👨‍👧‍👧Measures of Central Tendency

A plant breeder weighs 100 sorghum ear-heads and records values ranging from 60 g to 140 g. To communicate these results in a single number — “the typical ear-head weight” — she needs a measure of central tendency. But which one should she use? The answer depends on the data’s nature and distribution.

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What Is Central Tendency?

• Any mathematical measure intended to represent the center or central value of a set of observations is a measure of central tendency (also called a measure of location or an average).
• It answers the question: “What is the typical or representative value of this dataset?”

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Characteristics of a Satisfactory Average

• Rigidly defined — no ambiguity in computation
• Easy to understand and calculate
• Based on all the observations
• Least affected by sampling fluctuations
• Capable of further algebraic treatment
• Not much affected by extreme values
• Easily located

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Overview of All Measures

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Arithmetic Mean (A.M.)

• The sum of observations divided by the number of observations — the most commonly used and widely understood measure.
• A.M. is measured in the same units as the observations.

Ungrouped Data — Direct Method

Let x₁, x₂, …, x_n be ‘n’ observations:

Ungrouped Data — Deviation Method

When observations are large, the Linear Transformation Method simplifies calculation:

Where:

• A = Assumed mean (usually the mid-point of the middle class or the class with highest frequency)
• d_i = x_i - A

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Grouped Data

Let f₁, f₂, …, f_n be frequencies corresponding to mid-values x₁, x₂, …, x_n:

By deviation method:

Where d_i = (x_i - A)/C, f = frequency, C = class interval, x = mid-values.

• Population mean is denoted by μ; sample mean by X (an estimate of μ). This distinction is fundamental in statistics.

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Properties of A.M.

1. The algebraic sum of deviations from the arithmetic mean is zero:

   ∑(x - A.M.) = 0

2. Combined mean (weighted mean) of k groups:

Sampling Fluctuation: The variation in sample statistics from sample to sample. For example, different random samples of 5 plants from a population of 30 will yield different sample means.

Merits and Demerits

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Examples

i) Ungrouped data: Weights of 7 sorghum ear-heads: 89, 94, 102, 107, 108, 115, 126 g.

• A = 102, AM = 741/7 = 105.86 g
• By deviation method: AM = 102 + (27/7) = 105.86 g

ii) Grouped Data: 405 soybean plant heights from a plot:

a) Direct Method:

b) Deviation Method: (C = 5, A = 30)

• Direct Method: A.M. = 12270/405 = 30.30 cm
• Deviation Method: A.M. = 30 + (24/405) x 5 = 30.30 cm

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Median

• The value of the middle-most item when data is arranged in ascending or descending order. It divides the dataset into two equal halves — 50% below, 50% above.
• Used for qualitative data such as intelligence, ability, and honesty. Especially useful for skewed distributions where the mean may be misleading.

Ungrouped Data

• Odd n: Median = the middle value after arrangement.
• Even n: Median = arithmetic mean of the two middle terms.
• Formula: Median = Size of (N+1)/2, where N = ∑f

Continuous Frequency Distribution

Where: l = lower limit of median class, f = frequency of median class, m = cumulative frequency of preceding class, C = class length, N = total frequency.

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Examples

Case i) Odd n: Runs scored by 11 players: 5, 19, 42, 11, 50, 30, 21, 0, 52, 36, 27

• Arranged: 0, 5, 11, 19, 21, 27, 30, 36, 42, 50, 52
• Median = 6th value = 27 runs

Case ii) Even n: Plant heights: 6, 10, 4, 3, 9, 11, 22, 18

• Arranged: 3, 4, 6, 9, 10, 11, 18, 22
• Median = Average of 4th and 5th = (9 + 10)/2 = 9.5 cm

Grouped Data: 180 sorghum ear-heads:

• Median class = 100-120, Median = 100 + ((90.5-69)/45) x 20 = 109.56 g

Merits and Demerits

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Mode

• The value occurring most frequently in a dataset — the most common or most popular value.
• Example: 4, 7, 6, 5, 4, 6, 4 → Mode = 4
• Used for: model size of shoes, readymade garments, business/meteorological forecasting

Continuous Frequency Distribution

Where: l = lower limit of modal class, C = class interval, f = frequency of modal class, f₁ and f₂ = frequencies of preceding and succeeding classes.

Examples

Ungrouped: 27, 28, 30, 33, 31, 35, 34, 33, 40, 41, 55, 46, 31, 33, 36, 33, 41, 33 → Mode = 33 (appears 5 times)

Grouped: Marks of 89 students:

• Modal class = 29.5-34.5, Mode = 30 + [(21-16)/(2x21-16-18)] x 5 = 33.63

Merits and Demerits

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Harmonic Mean

• The reciprocal of the arithmetic average of the reciprocals of given values. Gives more weight to smaller values.

• Used to find average speed, distance, and rate. For example, if you travel equal distances at different speeds, the harmonic mean gives the correct average speed.

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Geometric Mean

• The n^th root of the product of n observations. Appropriate for data that is multiplicative or involves growth rates.

• If any observation is zero, G.M. = zero.
• Used in: bacterial growth, cell division, and wherever values grow by multiplication.

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Key Relationships

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Summary Table

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🟢 Central Tendency — Video Explanation

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Summary Cheat Sheet