BSc CSIT (TU) Science Statistics I (BSc CSIT, STA164) Question Paper 2081 Nepal

Index Numbers

Definition. An index number is a statistical measure designed to show changes in a variable (or a group of related variables) such as price, quantity, or value over time, geographically, or against some other characteristic. It is expressed as a percentage relative to a chosen base period taken as 100. Index numbers are sometimes called economic barometers because they measure the relative change in economic activity.

Methods of Constructing Weighted Index Numbers

Let $p_0, q_0$ be the price and quantity in the base year and $p_1, q_1$ in the current year.

1. Laspeyres' Price Index (base-year weights):

Uses base-year quantities as weights. It tends to overstate the rise in prices.

2. Paasche's Price Index (current-year weights):

Uses current-year quantities as weights. It tends to understate the rise in prices.

3. Fisher's Ideal Index (geometric mean of the two):

It is called ideal because it satisfies both the time-reversal and factor-reversal tests.

Numerical Illustration

Compute the required sums:

• $\sum p_0 q_0 = 16+50+56+38 = 160$
• $\sum p_1 q_0 = 32+60+70+38 = 200$
• $\sum p_0 q_1 = 12+25+40+26 = 103$
• $\sum p_1 q_1 = 24+30+50+26 = 130$

Laspeyres: $P^{L}=\dfrac{200}{160}\times100 = 125$

Paasche: $P^{P}=\dfrac{130}{103}\times100 = 126.21$

Fisher: $P^{F}=\sqrt{125\times126.21}=\sqrt{15776.25}=125.60$

Result. Prices have risen by about 25.6% over the base year (Fisher's ideal index $=125.60$).

Components of a Time Series

A time series is a set of observations recorded at successive, usually equally spaced, points of time. Its variations are classified into four components:

1. Secular Trend (T): the long-term, smooth, general tendency of the data to increase or decrease over a long period (e.g., steady growth in population).
2. Seasonal Variation (S): regular, periodic fluctuations that complete themselves within a year or less, caused by seasons, customs, or festivals (e.g., higher sales during festivals).
3. Cyclical Variation (C): wave-like movements lasting more than a year, repeating in oscillating cycles of prosperity, recession, depression, and recovery (the business cycle).
4. Irregular / Random Variation (I): unpredictable, erratic fluctuations due to chance causes such as floods, strikes, or earthquakes.

These combine in the multiplicative model $Y = T\times S\times C\times I$ or the additive model $Y = T+S+C+I$.

Method of Moving Averages

This method smooths a series by replacing each value with the average of a fixed number ($k$, the period) of surrounding values, thereby averaging out short-term fluctuations and exposing the trend.

For a period $k$, the moving average centered at $t$ is:

For an odd period the averages fall against existing time points. For an even period (e.g., 4 yearly), a second 2-period averaging (centering) is applied so values align with the original periods.

Merits: simple, flexible, no rigid assumption about trend shape. Demerits: trend values cannot be obtained for the first and last $(k-1)/2$ periods; not expressible as a mathematical equation; cannot be used for forecasting.

Method of Least Squares

This fits a mathematical trend line $Y_c = a + bX$ such that the sum of squared deviations $\sum (Y - Y_c)^2$ is minimum. The constants are found from the normal equations:

If the time origin is chosen at the middle so that $\sum X = 0$, these simplify to:

Merits: gives a unique, objective trend line usable for forecasting and obtainable for every period. Demerits: rigid (adding new data changes the whole line); assumes a fixed mathematical form for the trend.

Binomial Distribution

A discrete random variable $X$ follows the binomial distribution if it counts the number of successes in $n$ independent Bernoulli trials, each with constant probability of success $p$ (and $q=1-p$). Its probability mass function is:

with mean $= np$ and variance $= npq$.

Poisson Distribution

A discrete random variable $X$ follows the Poisson distribution if it counts the number of occurrences of a rare event in a given interval of time or space, with constant average rate $\lambda$. Its p.m.f. is:

with mean $=$ variance $= \lambda$.

Distinction

Poisson as a Limiting Case of the Binomial

Let $n\to\infty$ and $p\to 0$ such that $np = \lambda$ remains finite. Then in the binomial p.m.f. substitute $p = \lambda/n$:

Group the terms:

As $n\to\infty$: the first factor $\to 1$, $\left(1-\frac{\lambda}{n}\right)^{n}\to e^{-\lambda}$, and the last factor $\to 1$. Therefore:

which is the Poisson distribution. Hence the Poisson is the limiting form of the binomial when $n$ is large, $p$ is small, and $np=\lambda$ is moderate.

An index number is a specialized statistical measure that expresses the relative change in the magnitude of a variable (or group of variables) — such as price, quantity, or value — over time or space, with respect to a chosen base period taken as 100. For example, a price index of 125 means prices have risen 25% above the base period. Because they summarize economic change, index numbers are often called economic barometers. They are typically computed as $I_{01}=\dfrac{\sum p_1}{\sum p_0}\times 100$ (simple aggregative) or by weighted methods.

A time series has four components:

1. Secular Trend (T): the long-term general tendency of the data to rise or fall over a long period.
2. Seasonal Variation (S): regular, short-term periodic fluctuations completing within a year (due to seasons, festivals, customs).
3. Cyclical Variation (C): wave-like oscillations spread over more than a year (business cycles of boom and slump).
4. Irregular / Random Variation (I): unpredictable fluctuations caused by chance events (floods, strikes, wars).

They combine as $Y = T\times S\times C\times I$ (multiplicative) or $Y = T+S+C+I$ (additive).

Fisher's Ideal Index is the geometric mean of the Laspeyres and Paasche index numbers:

where $p_0,q_0$ are base-year price and quantity and $p_1,q_1$ are current-year price and quantity. It is called ideal because it uses both base- and current-year weights and satisfies the time-reversal test ($P_{01}\times P_{10}=1$) and the factor-reversal test ($P_{01}\times Q_{01}=\dfrac{\sum p_1 q_1}{\sum p_0 q_0}$).

The method of moving averages is a technique for measuring the trend of a time series by smoothing out short-term fluctuations. Each observation is replaced by the arithmetic mean of a fixed number $k$ of consecutive values (the period of the moving average):

When $k$ is odd, the averages line up directly with existing time points. When $k$ is even, a further 2-item averaging (centering) is done so values align with original periods. As the period increases, the smoothed series shows the trend more clearly.

Limitations: trend values are lost for the first and last few periods, it gives no equation, and it cannot be used for forecasting.

The Poisson is the limiting case of the binomial when $n\to\infty$, $p\to0$, with $np=\lambda$ finite.

For a binomial distribution with $n$ independent trials and success probability $p$ (with $q=1-p$), the probability mass function is $P(X=x)=\binom{n}{x}p^{x}q^{n-x}$.

Mean:

Variance:

The standard deviation is $\sigma=\sqrt{npq}$. Since $0<q<1$, the variance $npq$ is always less than the mean $np$; that is, for the binomial distribution mean > variance.

A secular trend (or simply trend) is the smooth, regular, long-term movement of a time series that shows the general tendency of the data to increase, decrease, or remain stable over a long period of time. It reflects the cumulative effect of persistent forces such as population growth, technological change, or improved productivity, rather than short-term seasonal or random factors.

For example, a steady rise in a country's population or GDP over several decades is a secular trend. It can be measured by the freehand (graphic) method, the method of moving averages, the method of semi-averages, or the method of least squares.

Both are weighted aggregative price index numbers. Let $p_0,q_0$ be base-year price and quantity and $p_1,q_1$ be current-year price and quantity.

Laspeyres' Index (base-year quantities as weights):

It measures the change in cost of buying the base-year basket at current prices and tends to overstate price rise.

Paasche's Index (current-year quantities as weights):

It measures the change in cost of the current-year basket and tends to understate price rise.

The geometric mean of the two gives Fisher's ideal index.

For a binomial distribution, the mean is $\mu = np$.

Given $n = 10$ and $p = 0.4$:

The mean of the distribution is 4.

(For completeness, the variance is $\sigma^2 = npq = 10\times0.4\times0.6 = 2.4$.)