A histogram is approximately symmetric when its left half roughly mirrors its right half. This doesn’t require perfect symmetry. If you drew a vertical line down the center of the histogram and the two sides looked like near-reflections of each other, the distribution qualifies. Several common histogram shapes meet this criterion, including bell-shaped curves, uniform (flat) distributions, and symmetric bimodal distributions.
What Makes a Histogram Symmetric
The core test is simple: the two halves of the histogram appear as mirror images of one another. In a perfectly symmetric distribution, the mean, median, and mode all fall at the same center point. In an approximately symmetric histogram, these three values sit very close together but may not be identical. The further apart they drift, the more skewed the distribution becomes.
A useful numerical check is the skewness value. A perfectly symmetric distribution has a skewness of zero. In practice, real-world data almost never hits exactly zero, so values close to zero (such as 0.03 or negative 0.05) indicate approximate symmetry. The closer the skewness is to zero, the more symmetric the histogram looks.
Bell-Shaped (Normal) Distributions
The most common example of a symmetric histogram is the classic bell curve, also called a normal distribution. Data clusters around a central peak, then tapers off evenly on both sides. Heights of adult men, standardized test scores, and repeated measurements of the same object all tend to produce this shape. The tallest bars are in the middle, and the bars get progressively shorter as you move in either direction.
Key features of a bell-shaped histogram: a single peak at the center, the mean and median and mode all located at or near that peak, and tails that extend roughly equally to the left and right. If you see this shape, the distribution is approximately symmetric.
Uniform (Flat) Distributions
A flat histogram where all bars are roughly the same height is also symmetric. This is called a uniform distribution. Rolling a fair die many times produces something close to this shape, with each outcome (1 through 6) appearing at roughly equal frequency. There’s no single peak, but the left and right halves still mirror each other. A uniform distribution doesn’t need to be perfectly flat to count as approximately symmetric. As long as the bars are relatively even without trending higher on one side, it qualifies.
Symmetric Bimodal Distributions
A bimodal histogram has two distinct peaks. When those peaks are roughly equal in height and positioned symmetrically around the center, the histogram is approximately symmetric. Imagine measuring the heights of a mixed group of adult men and women. You might see one peak near the average female height and another near the average male height, with a dip in between. If those two peaks are similar in size and spacing, the overall shape is symmetric even though it has two humps instead of one.
What Symmetric Histograms Are Not
Skewed distributions are the opposite of symmetric ones. In a right-skewed histogram, the tail stretches out to the right, meaning a few unusually high values pull the distribution in that direction. Income data is a classic example: most people earn moderate amounts while a small number earn vastly more, creating a long right tail. In a left-skewed histogram, the tail extends to the left, as with age at retirement in a population where most people retire around 65 but some retire much earlier.
In skewed distributions, the mean gets pulled toward the tail. For a right-skewed histogram, the mean is higher than the median. For a left-skewed histogram, the mean is lower than the median. This separation between mean and median is one of the quickest ways to detect that a histogram is not symmetric.
Why Bin Width Can Fool You
The same dataset can look symmetric or skewed depending on how you construct the histogram. Changing the bin width (the range each bar covers) regroups the data, and different groupings change the visual shape. A histogram with one set of bins might look perfectly symmetric with a central peak, while the same data with different bins can appear slightly skewed to the right.
Bins that are too wide compress everything into a few tall bars, hiding the true shape. Bins that are too narrow spread the data so thin that random variation creates a jagged, unreadable pattern. When judging symmetry visually, use a moderate number of bins and focus on the overall pattern rather than the exact height of any single bar. If the general outline mirrors itself, the distribution is approximately symmetric regardless of minor bar-to-bar differences.
How to Confirm Approximate Symmetry
Start with a visual check: does the histogram look roughly mirrored? Then compare the mean and median. If they’re close together, the distribution is likely symmetric. If one is noticeably larger, the data is skewed in that direction.
For a more precise answer, calculate the skewness coefficient. Values near zero confirm symmetry. You can also use a box plot as a quick visual tool. A symmetric box plot has the median line near the center of the box, with whiskers extending roughly equal distances on both sides. If the box plot looks balanced, the histogram will too.
In summary, three histogram shapes are approximately symmetric: bell-shaped distributions with a single central peak, uniform distributions with roughly equal bar heights, and bimodal distributions with two peaks that mirror each other. The common thread is that mirror-image quality, where neither side has a longer or heavier tail than the other.