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.

Sharpe Ratio

Measure the risk-adjusted performance, i.e. excess return per unit of risk. by taking into account an investor can earn the risk-free rate. The higher values, the better the return-risk tradeoff.

Sharpe ratio = (r_p- r_f)/ σ_p

Where:
r_p – portfolio return
r_f – risk free rate
σ_p– portfolio standard deviation

Coefficient of variation (CV)

A normalized measure of dispersion of a probability distribution, only for non-zero mean and most useful for variables that are always positive. The lower an asset’s CV, the more attractive it is in risk per unit of return.

CV = σ / μ

Advantages:

• Enable comparison between different distribtuions and means

Disadvantages:

• When the mean is near zero, the coefficient of variation is sensitive to small changes in the mean, limiting its usefulness.
• Unlike the standard deviation, it cannot be used to construct confidence intervals for the mean.
  

Chebyshev’s inequality

% of observations within k standard deviation is at least:

1-1/k2
Where: k>1 regardless of the shape of the distribution, represents as the deviations of the mean.

Example:
If using k=2, 75% of observations lie within ±2 standard deviation of mean.

Variance and Standard Deviation

The variance and standard deviation are two very popular measures of variation. Their formulations are categorized into whether to evaluate from a population or from a sample.

Population variance
The average of the population’s squared deviations from the mean

σ² = Σ[f_i( x_i - μ)²]/ Σf_i

Where:
f_i - the frequency of the ith item;
x_i - the value of the ith item or class mark;
μ- the population arithmetic mean.

Population standard deviation σ is defined as σ = σ^1/2

Sample variance
s² = Σ[f_i( x_i - x_a)²]/( Σ f_i – 1) = Σ[f_i( x_i - x_a)²]/(n – 1)

Where: f_i = n

Sample standard deviation is the square root of sample variance.


My tips:

• The formula of variance and standard deviation are complicated. Better to use calculator to input the data and get the answer directly.
• Be remembered that the standard deviation is the square root of varaince.
  
  

Mean Absolute deviation (MAD)

Mean absolute deviation is the mean of the absolute values of all deviations from the mean.

MAD = Σ(f_i x_i - μ)/ Σf_i

Where:
f_i - the frequency of the ith item;
x_i - the value of the ith item or class mark;
μ- the arithmetic mean.

Range

Range is the difference between two extreme values. The range is easy to calculate but can not be obtained if open ended grouped data are given.

Range = Max. value – Min. Value