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# Average|Definition & Meaning

## Definition

Average is a statistical representative measure of the central value of a set of numbers. It is obtained by dividing the sum of all the numbers by the count of numbers. For a set of numbers a, b, and c the average is given by (a + b + c)/3.

Figure 1 – Average of three numbers

Figure 1 illustrates the **average of three numbers** 2, 7, and 9. First, the numbers are added and then divided by the **total number of numbers.**

In layman’s terms, an **average is a single number** that symbolizes a collection of numbers; it is often the sum of the numbers divided by the total number of numbers in **the collection (arithmetic mean)**. The average of the 25 digits 2, 3, 4, 7, and 9 is 5, for instance. The mean may also be referred to as the **median or mode**, depending on the situation.

For instance, the median (a value between 50% and 100% of personal income) is frequently used to describe the typical personal income. The word “average” should be avoided when discussing measurements of central tendency because of this.

The** average of the list’s numbers** is the same as the number if all of them are the same. The majority of ordinary types have this characteristic. Another universal feature is monotonicity, which states that if two lists of integers A and B have the same length, and each element in** list A is at least as large as the corresponding entry in list B,** then list A’s average is at least as good as the list’s average. Linear homogeneity is likewise satisfied by all B ways.

**If ****you** **multiply** all **the** numbers **in** **the** list by the same positive number, **their** average **will** **change** by the same factor.

Before calculating the average, certain forms of averages give the list’s items various weights. The weighted median, weighted geometric mean, and weighted arithmetic mean are a few examples. In certain forms of moving averages, an item’s weight is also based on where it is on the list.

**However, ****most** **kinds** of **averages** satisfy **permutation** **independence.** **All** **elements** **are** **counted** equally in determining **the** **average. Their** **position** in the list **does** **not** **matter.** **The** **mean** of (1, 2, 3, 4, 6) is the same as **the** **mean** of (3, 2, 6, 4, 1).

The first **record** that the arithmetic means **for** **using** **estimates** was extended from 2 **cases** to n cases was in the **16th** century. **Since** the late **16th** **century,** it **has** **become** a **popular** method for reducing measurement **errors** in various **fields.** **At** the time, astronomers wanted to **get** **the** **true** value from noisy **measurements** such **as** **B.** **Planetary** position or **moon** **diameter.**

**Scientists**

assumed that the **average** **of** **several** **measurements** **would** **yield** **a** **number** **with** a relatively small **error** compared to the **sum** of all **measurements.** **In** **fact,** **methods** of **averaging** **to** **reduce** observation errors **were** **primarily** developed in astronomy.

**It ****is** **the** **intermediate** **range** **(the** **average** **of** **the** **two** **extreme** **values)** **that** **is** **considered** **a** precursor to the arithmetic **mean** **and** is **used,** **for** **example,** **in** **Arabic** **astronomy** **from** the **9th** **century** to **the** **11th** **century,** but also in metallurgy and **navigation.**

**Median**

The center value, if the groups were lined together, would be the median. (If the sum is even, use the average of the middle two.) The greatest and lowest value pairings should be continually removed until only one or two values are left in order to get the median. Sort the list according to the size of its items to do this.

If exactly one value **remains,** it is the **median.** **If** **there** **are** two values, the median is the arithmetic mean of these two. This method takes **lists** 1, 7, 3, 13 and **tells** it to read 1, 3, 7, 13. Then 1 and 13 are removed to **get** **Listings** **3** **and** **7.** **This** **remaining** list **has** two **items,** **so their arithmetic mean (3 + 7)/2 is 5, which is the median.**

**Mode**

In a list, the most frequently occurring number is the mode. For instance, the list’s mode is 3 for (1, 2, 2, 3, 3, 4). Two or more numbers may appear the same number of times, more often than **any** **other** **number**. There is no **agreed** **mode** **definition** in this case. Some authors say they are all modes, while others say there are no modes.

**Steps to Calculate Average **

**You** can easily calculate the average **of** a given set of values. **Just** **sum** all the values **and** divide the **result** by the number of **values** **given.** Averages can be calculated **in** three **easy** steps. **They** are:

### Step 1: Sum **the** **Numbers**

Finding the total of all provided numbers is the first step in calculating the average of a set of numbers.

### Step 2: Observations

Then count **the** **number** **of** **digits** in the given dataset.

### Step 3: Average **Calculation**

The total is divided by the quantity of observations as the last step in determining the average.

**Arithmetic Mean**

Arithmetic mean is the most common type of **average.** **Given** n **numbers,** each number **is** denoted by $a_i$ **(where** i = 1,2, …, **n),** **and** the arithmetic mean is the sum divided by **n.**

**Geometric Mean**

By determining the nth root of the product of n numbers, the geometric mean is a technique for determining the central tendency of a set of numbers. **This** is **quite** different from the arithmetic mean, **which** **adds** observations and then **divides** the sum by the number of observations. **However,** **the** geometric **mean** **finds** the product of all observations and the product’s nth root, where n is the total number of observations.

**Harmonic Mean**

According to its definition, the harmonic mean is the reciprocal of the averaged reciprocals of the provided data values.

**Average Of Negative Numbers**

The process or formula for finding the **average** **is** **the** **same** if there are negative numbers in the list.

Figure 2 – Formula for Arithmetic Mean

Figure 2 illustrates the formula for the arithmetic mean

**Some Examples of Average **

**Example 1**

Find the average of following

a) 3, -7, 12, 6, -2

b) 2, 6, 9, 10

c) 1, 3, 5, 8, 3

**Solution**

a) In order to evaluate the average, we have to find the sum, which is:

3 – 7 + 12 – 2 + 6 = 12

the total unit is equal to five, so the average will be equal to:

Average = $\dfrac{12}{5}$

Average = 2.4

So the average will be equal to 2.4.

b) In order to evaluate the average, we have to find the sum, which is:

2 + 6 + 9 + 10 = 27

the total unit is equal to four, so the average will be equal to:

Average = $\dfrac{27}{4}$

Average = 6.75

So the average will be equal to 6.75.

c) In order to evaluate the average, we have to find the sum, which is:

1 + 3 + 5 + 8 + 3 = 20

the total unit is equal to five, so the average will be equal to:

Average = $\dfrac{20}{5}$

Average = 4

So the average will be equal to 4.

### Example 3

Illustrate the average of 1, 3, 5 on the real line.

### Solution

The sum of the numbers 1 + 3 + 5 is equal to 9, and the total units are equal to 3, as illustrated in figure 3.

Figure 3 – Mean of Real numbers

*All images were created using GeoGebra.*