Measures of central tendency: mean, median and mode

Detail

What a measure of central tendency is

• Measure of central tendency (also called a statistical average): one value that stands for a whole set of data. It shows the centre point, the point around which the values gather.
• Geographers use it to sum up long lists of numbers in one figure. Examples:
• average rainfall of a district
• average elevation of a region
• average population density

• average years of schooling (educational attainment)

• There are three common measures:

Measure	How it is found	Type of average	Uses every value?	Pulled by extreme values?	Works for category data?
Mean (X̄)	Add all values, divide by their number	Calculated average	Yes	Yes, strongly	No
Median (M)	Middle item of an ordered list	Positional average	No (only its position)	No	No (needs an order)
Mode (Z or M₀)	Most frequent value	Frequency-based	No	No	Yes

Mean: definition and the direct method (ungrouped data)

• Mean: the sum of all values divided by the number of values. Formula: X̄ = Σx / N (Σ = "sum of", N = number of observations).
• Ungrouped data: raw values listed one by one, not put into classes.
• Direct method: add every value and divide by N.
• NCERT example: the normal rainfall of seven districts of the Malwa plateau (Madhya Pradesh). The values run from Mandsaur, 825 mm (lowest) to Dewas, 1,083 mm (highest).

District (Malwa plateau, M.P.)	Normal rainfall (mm)	d = x − 800
Indore	979	179
Dewas	1,083	283
Dhar	833	33
Ratlam	896	96
Ujjain	891	91
Mandsaur	825	25
Shajapur	977	177
Total	Σx = 6,484	Σd = 884

• Direct method: 6,484 ÷ 7 = 926.29 mm.
• A "normal" is itself a mean. A district's "normal rainfall" is the average of rainfall over many years.
• IMD defines the LPA (Long Period Average) as the rainfall of a region for a given month or season, averaged over a long period such as 30 or 50 years. It is the benchmark against which forecasts are judged [2].
• Example: the LPA of south-west monsoon rainfall over Kerala is 556 mm (June), 659 mm (July), 427 mm (August) and 252 mm (September) [2].
• All-India LPA for June–September is 87 cm, based on 1971–2020 [3]. An older base gave 880.6 mm for 1961–2010 [2].
• The WMO (World Meteorological Organization) defines climatological normals as averages of climate data over 30 consecutive years. The current standard is the latest 30-year period ending in a year with 0, e.g. 1991–2020. 1961–1990 is kept as the reference for long-term climate change studies [4].

Indirect method of mean (assumed mean, or "coding")

• Why it is used: when the values are large or many, adding them is slow and mistakes are easy.
• How it works, step by step:
• choose an assumed mean (A), any convenient round number near the centre
• subtract A from each value to get a small deviation, d = x − A (this step is called coding)

• add the deviations (Σd), divide by N, and add the result back to A

• Formula: X̄ = A + Σd / N

• Malwa example with A = 800: 800 + 884/7 = 800 + 126.29 = 926.29 mm. This is the same answer as the direct method.
• Exam point: the indirect method only makes the arithmetic shorter. It never changes the answer.

Mean of grouped data

• Grouped data: values sorted into classes (ranges such as 80–90), each with a frequency (f), the number of items in that class.
• The exact values inside a class are not known. So each class is stood for by its midpoint (x) = (lower limit + upper limit) ÷ 2.
• Direct method: X̄ = Σfx / N (multiply each midpoint by its frequency, add them up, divide by total frequency).

• NCERT example, daily wages of 99 workers: 10,160 ÷ 99 = Rs 102.6 a day.

• Indirect method: X̄ = A ± Σfd / N, where d = midpoint − A.

• Same example with A = 100: 100 + 260/99 = Rs 102.6.

• The indirect method works with both equal and unequal class intervals.

Median

• Median: the value that splits an ordered series into two equal halves. Half the items lie below it and half lie above it.
• It is a positional average: it is found by the position of an item in the ordered list, not by adding values.
• First step always: arrange the values in order, from smallest to largest or the reverse.

Ungrouped data

• Median = value of the (N + 1)/2th item.
• NCERT example: seven Himalayan peaks, from 7,817 m to 8,848 m (8,848 m = Mount Everest).
• N = 7, so (7 + 1)/2 = 4th item

• Median = 8,172 m (NCERT misprints this as "8.172 m")

• If N is even, (N + 1)/2 falls between two items. The median is then the average of those two middle items.

Grouped data

• Formula: M = l + (i / f)(N/2 − c)
• l = lower limit of the median class (the class that holds the N/2th item)
• i = class interval (width of the class)
• f = frequency of the median class

• c = cumulative frequency (running total of frequencies) of the class just before the median class

• Steps:

• work out the cumulative frequencies
• find N/2
• pick the first class whose cumulative frequency reaches N/2; this is the median class

• put the values into the formula

• NCERT example: N = 50, so N/2 = 25. The median class is 80–90 (NCERT misprints it as "40–44"), so l = 80, i = 10, f = 16 and c = 21.

• M = 80 + (10/16)(25 − 21) = 80 + 2.5 = 82.5

Mode

• Mode (symbol Z or M₀): the value that occurs most often in a series.
• No calculation is needed in ungrouped data. You count how often each value occurs.
• Types of series by number of modes:

Type	Meaning	Illustration
Unimodal	One mode	2, 3, 3, 5 → mode 3
Bimodal	Two modes	2, 2, 4, 6, 6 → modes 2 and 6
Trimodal	Three modes	Three values tie for most frequent
Multimodal	More than three modes	Many values tie
Without mode	No value repeats	1, 4, 7, 9

Properties: which average to use when

Mean

• It uses every value in the series. This makes it the most complete average and the easiest to use in further calculations.
• But it is pulled by extreme values:
• a few very large or very small values drag the mean towards them

• the mean may then describe almost no real item

• River-depth fallacy:

• a river's mean depth may be shallow, say 1 m
• a person who cannot swim may try to wade across, trusting this average
• at some points the river is much deeper than the mean, and the person may drown
• lesson: the mean hides the extremes that matter for safety and planning

Median

• It is not affected by extreme values. Only the middle position matters (NCERT exercise). Britannica says the same: unlike the mean, the median is not affected by extremely large or extremely small values [5].
• So it is the better average for skewed data. Skewed data have a long tail on one side, such as incomes or landholding sizes.

Mode

• It always sits at the hump (peak) of a frequency distribution, because that is where the frequency is highest.
• It is the only one of the three that works with category (non-numerical) data [5]. NCERT examples:
• the most common crop of a district

• the most common house type in a village

• In a perfectly symmetric, single-humped distribution, the mean, median and mode are the same value. In a skewed distribution they separate, and the mean moves furthest towards the tail.

Mean in practice: how IMD labels rainfall

• IMD's statistical idea of "normal":
• take the mean (m) and standard deviation (d) of a long rainfall record; standard deviation tells how far values usually spread from the mean
• if the data follow a normal (bell-shaped) distribution, 68% of years fall between m − d and m + d

• a year in this band is "normal", below it is "below normal", above it is "above normal" [2]

• Categories for all-India June–September rainfall (as % of LPA) [3]:

Category	% of LPA
Deficient	< 90
Below normal	90–95
Normal	96–104
Above normal	105–110
Excess	> 110

• 2025 forecast: 105% of LPA, with a model error of ±5% [3].
• District and state level (daily, weekly or monthly rainfall), measured as departure from LPA [2]:
• large excess: ≥ +60%
• excess: +20% to +59%
• normal: −19% to +19%
• deficient: −20% to −59%
• large deficient: −60% to −99%

Prelims Hooks

• The median is a positional average. The mean is a calculated average. The mode is a frequency-based average.
• The median is not affected by extreme values. The mean is. This is the classic "which of the following" trap.
• The mode is the only one of the three averages that works with category data, such as the most common crop or house type [5].
• Median of ungrouped data = value of the (N + 1)/2th item. The values must be put in order first.
• Grouped median: M = l + (i/f)(N/2 − c). Here c is the cumulative frequency of the class before the median class, not of the median class itself.
• The indirect (assumed-mean) method gives exactly the same mean as the direct method. It works with equal and unequal class intervals.
• Symbol of mode: Z or M₀. A series can be unimodal, bimodal, trimodal, multimodal or without mode.
• IMD's all-India monsoon LPA = 87 cm (1971–2020). "Normal" = 96–104% of LPA [3]. The earlier LPA was 880.6 mm (1961–2010) [2].
• WMO climatological normals are 30-year averages, e.g. 1991–2020. 1961–1990 is kept as the base for climate change studies [4].
• River-depth fallacy: it shows that a mean can hide dangerous extremes.

Mains Points

• Choosing the right average is a policy choice.
• Mean income or mean landholding is pulled up by a few rich households.
• The median gives a truer picture of the typical household in skewed data, for example in Census and NSS surveys.

• The mode suits category data, such as the main crop or the main house type, when planning at district level.

• Means in monsoon management (GS-III: disasters, agriculture).

• IMD's LPA (87 cm, 1971–2020) and its categories decide how a season is labelled: deficient, normal or excess [2][3].

• These labels in turn drive drought declarations, crop advisories and reservoir planning.

• Averages hide spatial and time differences.

• A "normal" all-India season can still include large-deficient districts. District departures range from +60% to −99% [2].

• Mean values must therefore be read together with measures of spread (standard deviation) and local data. This is the river-depth fallacy applied to policy.

• Normals shift with climate change.

• Normals are updated from one 30-year window to the next (WMO; IMD moved from 1961–2010 to 1971–2020) [2][3][4].
• So the "average" itself changes over time. Comparing data across periods needs a stated base period.