Statistics handouts by Sir Adnan Sial

1. Properties of Normal Distribution
i. The curve is a bell shaped and has a single peak.
ii. The mean of a normally distributed population lies at the center of its normal curve.
iii. Because of the symmetry of the normal probability distribution, the mean, median and
mode are the same value.
iv. The two tails of the normal probability distribution extend indefinitely and never touch the
horizontal axis. (Graphically it is not possible to show).
v. No matter what the values of μ and δ are, areas under normal curve remain in certain fixed
proportions within a specified number of standard deviations on either side of μ. For
example
a. μ ± δ will always contain 68.26%
b. μ ± 2δ will always contain 95.44%
c. μ ± 3δ will always contain 99.7%
Assumptions for OLS
i. Error term i.e. ε is a random variable.
ii. E(εi) = 0 i.e. The expected value of error term is zero. It implies that the expected value of
Y is related to X in the population by a straight line.
iii. Var (εi) = E (εi2) = δ2 for all i. i.e. the variance of error term is constant. It means that
distribution of error has same variance for all values of X (Homoscedasticity assumption).
iv. E ( εi, εj) = 0 for all i ≠ j i.e. error terms are independent of each other (assumption of no
serial or auto correlation b/w εs)
v. E (X, εi) = 0 i.e. X and ε are independent of each other.
vi. εis are normally distributed with a mean of zero and a constant Variance δ2. This implies
that Y values are also normally distributed. The distribution of y and ε are identical except

2. that they have different means. This assumption is required for estimation and testing of
hypothesis on linear regression.
Properties of Ordinary Least Squares:
i. The least squares regression line always goes through the point (Ȳ, ), the means of the
data.
ii. The sum of the deviations of the observed values Yi from the least squares regression line
is always equal to zero, i.e. Ʃ (Yi – Ŷ) = 0.
iii. The sum of squares of the deviations of the observed values from the least-squares
regression line is a minimum, i.e. Ʃ (Yi – Ŷ)2 = minimum
iv. The least squares regression line obtained from a random sample is the line of best fit
because a and b are the unbiased estimates of the parameters α and β.