BECS 184 Data Analysis Solved Question Paper June 2026 for IGNOU Bachelor of Arts (BAG, BAECH, BAM)

Steps in constructing a composite index, illustrated with the HDI

Question 4: Bring out the important steps in construction of a composite index. Illustrate your answer with the Human Development Index (HDI). (20)

Introduction. A composite index combines several separate indicators into one number that sums up a concept with many dimensions, such as development, poverty or well-being. No single variable can capture such a concept. Income alone tells us little about health or schooling, so a composite index brings different dimensions onto one comparable scale.

Composite indices are used to rank countries, states or districts and to track progress over time. The best-known example is the Human Development Index (HDI) published by the United Nations Development Programme (UNDP), and its construction follows the same general steps that any well-built composite index should follow.

Important steps in constructing a composite index

1. Defining the concept and objective. State clearly what the index will measure and why, then break the concept into dimensions. Human development, for example, is split into a long and healthy life, knowledge, and a decent standard of living. A vague concept always produces a vague index.
2. Selecting indicators. Each dimension needs one or more measurable indicators. Good indicators are relevant to the concept, available for every unit being compared, reliable over time, and not heavily overlapping. Too many closely correlated indicators quietly give one dimension extra weight.
3. Collecting data and treating gaps. Data are gathered from consistent sources, usually official statistics, for the same reference year. Missing values are filled by suitable imputation or the unit is dropped, and outliers are examined because extreme values can stretch the scale and distort every other score.
4. Normalisation. Indicators are measured in different units, such as years, rupees or percentages, so they must be converted to a common scale with no units. The main methods are:
   • Range equalisation (min-max): index = (actual value − minimum) ÷ (maximum − minimum), giving a score between 0 and 1.
   • Mean standardisation: each value is divided by the mean of that indicator.
   • Z-score: (value − mean) ÷ standard deviation.
   • Simple ranking: units are ranked on each indicator and the ranks are added or averaged.
   Direction also matters. For a negative indicator such as the infant mortality rate, the scale is reversed so that a higher score always means a better position.
5. Weighting. Weights decide how much each indicator or dimension contributes to the final score. Equal weights are simple and transparent. Weights can also be derived statistically, for example through principal component analysis, or from expert opinion. Since weights carry value judgements, the choice should be stated openly.
6. Aggregation. The normalised and weighted values are combined into one figure. An arithmetic mean allows full substitution, so a very high score in one dimension can hide a very low score in another. A geometric mean reduces this substitution and rewards balanced performance.
7. Testing robustness. The index is recalculated with a different normalisation method, weighting scheme or aggregation formula to see whether rankings change sharply. If the rankings stay broadly stable, users can trust the index more.
8. Presentation and interpretation. Results are presented as scores, ranks and categories, often with tables and maps, and interpreted against the original objective. A good index can also be broken back into its components, so users see which dimension pulls a score up or down.

Illustration: the Human Development Index

The HDI was introduced in UNDP's first Human Development Report in 1990. It was designed under the leadership of Mahbub ul Haq, drawing on Amartya Sen's idea that development means expanding people's capabilities, not only raising income. Each step above can be traced in its method.

Concept, dimensions and indicators (Steps 1 and 2). Human development is measured through three dimensions and four indicators, each with fixed minimum and maximum values called goalposts:

Normalisation (Step 4). The HDI uses range equalisation: dimension index = (actual − minimum) ÷ (maximum − minimum). For income, natural logarithms are used:

Income index = [ln(actual) − ln(100)] ÷ [ln(75,000) − ln(100)]

The log reflects that each extra dollar adds less to capabilities as income rises. The education index is the arithmetic mean of the two schooling indices, so both indicators inside the knowledge dimension count equally.

Weighting and aggregation (Steps 5 and 6). The three dimensions get equal weight. Since 2010 the HDI has been the geometric mean of the three dimension indices:

HDI = (I_Health × I_Education × I_Income)^1/3

Earlier versions used an arithmetic mean, along with adult literacy and gross enrolment instead of the present schooling indicators.

Worked illustration. Take a hypothetical country with life expectancy of 72 years, expected schooling of 12 years, mean schooling of 7 years and GNI per capita of 7,000 (2017 PPP $).

An arithmetic mean of the same three indices would give 0.669. The geometric mean is slightly lower because the education index lags behind health, which shows how the method penalises uneven progress across dimensions.

Presentation and interpretation (Step 8). UNDP places countries in four groups: very high human development (0.800 and above), high (0.700 to 0.799), medium (0.550 to 0.699) and low (below 0.550). Our hypothetical country, at 0.663, falls in the medium group, and its weakest dimension, education, points to where policy attention is needed.

Limitations. The HDI leaves out inequality, gender gaps, environmental damage and political freedom. UNDP therefore publishes companion measures, including the Inequality-adjusted HDI, the Gender Development Index and the Multidimensional Poverty Index, to fill some of these gaps.

Conclusion. Constructing a composite index means moving step by step from a clear concept to indicators, normalised scores, weights and a final aggregate that is tested and explained. The HDI shows these steps in practice and also shows why choices such as the geometric mean and the log of income shape the final ranking.

Why a sample survey may be preferred to a census

Question 9: Explain why a sample survey may be preferred to a census. (12)

Introduction. A census is the complete enumeration of every unit in the population, for example every household in a district. A sample survey collects data from only a part of the population, chosen by a scientific method, and uses the results to estimate the characteristics of the whole population. In most practical studies a sample survey is preferred, for the following reasons.

1. Lower cost. Fewer units are contacted, so spending on investigators, travel, printing and data processing falls sharply. For a large population, a census may simply be unaffordable.
2. Saving of time. Data from a sample can be collected, processed and published much faster. This matters when results are needed quickly, for example to estimate crop damage after a flood or to measure unemployment every quarter.
3. Greater accuracy. A smaller operation can use well-trained investigators, closer supervision and careful checking, so non-sampling errors such as wrong entries, non-response and interviewer bias are reduced. The sampling error that remains can be measured through the standard error, which is not possible for the hidden errors of a badly run census.
4. More detailed information. Because fewer units are covered, each respondent can be asked more questions in greater depth. Detailed questions on consumption, health or employment are practical in a sample but would overload a census.
5. Administrative convenience. A census needs a very large field staff, long training and complex coordination. A sample survey is far easier to organise, supervise and repeat at regular intervals.
6. Destructive testing. When testing destroys the item, a census is impossible. The life of electric bulbs, the strength of cement or the quality of tinned food can only be checked on a sample, otherwise nothing would remain to sell.
7. Infinite or inaccessible populations. Some populations cannot be fully counted, such as fish in a river, future output of a factory or people in remote areas. Here a sample survey is the only workable method.
8. Scientific basis. With random selection, the laws of statistical regularity and inertia of large numbers ensure that a reasonably large sample reflects the population. Results can be stated with a confidence interval, which tells users how reliable the estimate is.

India's National Sample Survey, running since 1950 and now part of the National Statistical Office, shows these advantages in practice. It estimates consumption, employment and other indicators for the whole country from carefully drawn samples of households.

When a census is still needed. A census remains necessary when the population is small, when information is required for every single unit (as in a voters' list or a full population count), or when the population is so varied that a small sample cannot represent it.

Conclusion. A sample survey is preferred because it is cheaper, faster, more detailed and often more accurate than a census, and because some studies cannot be done any other way. Its reliability, however, depends on a sound sampling design and an adequate sample size.