৪৯তম বিসিএস ⎯ পরিসংখ্যান [৯৮১]

Trend, seasonal and cyclic are classical components of time series. Regression is a statistical technique, not a component of time series.
  

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​Source: Live MCQ class lecture, Geekforgeeks.

Secular Trend or normally "Trend" reflects persistent long-term growth or decline in a series over several years, such as population growth.

When we talk of trend, we mean smooth, regular, long term movement of the data- sudden and erratic movements either in upwards or in downward direction have nothing to do with the trend.  
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Source: Business Statistics (SP Gupta, MP Gupta)

There are broadly two types of trend -Linear trend, Non-linear trend.

For measuring straight line trend, we use Semi average method, Method of least square etc.

On the other hand, Moving average method is used to measure non-linear trend.

The moving average method is a technique used to analyze data trends by calculating the average of a specific set of data points over a defined period, which is then "moved" through the data set.

Moving averages smooth out short-term fluctuations, isolating the long-term trend.

​Source:  Business Statistics (MP Gupta, SP Gupta), Investopedia.

Increasing sample size primarily minimizes sampling error.

Sampling error is the difference between a sample statistic and the true population parameter.

Larger samples are more representative of the population, leading to more accurate estimates and reduced variability.

So, Random sampling error decreases as sample size becomes larger.​

Deseasonalizing time series data involves removing the seasonal component to reveal underlying trends and make it easier to analyze the data.

This process is useful for identifying patterns and making predictions, especially when seasonal variations obscure the true trend. 

Cyclic variation in time series data refers to recurring upward or downward movements that are not of a fixed period and typically last longer than a year.

These fluctuations are distinct from seasonal variations, which are regular and predictable patterns within a year.

A classic example of cyclic variation is the business cycle, which involves periods of expansion, peak, contraction, and trough. 

​Seasonal variations repeat within a year, while cyclical variations last longer than a year.
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Source: https://moirabaricollege.com.in
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A time series can be broken down into four key components: trend, seasonality, cyclicality and irregularity.

These components help to understand patterns and make predictions about future values in a dataset.
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Irregular variation represents unpredictable, random shocks like wars or natural disasters.
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Source: Live MCQ lecture pdf
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Every sampling unit gets equal probability in SRS.

This sampling might give erroneous result when population is heterogeneous.

In this situation, we use Stratified random sampling. 
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Source: Business Statistics (Md. Abdul Aziz.) 

In SRS, each individual gets equal chance of being selected.

But this may give erroneous result.

To increase precision, stratified sampling divides heterogeneous population into homogeneous strata.
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Source: Live MCQ class. 

A starting point is selected at random, then every k-th unit is chosen.

This sampling method is also called quasi--random sampling because only the first element is selected at random, others are not.

Systematic sampling has two types- Linear, circular. 
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Source: Business Statistics (Md. Abdul Aziz)

In judgmental or purposive sampling, researcher selects samples based on judgment; hence, probability of selection not known.  

​Advantage:
a) Efficiency and Cost-Effectiveness,
​b) It allows researchers to focus on specific individuals or groups with desired characteristics or expertise. 
c) Useful for Exploratory Research such as pilot survey. Judgmental sampling can be valuable in pilot studies or when gathering preliminary data. 
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Disadvantage:  
​a) High Risk of Bias
​b) Subjectivity: The quality of the sample relies heavily on the researcher's judgment, which can be subjective and potentially inaccurate. 
​c) Difficult to Assess Representativeness: It's challenging to determine how well the sample reflects the overall population. 

Sampling is only a tool which helps to know the characteristics of the population by examining only a small part off it.

Sampling enables efficient study of large populations cost-effectively. It saves cost, time. 

Firstly, the researcher choose cluster (tract), then sample units inside cluster (households).
This type of sampling is called two stage sampling. 

​Source: Live MCQ lecture pdf

By separating population into homogeneous strata, variability is reduced in stratified sampling. It gives us→ higher precision for same cost. 
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Source: Business Statistics (SP Gupta,  MP Gupta.) 

Stage1 fraction = 4/20 = 1/5

Stage2 fraction = 5/40 = 1/8 

​Overall sampling fraction= 1/5 ×1/8 = 1/40

we know,
The number of smoothed points can be determined using the following formula: 
Number of smoothed points=Total number of data points−Moving average period+1
​Smoothed points 
​= 10 – 5 + 1 = 6

n_h = n × ( N_h / N)  
​= 100×300/1000 =30. ​

( Here, n= sample size=100,
​N_h=  N2= 300
​N= N1+N2+ N3=100)
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​source: Live MCQ class
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Stratified sampling is useful when the population can be divided into distinct, internally homogeneous but mutually heterogeneous subgroups called strata (like gender, age groups, income levels).

Sampling is done separately from each stratum to improve precision and ensure representation.

If the population is perfectly homogeneous (A), SRS is sufficient. Lack of sampling frame (C) blocks both methods. 

Time/cost (D) is not the key consideration.

Cluster sampling is most efficient when- 

The population is geographically scattered, making it costly and impractical to list or reach every unit individually.

There exist naturally occurring clusters (like villages, schools, hospitals) where units within a cluster are similar, but clusters differ from each other.

Option A correctly reflects this situation and captures the practical motivation: cluster sampling minimizes cost and logistical complexity by sampling groups (clusters) rather than every unit directly.

Option B favors simple random sampling, where listing every unit is easy.

Option C is more aligned with stratified sampling, which increases precision by controlling variability within strata.

Option D favors SRS, since low cost of contacting units removes the need for clustering.

Purposive sampling is a type of non-probability sampling where the researcher intentionally selects units based on specific characteristics or their own judgment, believing those units are most representative of the population.
Since selection is subjective and not based on chance, results may suffer from selection bias and are not generalizable to the whole population.

Options A, B, and C are probability sampling methods, where each unit has some known chance of selection — which helps reduce bias.

In contrast, non-probability sampling methods like purposive, convenience, and quota sampling do not rely on randomness, increasing the potential for systematic bias.

​Source: Live MCQ class

Non-sampling errors occur regardless of sample size or sampling technique—they relate to defects in data collection process (e.g., data entry error, response bias, measurement mistakes).

Option B clearly reflects that.

Options A, C, and D are flaws related to the sampling design itself, hence are sampling errors.

​Source: Live MCQ lecture pdf

A parameter represents a fixed but usually unknown characteristic of the population (e.g., population mean μ).

A statistic (e.g., sample mean x^-) is a value calculated from sample data and used to infer the parameter.

Note that statistics vary from sample to sample (sampling variability), while the parameter is fixed.

Multistage sampling involves selecting samples in multiple stages (e.g., selecting districts → villages → households).

It is practical when a complete list of population is not available and direct sampling is costly or difficult due to wide geographic spread.

It reduces field cost without compromising representation much.

​Source: Business Statistics (SP Gupta, MP Gupta)

Snowball sampling is a non-probability sampling method where existing subjects recruit future subjects among their acquaintances.
Used when population is difficult to locate or stigmatized (e.g. drug users, sex workers). 
It “snowballs” as sample grows through social networks.
It does not use randomization, so risk  of bias.
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Sampling error arises due to observing a sample instead of population, so it is quantifiable using probability theory, and can be reduced by larger/sample probability sampling.

Non-sampling errors (like response bias, measurement flaws) are harder to detect or measure precisely.

Options A and D are incorrect.

Option C misunderstands inference: parameter estimates rely directly on sample statistics.

In a 5-year moving average, the average is taken over 5 consecutive years to smooth out short-term fluctuations.

Since it is a centered moving average, the trend value is plotted at the middle year, but in practice, the calculated trend lags behind the actual series by half the period (here 2 years) when interpreted for forecasting or comparison.

Option A is incorrect because the first two and last two years cannot have 5-year averages.

Option C is partly true but misleading: moving average removes short-term irregularities, including some seasonal effects, but not all.

Option D is conceptually wrong; trend values depend on the average of consecutive points, not just first and last year.

The Semi-Average Method splits the series into two equal halves, computes the average of each half, and connects these averages with a straight line.

This inherently assumes that the trend is linear, which can lead to inaccurate estimates if the actual trend is non-linear or exponential.

Option A is not strictly true; the method works even for small series.

Option C is false; the method works for both odd and even numbers of observations (splitting may be slightly adjusted for odd-length series).

Option D is incorrect; semi-average provides trend estimates, not exact forecasts.
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​Source: Business Statistics (SP Gupta, MP Gupta.), Live MCQ class lecture

Step-1: Since we have even number (6) of years, split into two equal parts:
First Half (2016-2018):
Values = 20, 24, 27 → Mean = (20+24+27)/3 = 23.67
Second Half (2019-2021):
Values = 32, 39, 43 → Mean = (32+39+43)/3 = 38

Step-2: These represent average trend at the “middle year” of each half →
Trend value at 2017 = 23.67
Trend value at 2020 = 38

Step-3: Fit a straight line between (2017,23.67) and (2020,38) and interpolate for 2019.
​​Slope = (38-23.67)/ (2020-2017)= 14.33/3=4.77
​Change from 2017 to 2019 is 2 years → Trend at 2019 =
23.67+(4.777×2)=23.67+9.554=33.224

Additive model: Y=T+S+C+I -the seasonal effect is constant, independent of trend level.

Multiplicative model: Y= T×S×C×I — seasonal variation increases as the trend level rises, making it suitable for series with proportional fluctuations.

Option A is wrong: additive assumes constant seasonal variation.

Option B is wrong: multiplicative depends on trend level, not independent.

Option D is wrong: the choice of model affects the trend and seasonal decomposition, so results differ.
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Source: "Time Series and Official Statistics" by MS University

ARIMA is widely used for forecasting time series where trends or autocorrelations exist, after making the series stationary.
ARIMA stands for Auto Regressive Integrated Moving Average, where:

AR (p): Number of autoregressive terms
I (d): Number of differences required to make the series stationary
MA (q): Number of moving average terms

Option A is incorrect: ARIMA can handle non-stationary series after differencing (the “I” part).

Option C is partly true; ARIMA can model non-seasonal series, while SARIMA extends it to include seasonality.

Option D is wrong: p, d, q are autoregressive order, differencing order, and moving average order, not population/deviation/quantile.
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​Source:  "Introduction to Time Series and Forecasting" by Brockwell and Davis

Time series forecasting projects future values assuming continuation of historical patterns.

Number of MA values = n – k + 1 = 5 – 3 + 1 = 3.

In the multiplicative model components multiply; additive means they sum.

Source: Business Statistics (SP Gupta, MP Gupta.) 

In Simple Random Sampling, complete sampling frame is required to give each unit equal chance.
Then we select each unit randomly without any condition applied.
We use lottery method or random number generation method to select sample. 

Source: Live MCQ class lecture.

Additive model: Y = T + S + C + I.

Here, Seasonal index 1.2 → seasonal effect = +20. (.2 gives us 20%)

So C = Y – (T+S) = 140 – (110+20) = 10. (Random effect=0 here).

Holt–Winters (multiplicative) handles both trend and seasonal effects especially for quarterly/ monthly data.

Eliminating trend from time series data is called detrending.
While eliminating seasonality from time series data is called deseasoning.
Additive model is a model where components add up.

Source: Business Statistics (Md. Abdul Aziz) 

Trend can be measured by mainly 4 methods--- 
1. Graphical method, 
2. Method of Semi-averages, 
3. Method of moving averages, 
4. Method of Least square.

Source: Business Statistics (Md. Abdul Aziz) 

The semi-average method involves dividing the data into two equal parts and finding averages.

It is easy and quick to compute, but ignores seasonality, cycles, and random variation, hence less accurate for long-term forecasts.

Moreover, this method is effected by extreme values.

Least squares gives an equation of trend, which is its strength.

But its limitation is that if the chosen form (e.g., straight line) does not actually fit the data, the projection will be misleading.

When the data show a clear linear growth, the least squares method provides the best fit through a line equation.

This method makes us able to obtain the rate of growth per annum.

Semi-average and graphical are rough estimates, and moving average smooths fluctuations but does not provide an explicit mathematical equation.

Source: Live MCQ class, Business Statistics (Md. Abdul Aziz) 

The moving average method is strong at removing seasonal/irregular effects and showing the underlying trend.

However, it does not provide a mathematical equation and is less suitable for long-term forecasting.

Source: Live MCQ lecture pdf

The graphical method is a visual approach where a trend line is drawn by inspection.

Its strength is simplicity, but its limitation is subjectivity — different people may draw slightly different lines, leading to inconsistent forecasts.

Source: Live MCQ class.

In the ratio-to-moving average method, the actual values are divided by their corresponding moving average (which represents the trend + cycle).

This ratio gives seasonal effect, which can be averaged to get the seasonal index used for deseasoning.

Source: Online notes Nepal

Actual = Deseasonalised ÷ Seasonal Index
            → 200 / 0.8 = 250.

Multiplicative model → Y = T × S × C × I;

cyclic-irregular factor= C×I

C×I = Y/(T×S)= 100 / (90×1.25) = 0.888 ≈ 0.89.

Actual = Deseasonalised × Index
            = 300 ×1.20
            = 360.

The simple averages method calculates seasonal indices by averaging values in each season (e.g., monthly/quarterly).

While simple, it ignores trend influence, which may bias the seasonal index if data has strong upward/downward trends.

Source: Business Statistics (Md. Abdul Aziz) 

Seasonal variation refers to periodic changes repeating within 12 months (e.g. ice cream sales in summer).

Source: Live MCQ class

Deseasoning removes seasonal fluctuations so the underlying trend and cycle can be clearly analyzed.

Random errors are not eliminated, and while ARIMA may need stationary data, deseasoning mainly focuses on removing seasonal impact.