A bar graph displays symmetry when the two halves of the graph are mirror images of each other. If you drew a vertical line down the center, the bars on the left side would match the bars on the right side in height. This concept comes up most often with histograms, which are bar graphs that show how frequently data values fall into different ranges.
What Symmetry Looks Like in a Bar Graph
In a symmetric distribution, the bars rise in height as they approach the center and then fall at the same rate on the other side. The tallest bar (or bars) sit in the middle, and the shorter bars fan out evenly on both sides. Think of it as a mirror image: if you folded the graph along that center line, the left half would land directly on top of the right half.
A skewed graph, by contrast, has one side that stretches out longer than the other. A “skewed right” graph has a longer tail trailing off to the right, meaning a few unusually high values pull the shape in that direction. A “skewed left” graph has a longer tail on the left side. In either case, the two halves are not mirror images, so the graph is not symmetric.
The Bell Curve: The Classic Symmetric Shape
The most well-known symmetric bar graph follows what’s called a normal distribution, often referred to as a bell curve. The bars form a smooth, mound-like shape with one peak in the center. Values cluster heavily around the middle and become less and less frequent as you move toward either end. When data follow this pattern, the mean, median, and mode all fall at the same central point.
You’ll see this shape in many real-world datasets: heights of adults in a population, standardized test scores, repeated measurements of the same object. The key visual feature is that the tallest bars are in the center, and the bars shrink gradually and equally on both sides until they approach zero at the far edges.
Other Symmetric Shapes Beyond the Bell Curve
A bell curve isn’t the only type of symmetric bar graph. A uniform distribution is also symmetric. In this case, every bar is roughly the same height across the entire graph, forming a flat, rectangular shape. There’s no peak at all, but if you folded it in half, the two sides would still match. Think of rolling a fair die many times: each outcome (1 through 6) appears about equally often, so the bars are all approximately equal.
A bimodal distribution can also be symmetric. This type of graph has two peaks instead of one. If the two peaks are the same height and are spaced equally on either side of the center, the graph is symmetric even though it doesn’t look like a bell curve. Imagine plotting the heights of a mixed group of adult men and women on one graph. You might see two humps, one for each group, and if they balance evenly, the overall shape is symmetric.
How to Check for Symmetry
When you’re looking at a bar graph on a test or assignment and need to identify which one is symmetric, use this quick checklist:
- Find the center. Identify the middle bar or the midpoint between bars. This is your fold line.
- Compare left and right. The first bar to the left of center should be about the same height as the first bar to the right. The second bar out on each side should match, and so on.
- Check the tails. Both ends of the graph should taper off at the same rate. If one side trails out much farther than the other, the graph is skewed, not symmetric.
Perfect symmetry is rare in real data. Most textbook questions are looking for approximate symmetry, where the overall shape is balanced even if individual bars aren’t pixel-perfect mirror images.
Why Symmetry Matters
Symmetry tells you something important about the data. In a symmetric distribution, the mean, median, and mode are all equal (or very close). That means the “average” is genuinely representative of the typical value in the dataset. No extreme values on one side are dragging the average away from center.
In a skewed distribution, these three measures of center pull apart. The mean gets tugged toward the longer tail, making it a less reliable summary of what’s “typical.” Recognizing whether a bar graph is symmetric or skewed is the first step in understanding whether the average is telling you the full story.