Shape of a distribution: normal curve and skewness

Detail

Why the shape of data matters

• A frequency distribution is a table or graph that shows how many times each value, or each group of values, appears in a data set.
• When this is drawn as a smooth line, it is called a frequency curve. Its shape tells us how the three measures of central tendency sit against each other:
• Mean: the arithmetic average (add all the values, then divide by how many there are).
• Median: the middle value once the data are put in order.

• Mode: the value that appears most often. It is the peak of the curve.

• The shape decides which average gives a true picture of a geographical data set, such as rainfall, income, landholding size or population.

The normal distribution (Fig. 2.1)

• Normal distribution: a symmetrical, bell-shaped curve. Most values gather around the middle. They thin out evenly on both sides towards rare extreme values.

• Symmetrical means the left half is a mirror image of the right half.

• Main features:

• Mean = median = mode. All three fall at the same central point (100 in NCERT Fig. 2.1).
• Half the values lie above the middle and half lie below it.

• Its skewness is zero, because it leans to neither side [2].

• Real examples: traits such as intelligence and achievement (test) scores come close to a normal curve.

• Spread (variability): a normal curve can be:
• High spread: a wide, flat bell. The values are scattered far from the mean.
• Low spread: a narrow, tall bell. The values are packed close to the mean.
• The curve stays symmetrical either way. Only its width changes.

Skewness (Fig. 2.2, 2.3)

• Skewness: how far a curve departs from symmetry, so that it is lopsided [2]. In a skewed curve, the mean, median and mode move apart.
• The tail is the long, thin end of the curve, where the rare extreme values lie. The side of the tail gives the name:
• Longer tail to the right → positive skew (skewed to the right) [2].
• Longer tail to the left → negative skew (skewed to the left) [2].

	Positive skew (Fig. 2.2)	Negative skew (Fig. 2.3)
Tail	long tail to the right (towards high values)	long tail to the left (towards low values)
Order of averages	mean > median > mode	mean < median < mode
Where the peak (mode) sits	towards the low-value side	towards the high-value side
Which average gets pulled	the mean is pulled up by a few very high values	the mean is pulled down by a few very low values
Examples	incomes; rainfall in a dry region with a few extreme storms; landholding size	marks in an easy test, where most students score high and only a few score very low

• Rule of thumb: if the mean is larger than the median, the data are right-skewed (positive). If the median is larger than the mean, the data are left-skewed (negative) [2].

Why positive skew happens: two worked cases

• Rainfall in a dry region (for example, the arid west of Rajasthan):
• Most days or years get little or no rain, so the mode is near zero.
• Now and then an extreme storm dumps a very large amount of rain.
• These few huge values stretch the curve to the right and pull the mean up.

• Result: the "average rainfall" is higher than what a normal year gets. The median is the better guide.

• Landholdings in India:

• Holdings are grouped into 5 size classes: Marginal (below 1 ha), Small (1–2 ha), Semi-medium (2–4 ha), Medium (4–10 ha) and Large (10 ha and above) [3].
• Agriculture Census 2015-16: small and marginal holders together made up about 86% of all operational holders [3][4].
• The average (mean) size of an operational holding fell from 2.28 ha (1970-71) to 1.84 ha (1980-81), then 1.41 ha (1995-96), and 1.08 ha (2015-16) [3].
• A huge number of very small farms sit at the low end. A small number of large holdings form the long right tail.

• So the mean is pulled above the size of a typical farm. This is why the median is the right average for landholdings.

• Incomes follow the same pattern. Most people earn modest amounts, while a few very rich earners stretch the right tail.

Karl Pearson's empirical relation (beyond NCERT)

• For moderately skewed data: Mode ≈ 3 Median − 2 Mean (Karl Pearson).
• This gives an estimate of the mode when only the mean and the median are known.
• Worked example: mean = 60 and median = 55 → mode ≈ 3(55) − 2(60) = 165 − 120 = 45.

• The order 60 > 55 > 45 is mean > median > mode, so the data are positively skewed.

• The median always lies between the mean and the mode in a skewed curve, whichever way the skew goes.

Exam use: choosing the right average

• Symmetrical data → use the mean. It uses every value, and it equals the median and the mode anyway.
• Skewed data (incomes, landholdings, rainfall in dry areas) → use the median. Extreme values do not drag it.
• Mode → use it for the "most common" value, such as the most common farm size class or crop.
• This links to NCERT exercise 3(i)–(ii) (comparing mean, median and mode).

Prelims Hooks

• Normal curve: mean = median = mode. Its skewness is zero [2].
• Positive skew: long tail to the right, so mean > median > mode.
• Negative skew: long tail to the left, so mean < median < mode.
• Trap: the median always lies between the mean and the mode. It is never at either end.
• Karl Pearson: Mode ≈ 3 Median − 2 Mean. It holds for moderately skewed data, not for every distribution.
• Income, landholding size and dry-region rainfall are standard positive-skew examples. Intelligence and achievement scores come close to normal.
• A normal curve can have high or low spread and still be symmetrical. Spread is not the same thing as skew.
• Agriculture Census 2015-16: small and marginal holders ≈ 86%; mean holding size 1.08 ha [3][4].
• Holding size classes: Marginal < 1 ha; Small 1–2 ha; Semi-medium 2–4 ha; Medium 4–10 ha; Large ≥ 10 ha [3].

Mains Points

• Averages can mislead policy:
• India's mean holding size of 1.08 ha (2015-16) hides the fact that about 86% of holders have under 2 ha [3][4].

• Median-based and size-class data are needed to target schemes at small and marginal farmers.

• Rainfall planning in arid and semi-arid regions (western Rajasthan, Kutch, the rain-shadow Deccan):

• A mean pushed up by rare extreme storms overstates how reliable the rain is.

• Planning for drought, irrigation and crops should use the median and measures of how much rainfall varies.

• Inequality analysis (GS-I society / GS-III economy):

• Income and asset data are positively skewed, so the mean shows "growth" while the typical household may not gain.

• Medians and how the data are spread out across groups give a truer picture.

• Data processing practice:

• Check the shape of the distribution before choosing a central value. Symmetrical data → mean; skewed data → median.
• This is basic to reading Census, NSS and Agriculture Census tables correctly.