Symmetrical Distribution vs. Non- Symmetrical Distribution

Symmetrical distribution
Normal distribution is symmetric, bell shaped with mean = median = mode.
% of observations within ±1 σ is 66%; ±2σ is 95%; ±3 σ is 99%

Where: σ - standard deviation

Non-symmetrical distribution
The distribution is asymmetry where the data points are skewed either to the left or to the right.

Skewness
Important for finance and investing as most stock prices and asset returns are either positive or negative skewed rather than following normal distribution where skewness is zero.

The measure is free of units but preserves the sign of the deviation of the observation from the mean.

Absolute skewness
Absolute skewness equal to the sum of cubed deviations from mean:

1/n [Σf_i(X_i - X_mean)²]

Relative skewness (SK)
Sk= Skewness / Standard deviation³

Sk = (1/n) (ΣX_i-X_mean)³ / s³

If SK =0, not skewed;
If SK▷0,+ve skewed;
If SK◁0,-ve skwed

Where:

▷ - grearer than

◁ - smaller than

Positive skewness
· Many outliers in the upper region
· Right tail is long relative to left tail
· Mean ▷ median ▷mode;

Negative skewness
· Many outliers in lower region
· Left tail is long relative to right tail
· Mean ◁ median ◁ mode

Kurtosis
Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. Kurtosis is the likelihood that an event occurring is extreme in relation to a given distribution and critical in risk management.

The measure is free of units but is always positive regardless of sign of the deviation of the observation from the mean.

Leptokurtic
More peaked than normal distribution (fat tails) and larger probability of having large positive and negative deviations from the mean. Its true risk is higher than the risk suggested in normal distribtution.

Excess kurtosis▷0

Platykurtic
Less peaked than normal distribution. Data sets with low kurtosis tend to have a flat top near the mean rather than a sharp peak.

Excess kurtosis◁0

Excess Kurtosis = Kurtosis - 3;
kurtosis = (1/n) (ΣX_i-X_mean)⁴ / s⁴

Notes:
· Kurtosis of normal distribution is 3 and excess kurtosis is 0.