What it means
Imagine plotting a month of daily sales as a bar chart. If the bars rise to a peak in the middle and fall away evenly on both sides, the data is symmetrical.
Roughly as many days fall far above the average as far below it. In a perfectly symmetrical distribution with a single peak, the mean (the average), the median (the middle value) and the mode (the most frequent value) are all the same number.
When data is skewed, meaning it stretches further to one side, these figures separate. Comparing the mean and the median is a quick test: a mean far above the median suggests a long tail of high values.
Symmetry matters because many financial tools assume it. Standard deviation as a measure of risk, many forecasting methods and the usual way of calculating value at risk all work best when gains and losses are about equally likely and similar in size.
Real investment returns are often not perfectly symmetrical, with large losses more common than large gains of the same size. A measure called skewness puts a number on the lack of symmetry.
A skewness of zero indicates a symmetrical distribution, a positive number indicates a longer tail on the right and a negative number a longer tail on the left. Analysts also check kurtosis, which describes how heavy the tails are.
For a manager, the practical question is whether the data behaves symmetrically enough for the tool being used. If it does not, applying methods designed for symmetrical data can understate the chance of extreme outcomes.
Symmetry has a direct effect on how results should be reported to a board. For symmetrical data, an average and a standard deviation tell most of the story, and the board can read a range of plus or minus two standard deviations as covering most outcomes.
For skewed data it is safer to show percentiles, such as the 5th and 95th, so that the shape of the tails is visible.
In practice
Real-world examples.
Example
A packaging plant measures the weight of boxes filled by a machine. The results cluster evenly around the target weight, with as many boxes slightly over as slightly under, so the quality team treats the process as stable.
Example
An investment analyst examines daily returns on a diversified index fund and finds them close to symmetrical over a calm year. She uses standard deviation to describe risk but adds a stress test for rare large falls.
Example
A sales manager looks at the distribution of deal sizes and sees a long right tail from a few very large contracts. The mean is well above the median, so he reports both figures to avoid overstating the typical deal.
Formula
Calculation
In a symmetrical single-peaked distribution: Mean = Median = Mode
Suppose a business records monthly profit over seven months: $10,000, $20,000, $30,000, $30,000, $30,000, $40,000 and $50,000.
Mean = ($10,000 + $20,000 + $30,000 + $30,000 + $30,000 + $40,000 + $50,000) / 7 = $210,000 / 7 = $30,000
Median = the 4th value in order = $30,000
Mode = $30,000, which occurs three times
All three are equal, and the values on each side of $30,000 mirror one another ($10,000 and $50,000 are each $20,000 away, and $20,000 and $40,000 are each $10,000 away), so the data is symmetrical.Case study
Seen in the real world.
Ridgeway Metals is an illustrative, fictional supplier of steel bars that promises a length of 2.00 metres within a narrow tolerance. The quality manager plotted the lengths of 500 bars cut in one week and shared the chart with the production team.
The chart was close to symmetrical around 2.00 metres, with the mean and median both within a millimetre of the target. That told her that the cutting machine had no systematic bias, and that the remaining variation was random.
In the illustrative follow-up a different week showed bars bunched at 2.01 metres with a tail to the right. The mean was then above the median, signalling a drift in the machine setting, and maintenance recalibrated it before any shipments were rejected.
Watch out
Common mistakes.
- Assuming that all bell-shaped data is perfectly symmetrical, when small skews and heavy tails can still be present.
- Using only the mean to describe skewed data, when the median is often a better guide to a typical value.
- Assuming financial returns are symmetrical and underestimating the risk of large losses.
Questions
People also ask.
What is the difference between symmetrical and skewed data?
Symmetrical data mirrors itself around the centre, while skewed data has a longer tail on one side.
Is every symmetrical distribution a normal distribution?
No, a normal distribution is symmetrical, but other shapes, such as a uniform distribution, can be symmetrical too.
Why do analysts check the mean and the median together?
If they are close, the data is probably roughly symmetrical, and if they differ a lot it is a sign of skew or outliers.
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