Average

The word “average” is used in everyday life to describe where the middle number of a data set is. It’s the typical number you would expect to find in a series of numbers. In statistics, the average is called the “arithmetic mean,” usually just shortened to the mean. Both the average and the mean use the same formula:

avg = total sum of all the numbers / number of items in the set.

Example

You earned $129, $139, $155 and $176 over the last 4 weeks. What is your average pay?

- Add up all of the numbers in the set. $129 + $139 + $155 + $176 = $599.
- Divide Step 1 by the total number of items in the set. There are 4 items in the set, so $599 / 4 = $149.75.

In probability and statistics, you’ll see the following formula used:

Airthmatic , geomatric and harmonic means

Airthmatic mean

The arithmetic mean is the simplest and most widely used measure of a mean, or average. It simply involves taking the sum of a group of numbers, then dividing that sum by the count of the numbers used in the series.

For example take 34, 44, 56 and 78. The sum is 212. The arithmetic mean is 212 divided by four, or 53.

The arithmetic mean maintains its place in finance, as well. For example, mean earnings estimates typically are an arithmetic mean. Say you want to know the average earnings expectation of the 16 analysts covering a particular stock. Simply add up all the estimates and divide by 16 to get the arithmetic mean.

The same is true if you want to calculate a stock’s average closing price during a particular month. Say there are 23 trading days in the month. Simply take all the prices, add them, up and divide by 23 to get the arithmetic mean.

The arithmetic mean is simple, and most people with even a little bit of finance and math skill can calculate it. It’s also a useful measure of central tendency, as it tends to provide useful results, even with large groupings of numbers.

Geomatric mean

The Geometric mean is defined as the nth square root of the product of n distinct data values. That is, consider, n data values as x1, x2, …, xn the geometric mean is given by,

Moreover, geometric mean is considered as a special type of average. Also, the value of geometric is less than the arithmetic mean and is considered as one of the measures of central tendency.

Also, if the n data values has repeated values x_{1}, x_{2},…, x_{m} with frequencies f_{1},f_{2},…,f_{m}, then the geometric mean is given by,

The other formula to calculate the geometric mean is,

Harmonic mean

The harmonic mean is a very specific type of average. It’s generally used when dealing with averages of units, like speed or other rates and ratios.

Example: What is the harmonic mean of 1,5,8,10?

Add the reciprocals of the numbers in the set: 1/1 + 1/5 + 1/8 + 1/10 = 1.425

Divide the number of items in the set by your answer.

There are 4 items in the set, so: 4 / 1.425 = 2.80702

Relation between arithmetic geometric and harmonic means

Let us consider two numbers x and y. Then, if x, a, y forms an arithmetic progression, then this ‘a’ is termed as the arithmetic mean. Likewise, if this sequence of x, a, y forms a geometric progression then it is termed a geometric mean and same goes for harmonic mean. Mathematically, the three means can be described as under:

Arithmetic Mean: For two numbers ‘a’ and ‘b’, the arithmetic mean is defined as:

(a+b)/2

√ab

Geometric Mean: Unlike arithmetic mean, the geometric mean takes into account the product of the numbers. For two numbers ‘a’ and ‘b’, the geometric mean is defined as

Harmonic Mean: The harmonic mean of two numbers ‘a’ and ‘b’ is defined as

If A.M denotes the arithmetic mean, G.M denotes the geometric mean and H.M, the harmonic mean, then the relationship between the three is given by:

A.M × G.M = H.M^{2}

Their relationship can also be illustrated using the inequality:

A ≥ G ≥ H

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