ARITHMETIC MEAN FORMULA

August 14, 2018

ARITHMETIC MEAN FORMULA

Sum of all of the numbers of a group, when divided by the number of items in that list is known as the Arithmetic Mean or Mean of the group. For example, the mean of the numbers 5, 7, 9 is 4 since 5+7+9 = 21 and 21 divided by 3 [there are three numbers] is 7.

\bar{X} = \frac{\sum_{i = 1}^{n} X_{i}}{N}

Where, N is total number of observations.

Please note that Average is different than an Arithmetic Mean.

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SOLVED EXAMPLES

Question 1: The marks obtained by 6 students in a class test are 20, 22, 24, 26, 28, 30. Find the mean ?

Solution:

\bar{x}

\frac{20 + 22 + 24 + 26 + 28 + 30}{6}

= 25

GEOMETRIC MEAN FORMULA

August 14, 2018

GEOMETRIC MEAN FORMULA

The geometric mean is a type of mean that indicates the central tendency or typical value of a set of numbers by using the product of their values. It is defined as the

n^{t h}

root of the product of

n

numbers. You cannot calculate geometric mean from the arithmetic mean. Geometric mean is well defined only for positive set of real numbers. Example calculating the central frequency $f_{0}$ of a bandwidth BW= $f_{2} - f_{1}$

Formula for evaluating geometric mean is as follows if we have n number of observations, then

$\bar{x}_{geom}=\sqrt[n]{\prod_{i=1}^{n}x_{i}}=\sqrt[n]{x_{1}\cdot x_{2}\cdot ...\cdot x_{n}}$

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SOLVED EXAMPLES:

Question: Find the geometric mean of 4 and 3 ?

Solution:

Geometric Mean =

\sqrt{4 \times 3}

2 \sqrt{3}

= 3.46

Question: What is the geometric mean of 4, 8, 3, 9 and 17 ?

Solution:

Step 1: n = 5 is the total number of values. Find 1/n.

1/5 = 0.2

Step 2: Find geometric mean using the formula:

$\left [ 4*8*3*9*17 \right ]^{(0.2)}$

Geometric Mean = 6.81

HARMONIC MEAN FORMULA

August 14, 2018

HARMONIC MEAN FORMULA

Harmonic mean is used to calculate the average of a set of numbers. The number of elements will be averaged and divided by the sum of the reciprocals of the elements.

It is calculated by dividing the number of observations by the sum of reciprocal of the observation.

The formula to find the harmonic mean is given by:

For Ungrouped Data:

H.M OF X = $\bar{X}$ = $\frac{n}{\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+...............+\frac{1}{x_{n}}}=\frac{n}{\sum (\frac{1}{x})}$

Where,

n = Total number of numbers or terms.

x1, x2, x3, …. xn = Individual terms or individual values.

Lets Work Out-

Example: Find the harmonic mean of the following data {8, 9, 6, 11, 10, 5} ?

Solution:

Given data: {8, 9, 6, 11, 10, 5}

So Harmonic mean = $\frac{6}{\frac{1}{8}+\frac{1}{9}+\frac{1}{6}+\frac{1}{11}+\frac{1}{10}+\frac{1}{5}}$

H = $\frac{6}{0.7936 }$ = 7.560

Harmonic mean(H) = 7.560

For Grouped Data:

H.M OF X = $\bar{X}$ = $\bar{X}=\frac{f_{1}+f_{2}+f_{3}+................+f_{n}}{\frac{f_{1}}{x_{1}}+\frac{f_{2}}{x_{2}}+\frac{f_{3}}{x_{3}}+................+\frac{f_{n}}{x_{n}}}=\frac{\sum f}{\sum (\frac{f}{x})}$

Where,

f = Individual entries.

x1, x2, x3, …. xn = Individual terms or individual values.

Lets Work Out-

Example:The table given below represent the frequency-distribution of ages for Standard 1st students.

Ages	4	5	6	7
Number of Students	6	4	10	8

Find the Harmonic Mean of the given class.

Solution:

Here the data given are distributed data. So the ages are the variables and the number of student is considered as the frequency.

Ages (x)	Number of Students (f)	$\frac{f}{x}$
4	10	2.5
5	6	1.2
6	8	1.33
7	4	0.57
Total	$\sum f$ =28	$\sum(\frac{f}{x})$ = 5.6

So Harmonic mean = $\frac{\sum f}{\sum(\frac{f}{x})}=\frac{28}{5.6}$ = 5 years

Therefore, Harmonic mean(H) = 5 years

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