BSc CSIT (TU) Science Statistics I (BSc CSIT, STA164) Question Paper 2075 Nepal

Dispersion

Dispersion (or variation) measures the extent to which individual observations in a data set are scattered about a central value (mean/median). A small dispersion means the data are clustered closely around the average; a large dispersion means they are spread out. Dispersion describes the consistency, reliability, and homogeneity of a series.

Measures of Dispersion

1. Absolute measures (expressed in the same units as the data):

• Range $= L - S$ (largest value minus smallest value). Simplest but unstable.
• Quartile Deviation (Semi-IQR) $= \dfrac{Q_3 - Q_1}{2}$. Based on the middle 50% of data.
• Mean Deviation $= \dfrac{\sum f|x-\bar{x}|}{N}$. Average of absolute deviations from the mean (or median).
• Standard Deviation $\sigma = \sqrt{\dfrac{\sum f(x-\bar{x})^2}{N}}$. The most important and widely used measure.

2. Relative measures (unit-free, used to compare two series):

• Coefficient of Range $= \dfrac{L-S}{L+S}$
• Coefficient of Quartile Deviation $= \dfrac{Q_3-Q_1}{Q_3+Q_1}$
• Coefficient of Variation $C.V. = \dfrac{\sigma}{\bar{x}}\times 100\%$

Worked Computation (illustrative frequency distribution)

Mean: $\bar{x} = \dfrac{\sum fx}{N} = \dfrac{990}{40} = 24.75$

Standard deviation:

Coefficient of variation:

Comment on Consistency

The series whose C.V. is smaller is more consistent / uniform / stable, while a higher C.V. indicates greater variability. (When comparing two distributions, compute C.V. for each and conclude that the one with the lower C.V. is more consistent.) Here, a C.V. of about $47\%$ indicates a fairly high degree of variability in the data.

Method of Least Squares

The method of least squares fits a line (or curve) to data so that the sum of squares of the vertical deviations of observed points from the fitted line is a minimum. For the line of regression of $Y$ on $X$, written as

we minimise $S = \sum (Y_i - a - bX_i)^2$. Setting $\partial S/\partial a = 0$ and $\partial S/\partial b = 0$ gives the two normal equations:

Solving these gives:

Here $b = b_{YX}$ is the regression coefficient of $Y$ on $X$, the average change in $Y$ per unit change in $X$.

Worked Example (fitting $Y$ on $X$)

For data $X: 1,2,3,4,5$ and $Y: 2,4,5,4,6$:

$n=5,\ \sum X=15,\ \sum Y=21,\ \sum XY=71,\ \sum X^2=55$

Regression line: $\;Y = 1.8 + 0.8X$

Estimating Y for a given X

For example, at $X = 6$:

So the estimated value of $Y$ when $X=6$ is 6.6. (Substitute the actual $X$ asked in the question into the fitted line to obtain the estimate.)

Random Variable

A random variable is a real-valued function $X$ that assigns a numerical value to each outcome (sample point) of a random experiment, i.e. $X: S \to \mathbb{R}$. It is discrete if it takes a finite or countably infinite set of values (e.g. number of heads in tosses) and continuous if it can take any value in an interval (e.g. height, time).

Probability Mass Function (PMF)

For a discrete random variable $X$, the PMF is

satisfying (i) $p(x) \ge 0$ for all $x$, and (ii) $\sum_x p(x) = 1$. It gives the probability that $X$ equals each specific value.

Probability Density Function (PDF)

For a continuous random variable $X$, the PDF $f(x)$ satisfies (i) $f(x) \ge 0$, (ii) $\int_{-\infty}^{\infty} f(x)\,dx = 1$, and the probability over an interval is

For a continuous variable $P(X=x)=0$; only interval probabilities are meaningful.

Mean and Variance (worked example)

Let $X$ have the distribution:

Mean (Expectation):

$E(X^2)$:

Variance:

Standard deviation $= \sqrt{0.81} = 0.9$.

For a continuous distribution the same idea is used with integrals: $E(X)=\int x f(x)\,dx$ and $Var(X)=\int x^2 f(x)\,dx - [E(X)]^2$.

Dispersion is the degree to which the values of a data set are scattered or spread about a central (average) value. It indicates the consistency and homogeneity of the data.

Its measures are of two kinds:

• Absolute measures (same units as data): Range, Quartile Deviation, Mean Deviation, Standard Deviation.
• Relative measures (unit-free, for comparison): Coefficient of Range, Coefficient of Quartile Deviation, Coefficient of Mean Deviation, and Coefficient of Variation $\left(\dfrac{\sigma}{\bar{x}}\times 100\%\right)$.

A smaller dispersion means more consistent/uniform data; a larger dispersion means more variability.

The range is the simplest absolute measure of dispersion, defined as the difference between the largest ($L$) and smallest ($S$) values in a data set:

For the data $12, 15, 20, 8, 25$: largest $L = 25$, smallest $S = 8$.

Regression coefficients are the slopes of the two lines of regression; each measures the average rate of change of one variable per unit change in the other.

• Regression coefficient of $Y$ on $X$: $\;b_{YX} = r\dfrac{\sigma_y}{\sigma_x} = \dfrac{\text{Cov}(x,y)}{\sigma_x^2}$ — change in $Y$ per unit change in $X$.
• Regression coefficient of $X$ on $Y$: $\;b_{XY} = r\dfrac{\sigma_x}{\sigma_y} = \dfrac{\text{Cov}(x,y)}{\sigma_y^2}$ — change in $X$ per unit change in $Y$.

Important properties:

• $r = \pm\sqrt{b_{YX}\cdot b_{XY}}$, so the correlation coefficient is the geometric mean of the two regression coefficients.
• Both have the same sign as $r$.
• Their product cannot exceed 1: $b_{YX}\cdot b_{XY} \le 1$.

The multiplication theorem of probability gives the probability of the joint occurrence of two events.

For dependent events $A$ and $B$ (general form):

where $P(B\mid A)$ is the conditional probability of $B$ given that $A$ has occurred.

For independent events, $P(B\mid A) = P(B)$, so the theorem reduces to:

This extends to $n$ events: $P(A_1\cap A_2\cap\dots\cap A_n) = P(A_1)P(A_2\mid A_1)\cdots P(A_n\mid A_1\cap\dots\cap A_{n-1})$.

A random variable is a real-valued function $X$ that assigns a numerical value to every outcome of a random experiment, i.e. $X: S \to \mathbb{R}$ where $S$ is the sample space.

It is classified as:

• Discrete random variable — takes a finite or countably infinite number of distinct values (e.g. number of heads when a coin is tossed three times: $0,1,2,3$).
• Continuous random variable — can assume any value within an interval (e.g. height, weight, time).

Example: Tossing two coins, let $X$ = number of heads. Then $X$ takes values $0, 1, 2$ with probabilities $\tfrac14, \tfrac12, \tfrac14$.

Correlation describes the direction and strength of the linear relationship between two variables.

• Positive correlation: both variables move in the same direction — as one increases, the other also increases (and as one decreases, the other decreases). The correlation coefficient $r$ lies in $(0, +1]$. Example: height and weight; income and expenditure.

• Negative correlation: the two variables move in opposite directions — as one increases, the other decreases. Here $r$ lies in $[-1, 0)$. Example: price and quantity demanded; speed and time taken for a fixed distance.

$r = +1$ is perfect positive and $r = -1$ is perfect negative correlation.

The median is the middle value of a data set when the observations are arranged in ascending (or descending) order.

Arrange the data $7, 3, 9, 5, 11$ in ascending order:

There are $n = 5$ (odd) observations, so the median is the $\left(\dfrac{n+1}{2}\right)^{\text{th}} = 3^{\text{rd}}$ value.

The probability density function (PDF) is the function $f(x)$ that describes the distribution of a continuous random variable $X$. It satisfies:

1. $f(x) \ge 0$ for all $x$ (non-negative),
2. $\displaystyle\int_{-\infty}^{\infty} f(x)\,dx = 1$ (total area under the curve equals 1),
3. The probability that $X$ lies in an interval is the area under the curve:

Note that for a continuous variable $P(X = x) = 0$; the PDF gives probabilities only over intervals, not at single points. The PDF is the derivative of the cumulative distribution function: $f(x) = \dfrac{d}{dx}F(x)$.

A scatter diagram (scatter plot) is a graphical method of representing bivariate data, in which each pair of observations $(x_i, y_i)$ is plotted as a point on the $XY$-plane (with $X$ on the horizontal axis and $Y$ on the vertical axis).

It gives a quick visual idea of the nature, direction, and degree of correlation between the two variables:

• If points cluster around an upward-sloping line → positive correlation.
• If points cluster around a downward-sloping line → negative correlation.
• If points lie exactly on a line → perfect correlation ($r = \pm 1$).
• If points are scattered randomly with no pattern → little or no correlation.

It is simple to draw and is usually the first step before computing a numerical correlation coefficient.