Skewness vs Symmetric Distribution

Skewness is a statement about the tail. A distribution is skewed right when the values thin out gradually toward the high end, and skewed left when they thin out toward the low end. The name follows the tail, never the peak, so a right-skewed graph has its tall bars on the left and its thin straggle on the right.

Take 1, 2, 3, 4, 5, 6, 21. The bulk sits between 1 and 6 with one value far above, so the shape is skewed right. The median is 4, the fourth of seven ordered values. The mean is $42/7 = 6$, and only the 21 sits above it. The long tail pulled the mean toward it and left the median alone, because the median counts only how many values lie on each side of it.

The sentence to unlearn is "it is skewed right because most of the data are on the right." That describes a left skew. The pile shows where values are common; the skew is named for where they run out. Trace the tail with a finger and read off the direction it points.

The mean landing above the median is a consequence of right skew, not its definition, and it is a strong tendency rather than a theorem: statisticians have built right-skewed distributions whose mean falls below their median. Use the comparison to confirm a shape you have already read off a graph, not to decide the shape without one.

Two boundaries. Skew is a word for single-peaked quantitative distributions, so a graph with two clear peaks gets called bimodal instead, and one distant point in an otherwise balanced set is usually reported as roughly symmetric with an outlier, since skew describes a tail that thins out gradually rather than a lone gap. And skew means nothing for categorical data: the bars of a bar graph can be reordered at will, so any lopsidedness you see there is an artifact of the order you chose. Shape vocabulary is Unit 1 topic 1.6.

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.