Use of Statistical Tools — Study Notes

1. Introduction

The earlier chapters of this book introduced the tools of statistics - measures of central tendency, measures of dispersion, correlation and index numbers - and explained how data are collected, organised and presented. This chapter shows how these tools are actually used to solve problems. It deals with the application of statistical methods to real data, with examples drawn from the measurement of inflation, the analysis of consumption expenditure, the comparison of sectors of the economy, and the drawing of meaningful conclusions from the economic data of India.

Statistics is a means to an end: the end is the understanding and solution of economic problems. The tools of statistics become meaningful only when they are applied. In this chapter we take up numerical examples - computing averages, measuring dispersion, examining correlation between variables, and constructing index numbers - and interpret the results. We also learn the correct way to report results and to recognise the limitations of the conclusions that can be drawn from data.

The chapter emphasises the importance of choosing the right tool for the right purpose. Averages summarise data, dispersion measures the reliability of the average, correlation reveals association between variables, and index numbers measure changes over time. Used together, these tools enable the economist to convert raw numbers into informed judgement and sound policy.

2. Choosing the Appropriate Statistical Tool

The selection of a statistical measure depends on the nature of the problem, the type of data available, and the purpose of the enquiry. The general guidelines are:

1. Central tendency: To get a single representative value, use the arithmetic mean when the data are symmetric and free from extreme values; use the median when the data are skewed or have open-end classes; use the mode when the most common value is needed.
2. Dispersion: Use the range for a quick idea of spread, the quartile deviation when extreme values are present, the standard deviation when precise and algebraically useful measures are needed, and the coefficient of variation to compare variability across different series.
3. Correlation: Use Karl Pearson's coefficient for quantitative linear data and Spearman's rank correlation for ranked or qualitative data.
4. Index numbers: Use simple methods for homogeneous items and weighted methods such as Laspeyres, Paasche or Fisher when items differ in importance.

3. Application 1: Measuring Inflation and the Cost of Living

The most common application of statistical tools is the measurement of price changes. The Wholesale Price Index and the Consumer Price Index, studied in the previous chapter, are constructed by applying the formulas of index numbers to the price data of a representative basket of commodities.

For example, suppose we wish to measure the rise in the cost of living of a household. We select the items in its consumption basket, obtain the base year and current year prices, and apply the weighted aggregative method with the household's consumption pattern as weights. The resulting index tells us the percentage by which the household's expenditure must rise to maintain the same standard of living.

The tools of central tendency are also applied here. The average price of a group of commodities, computed as a weighted mean, gives a quick idea of the general price level. The dispersion of individual price changes around this average - measured by the standard deviation - shows whether the price rise is uniform across items or concentrated in a few commodities. If the dispersion is large, some essential commodities may be rising much faster than the average, which is important for policy.

4. Application 2: Analysing Consumption Expenditure

The National Sample Survey collects data on the monthly per capita consumption expenditure (MPCE) of households. These data are analysed using the tools we have studied:

1. Measures of central tendency: The mean MPCE tells us the average level of living; the median MPCE tells us the level below which half the households live; the mode shows the most common expenditure level.
2. Measures of dispersion: The standard deviation and coefficient of variation of MPCE across households measure the inequality of consumption. A larger CV implies greater inequality.
3. Correlation: The relationship between household income and consumption expenditure can be measured by the coefficient of correlation. Keynesian economics predicts a strong positive correlation between the two.
4. Presentation: The data are presented as frequency distributions, histograms and ogives, which we studied in the presentation chapter.

5. Application 3: Comparing Sectors of the Economy

Statistical tools are used to compare the performance of the agricultural, industrial and service sectors, or to compare the Indian economy with other economies. The comparisons use:

1. Relative measures of dispersion: The coefficient of variation allows comparison of the variability of output of two sectors even when their means and units differ. The sector with the smaller CV is more stable.
2. Index numbers: Index numbers of industrial production and agricultural output show the growth of each sector over time.
3. Measures of central tendency: Average productivity, average wages and average growth rates are computed for different sectors and regions and compared.
4. Correlation: The correlation between inputs and outputs - for example, between the area under irrigation and agricultural output - is examined to guide investment decisions.

6. Application 4: Analysing Distribution of Income and Wealth

The tools of dispersion are used to measure inequality in the distribution of income and wealth. A common procedure is the construction of the Lorenz curve, which plots the cumulative percentage of income against the cumulative percentage of households. The greater the deviation of the Lorenz curve from the line of equal distribution (the 45-degree line), the greater the inequality.

The following statistical steps are used in such an analysis:

1. Arrange the income data in ascending order and compute cumulative frequencies and cumulative incomes.
2. Present the data as cumulative frequency series (less than series).
3. Plot the Lorenz curve on a graph.
4. Measure dispersion to quantify inequality.

A related measure is the comparison of the mean and the median. When the mean exceeds the median substantially, the distribution is positively skewed, indicating that a few very high incomes pull up the average.

7. Application 5: Forecasting and Decision Making

Business firms and governments use statistical tools for forecasting. The analysis of time-series data - the trend and seasonal variations in sales, prices and output - helps in forecasting future values. Index numbers adjust historical data for price changes, and correlation analysis identifies the variables that most influence the quantity being forecast.

In decision making, statistical tools help in comparing alternatives. For example, a firm comparing the profits of two plants may find that although their average profits are equal, one plant is much more variable (larger standard deviation). The firm would prefer the plant with the smaller variability because it is more dependable. Similarly, an investor choosing between two securities would prefer the one with the lower coefficient of variation for the same average return.

8. Interpreting Results and Their Limitations

The final and most important step is the correct interpretation of statistical results. The following cautions must be observed:

1. Statistical results are valid only for the data used; extending them to other populations requires care.
2. Correlation does not imply causation.
3. Averages can be misleading if the dispersion is large; always report a measure of dispersion alongside an average.
4. The choice of the base year, weights and method affects the value of an index number.
5. Sample results are subject to sampling errors; the reliability of estimates depends on the size and design of the sample.
6. Statistical evidence must be combined with economic reasoning; statistics never decides policy by itself.

Quick Revision Tables

Mind Map

Important Diagrams (SVG)

Diagram 1: The Statistical Analysis Workflow

Diagram 2: Lorenz Curve of Income Distribution

Common Mistakes

1. Reporting an average without a measure of dispersion; the same mean can hide very different levels of spread, and the average may not be representative.
2. Using the mean for skewed or open-end data, where the median is more appropriate.
3. Concluding causation from a high correlation coefficient without further evidence.
4. Comparing the variability of two series with the standard deviation when their units or means differ; the coefficient of variation must be used.
5. Using an absolute measure of dispersion where a relative measure is required.
6. Drawing conclusions beyond the data; statistical results are valid only for the population studied.
7. Ignoring the influence of the base year and weights on the value of an index number when interpreting it.

Exam Tips

1. Explain how the appropriate statistical tool is selected for a given problem.
2. Show how index numbers are used to measure inflation and the cost of living.
3. Describe how the NSS data on consumption expenditure are analysed using mean, median and coefficient of variation.
4. Explain how the coefficient of variation is used to compare the stability of two sectors or firms.
5. Describe the construction and use of the Lorenz curve for measuring inequality.
6. State why statistics must always be interpreted with economic reasoning.
7. Give an example of forecasting using time-series data and index numbers.

Conclusion

This chapter brought together all the statistical tools studied in the previous chapters and applied them to real economic problems. We learned how to select the appropriate measure for a given purpose - averages for representative values, dispersion measures for reliability and consistency, correlation for association, and index numbers for changes over time. We applied these tools to the measurement of inflation and the cost of living, the analysis of consumption expenditure, the comparison of sectors of the economy, the measurement of income inequality through the Lorenz curve, and forecasting and decision making. We also stressed the importance of correct interpretation and the limitations of statistical evidence. With this practical orientation, we now turn to the second book of the course, Indian Economic Development, beginning with the state of the Indian economy on the eve of independence.