Average

1      Arithmetic Mean:

In simple language, an average is the sum of a list of numbers divided by the number of elements in the list.

Example: The average of four observations {50, 55, 70, 85} = (50 + 55 + 70 + 85)/4 = 65.

Although the arithmetic mean is not the only “mean”, it is by far the most commonly used. Therefore, if the term “mean” is used without specifying whether it is the arithmetic mean, the geometric mean, or some other mean, it is assumed to refer to the arithmetic mean.

Important facts about Arithmetic mean:

  • If each of the given element is increased or decreased by x, then their average is also respectively increased or decreased by x.

Example: The average of 45, 55, 80 is 60. If 20 is added to each of the elements the new set is 65, 75, 100 whose average is 80. Clearly 80 is 20 more than the initial average 60.

  • If each of the given elements is multiplied by x, then their average gets multiplied by x.
  • If each of the given elements is divided by a nonzero x, then their average gets divided by x.
  • If x1and x2 are the arithmetic means of two samples of sizes n1and n2 respectively, then the arithmetic mean z of the distribution combining the two samples can be calculated as

z = (x1n+ x2n2)/2

  • The value of the arithmetic mean of two elements is always in between the value of the two elements.

2      Geometric Mean:

The geometric mean is defined as the nth root of the product of n numbers.

That is, to find the geometric mean of a set of ‘n’ numbers, multiply the ‘n’ numbers and then take the nth root of the product.

Example 1: Geometric mean of two numbers, say 3 and 27 is the square root of their product; i.e.,9.

Example 2:  Geometric mean of three numbers, say 4, 6 and 9 is the cube root of their product; i.e.,6.

Note: Geometric mean is always less than or equal to the Arithmetic mean.

3         Median

Median is the middle value that separates the higher half from the lower half of the set of observations.

To find the median, we arrange the observations in the increasing (or descending) order. If there is an odd number of observations, the median is the middle value. If there is an even number of observations, the median is the average of the two middle values.

Example 1: To find the median of the set of observations {10, 15, 12, 3, 55}, arrange the set of observations in the increasing order as {3, 10, 12, 15, 55}.

Since there are 5 observations (odd), the median is the 3rd term, that is 12.

Example 2: To find the median of the set of observations {5, 20, 10, 35}, arrange the set of observations in the increasing order as {5, 10, 20, 35}.

Since there are 4 observations (even), the median is the average of 2nd and 3rd terms.

That is, Median = (10+20)/2  = 15.

4      Mode:

The mode is the most frequently appearing value in the set of observations.

Example: The mode of the set of observations {1, 5, 4, 5, 7, 2, 2, 5}

Here 5 is the most frequently appearing value.

Hence, 5 is the mode.

5      Standard Deviation:

Standard deviation is a measure that is used to quantify the amount of variation or dispersion of a set of data values. A standard deviation close to 0 indicates that the data points tend to be very close to the mean of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.

Variance for N observations is defined as:

Variance = (xi – Mean)2

Standard Deviation = root(variance) 

Example: Consider the set of observations {1, 3, 5, 7}

Here the Mean = 4.

Variance = ((1– 4)2 + (3 – 4)2 + (5 – 4)2 + (7 – 4)2)/4  = 5.

Standard Deviation = root(5)

“For detailed theory, refer the book “CSIR-NET General Aptitude – A New Outlook”

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