# Left Skewed Distribution: Meaning and Examples

A left skewed distribution is a distribution whose long tail stretches toward the small values on the left side of a graph. Most of the data piles up on the right, so the mean is pulled down below the median. The curve looks like it leans right even though the skew is named after the left tail.

## Quick Answer

- A left skewed distribution has a longer tail on the left and most values concentrated on the right [1].
- Its skewness value is negative, while a right-skewed distribution has a positive value [2].
- The typical order of the three averages is mean < median, with the mode usually sitting at the highest peak.
- The mean is dragged toward the tail, so it understates the typical value when the skew is strong [3].
- The median is usually the better summary of a typical observation in skewed data [3].

## What Left Skewed Distribution Means

In plain terms, a left skewed distribution is lopsided. Imagine a histogram of exam scores where almost everyone scored in the 80s and 90s but one student scored 55. The bulk of the bars sits on the right, and a thin tail of low values trails off to the left. That trailing tail is what gives the distribution its name.

The precise statistical definition is about the tails. A distribution is left-skewed when its left tail is longer or more drawn out than its right tail, and the mass of the distribution is concentrated on the right [1]. The same source notes that a left-skewed distribution usually appears as a right-leaning curve, because "left" refers to the tail, not to the direction the curve leans [1].

You will see this shape written several ways. People search for "distribution left skewed", "skewed left", "left skewness" or "histogram left skewed", and they all describe the same thing. The opposite shape, where the long tail points right, is a positively skewed distribution, also called right-skewed [3].

## How It Works

Skewness is a single number that measures how asymmetric a distribution is. The most common version is the Fisher-Pearson coefficient, which is built from the third central moment of the data [2][4].

$$\text{skewness} = \frac{\frac{1}{N}\sum_{i=1}^{N}(x_i - \bar{x})^3}{s^3}$$

Each symbol means the following.

- $x_i$ is one observation in the data set.
- $\bar{x}$ is the sample mean.
- $N$ is the number of observations.
- $s$ is the standard deviation computed with $N$ in the denominator.
- The cube in the numerator is what makes the sign matter. Values far below the mean contribute large negative terms, and values far above it contribute large positive terms.

If the low values pull harder, the sum is negative and the skewness is negative. Negative skewness means the data are skewed left, and positive skewness means the data are skewed right [2]. A normal distribution has a skewness of zero, and any symmetric data should have a skewness near zero [2]. SciPy computes the same Fisher-Pearson coefficient through its `skew` function, and it notes that for normally distributed data the skewness should be about zero [4].

One caution about the mean and median. Skewness is not directly tied to the relationship between the mean and the median, and a negatively skewed distribution can have its mean above or below its median [1]. The familiar ordering is a tendency, not a law.

## Worked Example

Here is a small data set of exam scores out of 100 for 12 students. The scores cluster high, with one low outlier at 55.

| student_id | score |
|---|---|
| 1 | 95 |
| 2 | 92 |
| 3 | 90 |
| 4 | 88 |
| 5 | 85 |
| 6 | 84 |
| 7 | 82 |
| 8 | 80 |
| 9 | 78 |
| 10 | 75 |
| 11 | 70 |
| 12 | 55 |

**Step 1. Count and sum.** There are n = 12 scores, and they add up to 974.

**Step 2. Mean.** Divide the sum by the count.

$$\text{mean} = \frac{974}{12} = 81.1667$$

**Step 3. Sort the scores.** Sorted values are 55, 70, 75, 78, 80, 82, 84, 85, 88, 90, 92, 95.

**Step 4. Median.** With an even count, average the two middle values, which are 82 and 84.

$$\text{median} = \frac{82 + 84}{2} = 83.0000$$

**Step 5. Mode.** Every score appears exactly once, so there is no repeated value. The software returns the smallest value, 55, as the mode.

**Step 6. Check the order.** The mean is 81.17, the median is 83.00, and the mode is 55.00. The mean sits below the median, which is the signature of a left skew.

**Step 7. Skewness.** The adjusted Fisher-Pearson skewness, which pandas `skew()` reports, is -1.2151. The unadjusted formula above gives -1.0576. The negative sign confirms the left tail.

```python
import pandas as pd
s = pd.Series([95,92,90,88,85,84,82,80,78,75,70,55])
print(s.mean(), s.median(), s.mode().iloc[0], s.skew())
```

Output: `81.16666666666667 83.0 55 -1.2151083938289724`

The single score of 55 does the damage. It sits 26 points below the mean, and because that gap is cubed in the skewness formula, it dominates the calculation. The median barely moves, because it only cares about position, not distance.

## How to Interpret It

Start with the shape. If the histogram has a long thin tail on the left and a tall cluster on the right, you are looking at a left skewed distribution. The peak of the curve is the mode, and it sits to the right of both the mean and the median.

Then read the averages in order. In a left-skewed distribution the mean is less than the median, and the mode is usually the largest of the three [3]. The mean is the balance point of the data, so extreme low values pull it left. The median is the middle position, so it resists that pull.

Finally, decide which average to report. When data is highly skewed, the mean may be misleading, and the median often provides a better representation of the typical value [3]. In the exam example, the mean of 81.17 makes the class look weaker than it is, while the median of 83.00 matches what most students actually scored.

If you want to see how the shape shows up in a five-number summary, a left-skewed box plot makes the asymmetry visible through the position of the median inside the box.

## When to Use It (and when not to)

Use skewness when you need to describe the shape of a distribution before choosing a summary statistic or a test. It tells you whether the mean is trustworthy and whether normality assumptions are reasonable. Many classical statistical tests and intervals depend on normality assumptions, and significant skewness is a warning sign [2].

Use it when comparing two groups. If one group has a skewness of -1.2 and another has a skewness near zero, you know the first group needs a different summary and possibly a different test.

Do not use it as a substitute for looking at the data. A single number cannot show you whether the distribution has two peaks, a gap, or a data entry error. Do not rely on it with very small samples either, since one extreme value can swing the value sharply, as the 12-score example shows.

Do not assume a negative skewness always means the mean is below the median. That relationship is common but not guaranteed [1].

## Left Skewed vs Right Skewed

The two shapes are mirror images. The table below puts them side by side.

| Feature | Left skewed | Right skewed |
|---|---|---|
| Long tail | Left side | Right side |
| Mass of data | Concentrated on the right | Concentrated on the left |
| Sign of skewness | Negative | Positive [2] |
| Typical order | Mean < median | Mean > median [3] |
| Curve appears to lean | Right | Left |
| Common example | Easy exam with a few low scores | Income with a few very high earners |

If you want a fuller walkthrough of the mirror case, see skew to the left vs skewed right. For the symmetric benchmark that both shapes depart from, the normal distribution has a skewness of zero [2].

## Common Mistakes

- **Confusing the direction of the name with the direction of the lean.** The name comes from the tail, not the bulk. A left-skewed curve looks like it leans right [1]. Fix: point at the long tail first, then name the skew.
- **Reporting the mean as the typical value.** The mean is pulled toward the tail and can mislead [3]. Fix: report the median alongside the mean, or instead of it.
- **Assuming the mode is always the peak.** In a data set with no repeated values, software may return the smallest value as the mode, as in the example above. Fix: check whether the mode is meaningful before quoting it.
- **Treating any negative skewness as proof of a mean below the median.** The two are not strictly linked [1]. Fix: compute both and compare them directly.
- **Reading skewness from a small sample as a stable property.** One outlier can dominate the value. Fix: look at the histogram and the sample size together.
- **Ignoring the shape when choosing a test.** Skewness affects the choice of statistical tests, particularly those that assume normally distributed data [3]. Fix: check skewness before running a test that assumes normality.

## Limitations

Skewness compresses the entire shape of a distribution into one number, and that number can hide a lot. It cannot tell you whether the distribution has one peak or two, and it cannot reveal gaps, clusters or data entry errors. Two very different histograms can produce similar skewness values.

The value is also sensitive to extreme observations. Because the formula cubes deviations from the mean, a single far-out point can dominate the result [2]. With small samples this makes the number unstable, and the adjusted version of the coefficient exists precisely to correct for sample size bias [2][4]. Skewness describes the data you have. It does not tell you why the shape occurred or what the underlying process looks like.

## Frequently Asked Questions

### What does a left skewed distribution look like?

It has a tall cluster of values on the right and a long thin tail stretching to the left. The peak sits to the right of the mean and median. The curve often appears to lean right even though the skew is named for the left tail [1].

### Is a left skewed distribution positive or negative?

It is negative. Negative values of skewness indicate data that are skewed left, and positive values indicate data that are skewed right [2]. A normal distribution has a skewness of zero [2].

### In a left skewed distribution, is the mean less than the median?

Usually yes. In a negatively skewed distribution the mean is less than the median [3]. The mean is pulled toward the long tail of low values while the median stays near the middle position. This is a tendency, not a strict rule, since skewness is not directly tied to the mean-median relationship [1].

### What is the difference between left skewed and positively skewed?

They are opposites. A left skewed distribution has its long tail on the left and a negative skewness. A positively skewed distribution, also called right-skewed, has its long tail on the right and a positive skewness [3][2]. The right skewed distribution article covers the mirror case in detail.

### Which measure of center should I use for skewed data?

The median is usually the safer choice. When data is highly skewed, the mean may be misleading, and the median often provides a better representation of the typical value [3]. Report the mean too if your audience expects it, but pair it with the median so the skew is visible. For a broader comparison of shapes, see probability distributions explained.

## References

1. [Skewness - Wikipedia](https://en.wikipedia.org/wiki/Skewness)
2. [1.3.5.11. Measures of Skewness and Kurtosis](https://www.itl.nist.gov/div898/handbook/eda/section3/eda35b.htm)
3. [13: Distribution - Applied Statistics for Quantitative Research: A Practical Guide with Jamovi](https://odp.library.tamu.edu/appliedstatswithjamovi/chapter/13-distribution/)
4. [skew, SciPy v1.18.0 Manual](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.skew.html)

## Further Reading

- [1.3.3.14.6. Histogram Interpretation: Skewed (Non-Normal) Right](https://www.itl.nist.gov/div898/handbook/eda/section3/eda33e6.htm)
- [NIST/SEMATECH e-Handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/index.htm)

## Related Articles

- [Skew to the Left vs Skewed Right: How to Tell the Difference](/blog/data-analysis/skew-to-the-left-vs-skewed-right)
- [Uniform Distribution: Definition, Formula and Examples](/blog/data-analysis/uniform-distribution)
- [Negative Binomial Distribution: Formula and Examples](/blog/data-analysis/negative-binomial-distribution-formula-examples)
- [Left-Skewed Box Plot: How to Read Skewness in Box Plots](/blog/data-analysis/left-skewed-box-plot-skewness)
- [Probability Distributions: Definition, Types and Examples](/blog/data-analysis/probability-distributions-explained)
- [Right Skewed Distribution: Meaning and Examples](/blog/research-skills/right-skewed-distribution-meaning-and-examples)
- [Poisson Distribution: Formula and Examples](/blog/research-skills/poisson-distribution-formula-and-examples)