmetric; exact symmetry belongs to models such as the normal curve. When a distribution is symmetric, the mean $\bar{x}$ ("x-bar") and the median sit in the same place.

The six values 3, 5, 7, 7, 9, 11 are symmetric about 7: the distances from 7 are $-4$, $-2$, $0$, $0$, $2$, $4$, each matched by its opposite. The mean is $42/6 = 7$ and the median is $(7 + 7)/2 = 7$. Now take 1, 1, 2, 9, 10, 10. That set is just as perfectly symmetric, about 5.5, with two clumps and a hole where its center is.

The second set kills the sentence "it looks symmetric, so use the empirical rule." Symmetric is not the same as normal. For 1, 1, 2, 9, 10, 10 the mean is 5.5 and the sample standard deviation is about 4.59, so one standard deviation either side spans roughly 0.91 to 10.09 and captures all six values, 100 percent rather than the 68 percent the empirical rule would predict. That rule needs the bell shape, and symmetry alone does not supply it.

The implication runs one way only. Symmetric gets you the mean equal to the median; equal centers do not get you symmetry. For 1, 2, 4, 4, 9 the mean and the median are both 4, and yet the largest value sits 5 above that center while the smallest sits only 3 below. Read symmetry off a graph, then check the centers, not the reverse.

Judge symmetry from a display with equal-width bins, and remember that a dozen values wobble too much for the word to be more than a description. Ask whether the departure from a mirror image is bigger than the bumpiness that sample size would produce anyway. Shape vocabulary is Unit 1 topic 1.6.