Dispersion and relationship: spread and association

Detail

Where this fits in NCERT

• Class 12, Ch 2 "Data Processing" lists three groups of statistical measures:
• measures of central tendency: mean, median, mode
• measures of dispersion: spread

• measures of relationship: association

• The chapter names dispersion and relationship but does not calculate them. The glossary defines dispersion, mean deviation, standard deviation and correlation coefficient.

• Why a geographer needs them: an average alone hides a lot.
• Two places can have the same mean rainfall.
• In one, the rain may come almost the same every year. In the other, it may swing between drought and flood.
• Dispersion shows that difference. Relationship measures show whether two things (rainfall and floods, fertiliser and yield) move together.

Measures of dispersion: how spread out the data is

• Dispersion: how far the values in a data set are spread out, usually measured around a central value (the mean or the median).
• Small dispersion = values stay close to the average. Large dispersion = values are scattered far from it.

Range

• Range = maximum value − minimum value.
• It is the simplest measure. It uses only the two extreme values.
• Weakness: one unusual year (a very heavy flood year, for example) can make it much bigger. It says nothing about the values in between.
• Use in mapping: it sets the classes of a choropleth map (a map that shades areas such as states or districts by value).
• Divide the range by 5 to get the class interval for five equal classes.
• Example: if district literacy runs from 40% to 90%, the range is 50 and each class is 10 points wide.

Mean deviation

• Mean deviation: the average distance of each value from the mean or the median, ignoring plus and minus signs.
• The signs are ignored because otherwise the gaps above and below the mean would cancel out (around the mean, they always add up to zero).
• Unlike the range, it uses every value.

Standard deviation (SD)

• Standard deviation: the positive square root of the average of the squared deviations from the mean.
• How it is worked out:
• find each value's distance from the mean
• square each distance, which removes the minus signs
• take the average of these squares

• take the square root

• It is the most widely used measure of dispersion. A bigger SD = more spread.

• Squaring gives more weight to values far from the mean. So SD is sensitive to extreme years.

Coefficient of variation (CV) (beyond NCERT)

• CV = (SD ÷ mean) × 100, shown as a percentage.
• Why it is needed: SD is in the same units as the data (mm, cm). So a wet place and a dry place cannot be compared fairly by SD alone. CV turns spread into a percentage of the average, so different data sets can be compared.
• Higher CV = less dependable (more variable) data series.

Worked example (made-up rainfall of 5 years: 80, 90, 100, 110, 120 cm)

Measure	Working	Result
Mean	500 ÷ 5	100 cm
Range	120 − 80	40 cm
Mean deviation	(20+10+0+10+20) ÷ 5	12 cm
Standard deviation	√[(400+100+0+100+400) ÷ 5] = √200	≈ 14.1 cm
Coefficient of variation	14.1 ÷ 100 × 100	≈ 14.1%

Rainfall variability in India: CV on the map

• Pattern on India's rainfall variability map:
• CV above 50%: western Rajasthan and Kutch (Gujarat), the driest parts of India

• CV below 25%: the west coast (Western Ghats side) and the north-east, the wettest parts

• The lesson, step by step:

• Low-rainfall areas get rain from a few monsoon spells that do not come every year.
• So their yearly totals swing widely → high CV.

• Result: the driest areas also have the least dependable rain. Low rainfall and high variability come together. This is why drought keeps coming back in Thar–Kutch (see indian-climate-monsoon).

• Season matters too (IMD data):

• South peninsular India gets a mean north-east monsoon (October–December) rainfall of 338.4 mm, with a CV of about 25% [2].
• That is far higher than the south-west monsoon (June–September) CV for India as a whole, about 10% [2].

• Within the south peninsula, the north-east monsoon CV is 30–50%. The east coast has smaller CVs than the interior [2].

• Hill state example: the CV of annual rainfall in Himachal Pradesh is 30% to 50% across the whole state [3].

• Season vs year (Kolkata, 1901–2019):
• winter rainfall had the highest CV, 72.89%
• annual rainfall had a low CV, 17.68% [4]
• The point: small seasonal totals vary much more than the yearly total.

Measures of relationship: how two things move together

• Measures of relationship: they show how closely two variables are linked. Examples: rainfall and floods, fertiliser use and crop yield.
• Correlation coefficient: one number from −1 to +1. It gives both the direction and the strength of the link.

Value of r	Meaning	Geographical example
+1 / close to +1	Positive: both rise together	Rainfall and area under paddy
−1 / close to −1	Negative: one falls as the other rises	Altitude and temperature (air cools as you go higher)
Close to 0	Little or no link	Two unconnected variables

• Scatter diagram: plot one variable against the other. The pattern of points shows the link at a glance before any calculation [5].

Two ways to calculate correlation (beyond NCERT geography)

• NCERT's statistics book lists three tools to study correlation: scatter diagrams, Karl Pearson's coefficient of correlation and Spearman's rank correlation [5].

Feature	Karl Pearson's r	Spearman's rank correlation (ρ, "rho")
Data used	Actual measured values (mm of rain, tonnes per hectare)	Ranks (1st, 2nd, 3rd…)
What it measures	Linear relationship (one that a straight line can show) [5]	Linear association between the ranks given to items [5]
Geography use	Rainfall vs yield across districts	Ranking states by literacy vs by urbanisation

Correlation is not causation

• Correlation measures covariation (how two things change together), not causation (one thing causing the other) [5].
• A high r shows only that when one variable changes, the other also changes. It does not prove cause and effect [5].
• Two common traps:
• a third factor may drive both variables

• the link may be pure chance (a spurious correlation)

• Geographical example: fertiliser use and yield can rise together because irrigation spreads at the same time (Punjab–Haryana, Green Revolution). Irrigation, the third factor, may be doing much of the work.

• Mains caution: say that correlation is "associated with". Do not say "caused by" unless the process is explained.

Prelims Hooks

• Range = maximum − minimum. It is the simplest measure of dispersion. Range ÷ 5 gives the class interval for a five-class choropleth map.
• Mean deviation ignores plus and minus signs. It can be taken around the mean or the median.
• Standard deviation is the positive square root of the mean of squared deviations from the mean. It is the most widely used measure of dispersion.
• CV = SD ÷ mean × 100. It is a relative measure. Use it to compare variability between series with different averages.
• Rainfall CV above 50%: western Rajasthan, Kutch. Below 25%: west coast, north-east India. Trap: the highest variability is in the lowest rainfall areas.
• Season trap: north-east monsoon CV over the south peninsula is about 25%, against about 10% for the all-India south-west monsoon. Within the south peninsula, the east coast has a lower CV than the interior [2].
• Correlation coefficient lies between −1 and +1. Altitude–temperature is negative. Rainfall–paddy area is positive.
• Pearson is used for measured values. Spearman is used for ranks [5].
• Statement trap: "A high correlation proves causation" is wrong [5].

Mains Points

• Rainfall variability and drought risk (GS-I/III):
• Western Rajasthan and Kutch have both low mean rainfall and a CV above 50%.
• So farming there faces repeated drought.

• The policy answer is drought-proofing: watershed works, dryland crops, and water storage in tanks and johads. The high-variability, high-reliability split also guides crop choice: paddy on the west coast, millets in Thar.

• Why averages alone mislead planners:

• The north-east monsoon CV over the south peninsula is about 25%, against about 10% for the south-west monsoon [2].
• Winter rainfall CV in Kolkata is 72.89% [4].

• So water budgets for Tamil Nadu reservoirs or rabi crops must plan for variability, not just the mean.

• Correlation in regional analysis:

• Linking fertiliser with yield, or rainfall with floods, helps target inputs and flood warnings.

• But correlation is not causation [5]. Confounding factors such as irrigation, soil and seed must be checked before drawing policy conclusions.

• Choosing the right tool:

• Range is quick but distorted by extreme values. SD is the standard. CV is for comparisons. Spearman is for ranked or ordinal data such as state rankings in development indices.
• Correct choice of measure improves district-level planning from Census and NSS data.