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4.3 Properties Of Arithmetic Mean

  1. The sum of the deviations, of all the values of x, from their arithmetic mean, is zero.

    Justification :

    Since is a constant,

  2. The product of the arithmetic mean and the number of items gives the total of all items.

    Justification :


  3. If and are the arithmetic mean of two samples of sizes n1 and n2 respectively then, the arithmetic mean of the distribution combining the two can be calculated as

    This formula can be extended for still more groups or samples.

    Justification : = total of the observations of the first sample

    Similarly = total of the observations of the first sample

    The combined mean of the two samples



Example The average marks of three batches of students having 70, 50 and 30 students respectively are 50, 55 and 45. Find the average marks of all the 150 students, taken together.

Solution :

Let x be the average marks of all 150 students taken together.

Batch - I   Batch - II   Batch - III


A. marks :           = 50           = 55            = 45

No. of students n1 = 70    n2 = 50       n3    = 30




4.1 Introduction
4.2 Arithmetic Mean
4.3 Properties of Arithmetic Mean
4.4 Median
4.5 Mode
4.6 Empirical relation between mean, median & mode

Chapter 5

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