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Chapter 16 Average

16.1 Introduction: an average or an arithmetic mean is defined as sum of n terms divided by n.

Average = sum of data/Number of data

Example: If a batsman scores 30, 45 and 51 runs in first, second and third innings respectively, then his average runs in 3 innings is

Avg = 30+45+51

The average score of the batsman is 42 runs

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16.2 Calculation:

16.2.1 Arithmetic mean: If n numbers are given, each number denoted by ai, where i = 1, ..., n, the arithmetic mean is the of the ai is divided by n

AM = 1/n Σai

The arithmetic mean, is often called as mean.

16.2.2 Geometric mean: The geometric mean of n numbers is determined by multiplying them together and then taking the nth root. In algebraic terms, the geometric mean of a1, a2, ..., an is defined as

Geometric mean

16.2.4 Inequality concerning AM, GM, and HM: Inequality concerning arithmetic, geometric, and harmonic means for any set of positive numbers is

AM ≥ GM ≥ HM

16.2.5 Median and Mode: Mode is defined as the most frequently occurring number in a given list.

For example: Mode in the given list of number?

21, 12, 34, 21, 16, 21, 35, 34, 21,16

Mode in the above list is 21, as it has occurred 4 times.

The mode has the advantage that it can be used with non-numerical data but averages cannot.

Median can be calculated by first ordering the list according to its elements' magnitude and then repeatedly removing the pair consisting highest and lowest values until either one or two values are left. If exactly one value is left, it is the median and if two values are left then the median is the arithmetic mean of these two.

For example: Median of the given list

1, 4, 3, 5, 12, 2

1, 2, 3, 4, 5, 12

2, 3, 4, 5

3, 4

Now you can see two values are left, so take the mean of the two

Median = (3+4)/ 2 = 3.5

16.4 Average of functions: In calculus, the average value of an integral function ƒ on an interval [a,b] is defined by b

f = 1f(x)dx

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