VINITHA'S RESEARCH AND STATISTICS NOTES
Normal
Distribution Or Gaussian Distribution Of Data
A normal distribution or Gaussian distribution of data is a statistical pattern that forms a symmetric, bell-shaped curve where most data points cluster around an average value, with fewer data points farther away from this average. It's a common way to describe how data is distributed in many real-world situations.
Symmetrical Distribution
A frequency distribution is said to be symmetrical if the frequencies are equally distributed on both the sides of central value. A symmetrical distribution may be either bell–shaped or U shaped.
Deviation from Normal Distribution
In the normal curve model, the mean, the median, and the mode all coincide and there is the perfect balance between the right and left values of the curve. However, the skewness and kurtosis coefficients can be used to measure how different a given distribution is from a normal distribution. Generally, two types of divergence occur in the normal curve: Skewness and Kurtosis
Skewness
Skewness refers to distortion or asymmetry in a symmetrical bell curve, or normal distribution, in a set of data. Real-life data rarely follow a perfectly normal distribution. The skewness measures the symmetry of a distribution.
The normal distribution is symmetric and has a skewness of 0. if the distribution of the data set has a skewness less than zero, or negative skewness, then the left tail of the distribution is longer than the right tail; positive skewness implies that the right tail of the distribution is longer than the left.
A distribution is said to be "Skewed when the mean and median fall at different points in the distribution and the balance i.e. the point of center of gravity is shifted to one side or the other to left or right.
Types of Skewness
There are two types of skewness which appear in the Normal Curve.: Negative skewness and positive skewness
Negative Skewness Distribution
Negative Skewness Distribution is said to be skewed negatively or to the left, when scores are massed at the high end of the scale, i.e., the right side of the curve, and are spread out gradually towards the low end i.e., the left side of the curve. In a negatively skewed distribution, the value of the median will be higher than that of the value of the mean.
Positive Skewness Distributions
Positive Skewness Distributions are skewed positively or to the right, when scores are massed at the low, i.e., the left end of the scale, and are spread out gradually toward the high or right end as shown in the figure below.
Comparison Between Types of Skewness
Kurtosis
The term Kurtosis refers to the divergence in the height of the curve, especially in the peakedness/ tailedness of a distribution. There are two types of divergence in the peakness of the curve and three types of kurtosis are leptokurtosis, platykurtosis and Mesokurtosis.
Leptokurtosis
In a Leptokurtic distribution, the frequency is more peaked at the center than in the normal distribution curve.
Example: Suppose you have a normal curve that is made up of a steel wire. Suppose you push both ends of the wire curve together. What would happen to the shape of the curve? Probably your answer may be that by pressing both the ends of the wire curve, the curve becomes more peaked i.e., its top becomes narrower than the normal curve and scatterness in the scores or area of the curve shrinks towards the centre.
Platykurtosis
A distribution of flatter peaks than the normal distribution is known as platykurtic distribution.
Example: Suppose we put a heavy pressure on the top normal curve made from the steel wire. What would be the change in the shape of the curve? Probably the top of the curve would become flatter than that of the normal.
Mesokurtosis
Mesokurtic distributions are the normal or symmetrical distributions. The values or scores are moderately distributed about the center of the distributions. It is neither too peaked nor too flat.
Value of Different Types of Kurtosis
· A kurtosis value = 3 (Mesokurtic) indicates a normal distribution.
· Kurtosis value > 3 indicates positive kurtosis (Leptokurtic)
· Kurtosis value <3 indicates negative kurtosis (Platykurtic)
COMPARISON BETWEEN SKEWNESS AND KURTOSIS
Aspect | Skewness | Kurtosis |
Definition | Measures the asymmetry of a distribution. | Measures the tailedness or peakedness of a distribution. |
Range | · Negative skew (left-skewed) · Zero skew (symmetric) · Positive skew (right-skewed) | · Negative kurtosis (light-tailed or platykurtic) · Zero kurtosis (mesokurtic or normal) · Positive kurtosis (heavy-tailed or leptokurtic) |
Interpretation | · Negative skew indicates a longer left tail, and data is concentrated on the right. · Positive skew indicates a longer right tail, and data is concentrated on the left. · Zero skew means the data is roughly symmetric. | · Negative kurtosis suggests that the distribution has thinner tails and is less peaked than a normal distribution. · Positive kurtosis suggests that the distribution has fatter tails and is more peaked than a normal distribution. · Zero kurtosis indicates a distribution similar to the normal distribution. |
Normal Distribution | A normal distribution has a skewness of 0. | A normal distribution has a kurtosis of 3. |
Application | Useful for understanding the direction of asymmetry in data. | Useful for understanding the shape of the tails and central peak of a distribution. |
Notation | Skewness is often denoted as "S" or "γ" (gamma). | Kurtosis is often denoted as "K" or "κ" (kappa). |
Conclusion
Skewness: Skewness measures the degree of asymmetry in a probability distribution.
Kurtosis: Kurtosis measures the shape of the tails and central peak of a probability distribution.
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