Chapter 5 — Measures of Central Tendency

Chapter 5, titled 'Measures of Central Tendency', explains how to summarise a large data set using a single representative value. It covers three main averages — Arithmetic Mean, Median, and Mode — and teaches students how to compute each for different types of data series.

Arithmetic Mean is defined as the sum of the values of all observations divided by the number of observations, denoted by X̄. The formula is X̄ = ΣX/N, where ΣX is the sum of all observations and N is the total number of observations. For example, the mean monthly income of six families earning Rs 1600, 1500, 1400, 1525, 1625, and 1630 works out to Rs 1,547.

In the Assumed Mean Method, a value A is assumed as the mean, deviations d = X – A are calculated for each observation, and the actual mean is obtained by X̄ = A + Σd/N. This method is used when the number of observations is large or figures are large, to simplify computation. Any value — whether present in the data or not — can be taken as the assumed mean, but a centrally located value is preferred.

The Step Deviation Method further simplifies calculation by dividing all deviations from the assumed mean by a common factor c, giving d' = (X – A)/c. The arithmetic mean is then X̄ = A + (Σd'/N) × c. This method avoids very large numerical figures and is especially useful for grouped data with equal class intervals.

Two key properties are stated in the chapter: (i) the sum of deviations of items from the arithmetic mean is always equal to zero, i.e., Σ(X – X̄) = 0; and (ii) arithmetic mean is affected by extreme values — any very large or very small value on either end can push the mean up or down significantly.

Median is the positional middle value that divides the distribution into two equal parts. For ungrouped data, observations are arranged in ascending order and the middle value is identified. Its position is given by (N+1)/2th item. When N is even, the median is the arithmetic mean of the two middle values — for example, for 20 observations with middle values 45 and 46, the median is (45 + 46)/2 = 45.5 marks.

For continuous series, the median class is located at the N/2th item using cumulative frequencies. The median is then calculated using the formula: Median = L + ((N/2 – c.f.) / f) × h, where L is the lower limit of the median class, c.f. is the cumulative frequency of the class preceding the median class, f is the frequency of the median class, and h is the class interval.

Quartiles divide the total data into four equal parts. The first quartile Q1 (lower quartile) has 25% of items below it, Q2 is the median with 50% on each side, and Q3 (upper quartile) has 75% of items below it. Q1 and Q3 together mark the limits within which the central 50% of the data lies. Q1 is located at the (N+1)/4th item and Q3 at the 3(N+1)/4th item for individual and discrete series.

Percentiles divide the distribution into 100 equal parts, giving 99 dividing positions denoted P1, P2, ..., P99. P50 is equal to the median value. If a student secures the 82nd percentile in an examination, their position is below 18% of the total candidates who appeared.

Mode is the value that occurs most frequently in a data set, derived from the French word 'la Mode' meaning most fashionable. It is most appropriate when you need to know the most typical value — for example, a shoe manufacturer wanting to know which shoe size has maximum demand would use Mode. Data can be unimodal (one mode), bimodal (two modes), or multimodal (more than two modes); it is also possible to have no mode.

For continuous series, the modal class is the class with the largest frequency. Mode is then calculated as: Mo = L + D1/(D1 + D2) × h, where L is the lower limit of the modal class, D1 is the difference between the frequency of the modal class and the preceding class, D2 is the difference between the modal class frequency and the succeeding class, and h is the class interval. For this calculation the class intervals must be equal and the series must be exclusive.

The chapter states that the Median always lies between the Arithmetic Mean and the Mode. This relationship can be expressed as: Arithmetic Mean > Median > Mode, or Arithmetic Mean < Median < Mode (the suffixes occur in alphabetical order: Me for Mean, Mi for Median, Mo for Mode).

Weighted Arithmetic Mean is used when different observations must be assigned different levels of importance. Weights (W) are assigned to each observation (X), and the weighted mean is calculated as X̄ = ΣWX/ΣW. The chapter gives the example of finding the average price of mangoes and potatoes, where the weight assigned to each is the share of that commodity in the consumer's budget.

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