Statistics — Moments of a distribution
Published in
3 min readJun 6, 2020
Moments in statistics are popularly used to describe the characteristic of a distribution.
1 Moment: Measure of central location
2 Moment: Measure of dispersion
3 Moment: Measure of asymmetry
4 Moment: Measure of peakedness
First moment- Mean
Measure the location of the central point.
Second moment- Standard Deviation (SD, σ(Sigma)):
Measure the spread of values in the distribution OR how far from the normal.
Small SD : Numbers are close to mean
High SD : Numbers are spread outFor normal distribution:
Within 1 SD: 68.27% values lie
Within 2 SD: 95.45% values lie
Within 3 SD: 99.73% values lieAdvantages over Mean Absolute Deviation(MAD):
1. Mathematical properties- Continuous, differentiable.
2. SD of a sample is more consistent estimate for a population- When drawing repeated samples from a normally distributed population, the standard deviations of samples are less spread out as compare to mean absolute deviations.
Third moment- Skewness
Measure the symmetry in the distribution.
Skewness=0 [Normal Distribution, Symmetric]Other Formulas:
1. Skewness = (Mean-Mode)/SD
2. Skewness = 3*(Mean-Median)/SD
(Mode = 3*Median-2*Mean)Transformations (to make the distribution normal):
a. Positively skewed (right): Square root, log, inverse
b. Negatively skewed (left) : Reflect and square[sqrt(constant-x)],
reflect and log, reflect and inverse
Fourth moment- Kurtosis:
Measure the amount in the tails.
Kurtosis=3 [Normal Distribution]
Kurtosis<3 [Lighter tails]
Kurtosis>3 [Heavier tails]Other Formulas:
Excess Kurtosis = Kurtosis - 3Understanding:
Kurtosis is the average of the standardized data raised to fourth power. Any standardized values less than |1| (i.e. data within one standard deviation of the mean) will contribute petty to kurtosis.
The standardized values that will contribute immensely are the outliers.
High Kurtosis alerts about attendance of outliers.
References:
SD and variance: https://www.mathsisfun.com/data/standard-deviation.html
Advantages of the mean deviation: http://www.leeds.ac.uk/educol/documents/00003759.htm
