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Negatively Skewed — Definition, Formula & Examples

Negatively skewed describes a distribution whose long tail stretches toward the smaller values on the left. Most of the data piles up on the right, and a few unusually low values pull the tail out to the left. It is also called left-skewed.

A distribution is negatively skewed when its coefficient of skewness is less than zero. The standard measure is the third standardized moment, E[(X−μσ)3]E\left[\left(\frac{X-\mu}{\sigma}\right)^3\right]. Cubing keeps the sign of each deviation, so large negative deviations from the mean dominate and make this value negative.

Key Formula

g1=1n∑i=1n(xi−xˉ)3(1n∑i=1n(xi−xˉ)2)3/2,g1<0 means negatively skewedg_1 = \frac{\frac{1}{n}\sum_{i=1}^{n}(x_i-\bar{x})^3}{\left(\frac{1}{n}\sum_{i=1}^{n}(x_i-\bar{x})^2\right)^{3/2}}, \qquad g_1 < 0 \text{ means negatively skewed}
Where:
  • g1g_1 = sample (moment) coefficient of skewness
  • xix_i = the i-th data value
  • xˉ\bar{x} = sample mean
  • nn = number of data values

Worked Example

Problem: Is the data set 2, 8, 9, 10, 11 negatively skewed? Compare the mean and median, then compute the skewness coefficient g1g_1.
Step 1: Find the mean. The data are already in order, so the median is the middle (3rd) value, 9. The mean is less than the median, because the low value 2 pulls the mean left.
xˉ=2+8+9+10+115=405=8,8<9\bar{x} = \frac{2+8+9+10+11}{5} = \frac{40}{5} = 8, \qquad 8 < 9
Step 2: Find the deviations from the mean: −6,0,1,2,3-6, 0, 1, 2, 3. Sum their cubes and their squares.
∑(xi−xˉ)3=−216+0+1+8+27=−180,∑(xi−xˉ)2=36+0+1+4+9=50\sum(x_i-\bar{x})^3 = -216+0+1+8+27 = -180, \qquad \sum(x_i-\bar{x})^2 = 36+0+1+4+9 = 50
Step 3: Divide each sum by n=5n = 5 and substitute into the formula.
g1=−180/5(50/5)3/2=−36103/2≈−1.14g_1 = \frac{-180/5}{(50/5)^{3/2}} = \frac{-36}{10^{3/2}} \approx -1.14
Answer: Yes. The skewness coefficient is g1≈−1.14<0g_1 \approx -1.14 < 0, and the mean (8) is less than the median (9). Both point to a negatively skewed data set.

Visualization

Why It Matters

On the AP Statistics exam, you must describe a distribution's shape, center, spread, and unusual features. Shape tells you which summary to report. For a skewed distribution, the median and IQR resist the tail better than the mean and standard deviation. Exam scores on an easy test and age at retirement are typical left-skewed variables.

Common Mistakes

Mistake: Calling a distribution right-skewed because its peak sits on the right side of the graph.
Correction: Skew is named for the direction of the tail, not the peak. A tall cluster on the right with a tail trailing left is negatively (left) skewed. Also, 'mean < median' is only a rule of thumb. It usually holds for left-skewed data, but some distributions break it, so check the graph or g1g_1 too.

Related Terms