Skewness Calculator

What is Skewness?

Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. It quantifies the extent to which a distribution "leans" to one side of the mean. A symmetric distribution, such as a normal distribution, has a skewness of zero.

Formula and Its Meaning

The formula for skewness is:

\[g_1 = \frac{\frac{1}{n}\sum_{i=1}^{n} (x_i - \bar{x})^3}{s^3}\]

Where:

• \(g_1\) is the skewness
• \(n\) is the number of data points
• \(x_i\) are the individual values
• \(\bar{x}\) is the mean of the values
• \(s\) is the standard deviation

Calculation Steps

1. Calculate the mean of the dataset.
2. Calculate the standard deviation.
3. For each data point, subtract the mean and cube the result.
4. Sum all the cubed differences.
5. Divide by the number of data points and the cube of the standard deviation.

Example Calculation

Let's calculate the skewness for the dataset: 2, 4, 6, 3, 1

1. Mean: \(\bar{x} = \frac{2 + 4 + 6 + 3 + 1}{5} = 3.2\)
2. Squared differences: \((2-3.2)^2, (4-3.2)^2, (6-3.2)^2, (3-3.2)^2, (1-3.2)^2\)
3. Variance: \(s^2 = \frac{1.44 + 0.64 + 7.84 + 0.04 + 4.84}{5} = 2.96\)
4. Standard Deviation: \(s = \sqrt{2.96} \approx 1.72\)
5. Cubed differences: \((2-3.2)^3, (4-3.2)^3, (6-3.2)^3, (3-3.2)^3, (1-3.2)^3\)
6. Sum of cubed differences: \(-1.728 + 0.512 + 21.952 - 0.008 - 10.648 = 10.08\)
7. Skewness: \(g_1 = \frac{10.08 / 5}{1.72^3} \approx 0.41\)

Visual Representation

This curve represents a positively skewed distribution. The tail extends more to the right of the mean (red dashed line), indicating a positive skewness.