What is the skew of the following data set X = [4; 5; 5; 6]?

To calculate the skewness of the dataset \( X = [4, 5, 5, 6] \), we can use the formula for sample skewness:

\[
\text{Skewness} = \frac{n}{(n-1)(n-2)} \sum_{i=1}^{n} \left(\frac{X_i - \bar{X}}{s}\right)^3
\]

Where:
- \( n \) is the number of data points,
- \( \bar{X} \) is the mean of the dataset,
- \( s \) is the standard deviation of the dataset,
- \( X_i \) represents each individual data point.

### Step 1: Calculate the mean (\( \bar{X} \))
\[
\bar{X} = \frac{4 + 5 + 5 + 6}{4} = \frac{20}{4} = 5
\]

### Step 2: Calculate the standard deviation (\( s \))
\[
s = \sqrt{\frac{\sum_{i=1}^{n} (X_i - \bar{X})^2}{n-1}}
\]
\[
s = \sqrt{\frac{(4-5)^2 + (5-5)^2 + (5-5)^2 + (6-5)^2}{3}}
\]
\[
s = \sqrt{\frac{(-1)^2 + 0^2 + 0^2 + 1^2}{3}} = \sqrt{\frac{1 + 0 + 0 + 1}{3}} = \sqrt{\frac{2}{3}} \approx 0.8165
\]

### Step 3: Calculate the skewness
\[
\text{Skewness} = \frac{4}{(4-1)(4-2)} \sum_{i=1}^{4} \left(\frac{X_i - 5}{0.8165}\right)^3
\]
\[
\text{Skewness} = \frac{4}{6} \left[\left(\frac{4 - 5}{0.8165}\right)^3 + \left(\frac{5 - 5}{0.8165}\right)^3 + \left(\frac{5 - 5}{0.8165}\right)^3 + \left(\frac{6 - 5}{0.8165}\right)^3\right]
\]
\[
\text{Skewness} = \frac{4}{6} \left[\left(\frac{-1}{0.8165}\right)^3 + 0^3 + 0^3 + \left(\frac{1}{0.8165}\right)^3\right]
\]
\[
\text{Skewness} = \frac{4}{6} \left[-1.2247^3 + 1.2247^3\right] = \frac{4}{6} \times [(-1.836) + 1.836] = \frac{4}{6} \times 0 = 0
\]

### Conclusion:
The skewness of the dataset \( X = [4, 5, 5, 6] \) is 0. This indicates that the data is perfectly symmetrical.

Would you like more details or have any questions?

### Relative Questions:
1. How is skewness different from kurtosis?
2. What does a positive skew indicate in a dataset?
3. Can skewness be negative? If so, what does that mean?
4. How does sample size affect the calculation of skewness?
5. What are some real-world examples where skewness is important?
6. How can we interpret skewness in financial data?
7. What happens to skewness if we add a constant to all data points?
8. How does skewness change if outliers are introduced to the dataset?

### Tip:
Skewness is highly sensitive to outliers, so even a small change in the dataset can significantly affect its skewness.

Learn how to calculate the skewness of dataset [4, 5, 5, 6] using the sample skewness formula. Step-by-step explanation with mean and standard deviation calculations included. Suitable for undergraduate students studying statistics.

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