Skewness Formula Types and Solved Examples
The measure of the asymmetry of a distribution of probability that is ideally symmetric and is given by the third standardized moment is skewness. In simple words, skew is the measure of how much a random variable's probability distribution varies from the normal distribution.
When both sides of the distribution are not distributed equally then this is known as Skewed Data. It is not a symmetrical distribution. To quickly see if the data is skewed, we can use a histogram.
A Skewed Histogram
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Types of Skewness
Well, the normal distribution is the distribution of the probability without any skewness. There are two types of skewness, apart from this:
Positive Skewness
Negative Skewness
Positive Skewness
A positively skewed distribution (often referred to as Right-Skewed) is a distribution type where most values are concentrated to the left tail of the distribution whereas the right tail of the distribution is longer. A positively skewed distribution is the complete opposite of a negatively skewed distribution.
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A Positively Skewed Curve
In contrast to normally distributed data, where all central trend measurements (mean, median, and mode) are equal to each other, with positively skewed data, the observations are dispersed. The general relationship between the central tendency measures in a positively skewed distribution can be expressed using the following inequalities:
Mean > Median > Mode
Negative Skewness
A negatively skewed distribution (often referred to as Left-Skewed) is a kind of distribution where more values are on the right side of the distribution graph whereas the left tail of its distribution graph is longer.
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A Negatively Skewed Curve
Apart from normally distributed data, where all central trend measurements (mean, median, and mode) are equal to each other, with negatively skewed data, the measurements are dispersed. The general relationship between central trend measures in the negatively skewed distribution can be displayed using the following inequality:
Mode > Median > Mean
How to Find Skewness of Data?
One measure of skewness would be to subtract the mean from the mode, then divide the difference by the Standard Deviation of the data. This is called Pearson's first coefficient of skewness. We have a dimensionless quantity as the explanation for dividing the difference. This explains why there is positive skewness in data skewed to the right. The mean is greater than the mode if the data set is skewed to the right, so subtracting the mode from the mean gives a positive number. A similar argument shows why there is negative skewness in data skewed to the left.
To calculate the asymmetry of a data set, Pearson's second coefficient of skewness is also used. We deduct the mode from the median for this value, multiply this number by 3 and then divide it by the Standard Deviation.
Note: If the data shows a strong mode, Pearson's first coefficient of skewness is useful. Pearson's second coefficient can be preferable if the data has a poor mode or several modes, as it does not depend on mode as a central tendency measure.
Uses of Skewed Data
In various contexts, skewed data arises very naturally. Incomes are skewed to the right because the mean can be significantly influenced by even a few people making millions of dollars, and there are no negative incomes. Similarly, details related to a product's lifetime, such as a light bulb brand, is skewed to the right. Here, zero is the smallest that a lifetime can be, and long-lasting light bulbs can give the data a positive skew.
What is Skewness in Statistics?
In statistics, if one asks what is skewness, it is the degree of asymmetry found in a distribution of probability. Distributions can exhibit to varying degrees right (positive) skewness or left (negative) skewness. Zero skewness exhibits a natural distribution (bell curve).
Conclusion
In a statistical distribution, data is considered skewed when the curve appears bent or skewed either to the left side or on the right. The graph shows symmetry in a normal distribution, implying that there are just as many data values on the left side of the median as on the right side.
FAQs on Understanding Skewness in Statistics
1. What is skewness in statistics?
Skewness is a measure of the asymmetry of a probability distribution around its mean. In a perfectly symmetric distribution (like the normal distribution), skewness is 0. If the distribution has a longer tail on one side, it is skewed.
- Positive skewness: Tail extends to the right.
- Negative skewness: Tail extends to the left.
- Zero skewness: Symmetrical distribution.
2. What is the formula for skewness?
The most common formula for population skewness is γ₁ = μ₃ / σ³, where μ₃ is the third central moment and σ is the standard deviation. For a sample, skewness is often calculated as:
- Skewness = (1/n) Σ[(xᵢ − x̄)³] / s³
- x̄ = sample mean
- s = sample standard deviation
- n = number of observations
3. What does positive and negative skewness mean?
Positive skewness means the distribution has a longer right tail, while negative skewness means it has a longer left tail. In detail:
- Positively skewed: Mean > Median > Mode.
- Negatively skewed: Mean < Median < Mode.
- Symmetric: Mean = Median = Mode.
4. How do you calculate skewness step by step?
To calculate skewness, use the third standardized moment formula. Follow these steps:
- 1. Find the mean (x̄).
- 2. Compute each deviation (xᵢ − x̄).
- 3. Cube each deviation: (xᵢ − x̄)³.
- 4. Find the average of these cubed deviations.
- 5. Divide by s³ (standard deviation cubed).
5. What is Karl Pearson’s coefficient of skewness?
Karl Pearson’s coefficient of skewness measures skewness using mean, mode, and standard deviation. The formula is:
- Sk = (Mean − Mode) / Standard Deviation
- Sk = 3(Mean − Median) / Standard Deviation
6. What is Bowley’s coefficient of skewness?
Bowley’s coefficient of skewness is based on quartiles and is given by Sk = (Q₃ + Q₁ − 2Q₂) / (Q₃ − Q₁). Here:
- Q₁ = first quartile
- Q₂ = median
- Q₃ = third quartile
7. Can you give an example of skewness calculation?
Yes, skewness can be calculated using the formula based on mean and standard deviation. Example: Consider data 2, 3, 4, 10.
- Mean (x̄) = 4.75
- Standard deviation (s) ≈ 3.30
- Using the skewness formula gives a positive value
8. How does skewness affect the mean and median?
Skewness affects the mean more than the median because the mean is sensitive to extreme values. In a skewed distribution:
- Positive skew: Mean is pulled right (Mean > Median).
- Negative skew: Mean is pulled left (Mean < Median).
9. What is the difference between skewness and kurtosis?
Skewness measures asymmetry, while kurtosis measures peakedness or tail heaviness of a distribution. Specifically:
- Skewness: Direction and degree of asymmetry.
- Kurtosis: Sharpness of peak and thickness of tails.
10. Why is skewness important in data analysis?
Skewness is important because it shows whether data is symmetric or distorted, which affects statistical analysis and interpretation. It helps in:
- Choosing appropriate measures of central tendency.
- Checking normality assumptions.
- Identifying outliers and extreme values.
- Improving decision-making in business, finance, and research.
