How do you tell if a distribution is skewed left or right

Skewness describes asymmetry in a frequency distribution. When most observations cluster on the left and a few large values stretch the right side, the shape is right‑skewed; the opposite pattern produces left‑skewness. Visual clues include a longer tail on one side and the location of the mode relative to the mean and median.

Numerical measures of skewness

The most common numeric indicator is the sample skewness ($g_1$ = $\frac{\frac{1}{n}$$\sum$ ($x_i$-\bar{x})^3}{\left(\frac{1}{n}$\sum$ ($x_i$-\bar{x})^2\right)^{3/2}}). Pearson’s first coefficient uses the difference between mean and mode, while the second coefficient uses mean and median: $\text{Skew}_2 = 3\frac{\bar{x}-\tilde{x}}{s}$. Positive results signal right‑skew, negative results left‑skew.

Compute sample skewness step‑by‑step

1. 1Arrange the data and find the mean $\bar{x}$ and standard deviation $s$.
2. 2Calculate each deviation $x_i-\bar{x}$ and raise to the third power, then sum.
3. 3Divide the summed cubes by $n$ and by $s^3$ as shown in the formula.
4. 4Interpret the sign of the resulting value.

Example: data = {4,5,7,8,9,12,30}. Mean $\bar{x}=11$. Standard deviation $s\approx9.2$. Deviations cubed sum to 5,400. Skewness $g_1 = \frac{5400/7}{9.2^3}\approx0.96$. Because the value is positive, the distribution is right‑skewed, matching the long tail created by the outlier 30.

Interpretation of skewness signs

• Positive skewness → right‑skewed, mean > median > mode.
• Negative skewness → left‑skewed, mean < median < mode.
• Zero (or near zero) skewness → symmetric, mean ≈ median ≈ mode.

Typical skewness ranges for common shapes

To decide quickly, compare mean and median first: if the mean exceeds the median, suspect right‑skew; if it is lower, suspect left‑skew. Then confirm with a calculated skewness statistic for precision. Remember that sample size influences stability; larger samples give more reliable skewness values.