Average

The concept of average plays a major role in mathematics, statistics, and real-life problem-solving. It provides a quick way to represent a group of numbers using a single value that reflects their overall tendency. Thus, the average is not only a mathematical tool but also an essential method for decision-making in science, economics, sports, and daily life. So, let’s explore this important topic of mathematics in this blog. 

1.0Average Definition

By mathematical definition, the average is a specific value of a group of numbers, obtained by dividing the sum of all the values by the number of values in that group. This single value gives an idea of all the values in the group; that is, it represents the overall tendency of that group. 

To understand it better, take a group of three values, say 3, 6, 9. The average for this group can be obtained by adding all these values, that is, 18, and dividing it by the number of values, that is, 3. Hence, the average we get will be 6. 

2.0Average Formula 

The average definition gives its standard formula, which can be applied to solve any problem in math. The average formula can be expressed as: 

$\text{ Average }=\frac{\text{ Sum of all the values }}{\text{ Number of Values }}$

The above-mentioned formula is the simplest formula to find the average of a given data set. Also, note that the average in mathematics can be symbolised as: 

$\bar{x}$

The average symbol is read as x-bar.

Therefore, if there are n values in a set, i.e., x₁, x₂, x₃, x₄, up to x_n, then their mean can be given by: 

$\bar{x}=\frac{x_1+x_2+x_3+x_4+\dots ..+x_n}{n}$

3.0Steps to Calculate the Average

Finding an average (mean) involves three simple steps and uses two basic operations: addition and division.

Step 1: Sum of the Numbers

First, find the total of all the values given.  

Let’s look at an example of the daily temperatures (in °C) recorded over 7 days:
32°C, 30°C, 31°C, 29°C, 33°C, 30°C, 31°C

Now, add them all together:
32 + 30 + 31 + 29 + 33 + 30 + 31 = 216

So, the sum of all observations is 216°C.

Step 2: Number of Observations

Next, count how many data points you have.
Here, the number of days = 7.

Step 3: Average Calculation

Now, apply the average formula:

$\begin{array}{rl}& \text{ Average }=\frac{\text{ Sum of all the values }}{\text{ Number of Values }}\\ & \text{ Average }=\frac{216}{7}\end{array}$

Therefore, the average temperature over the week is 30.9°C.

4.0Properties of Average

The average in mathematics has certain unique properties that render it an effective method to simplify and analyse any data set. Certain of these properties of the average are:

1. The mean is always between the largest and smallest values in a particular data set. Example: For 2, 4, 6, average = 4, which lies between 2 and 6.
2. If all values in a set of given data are the same, then the average is unchanged. Example: For 10, 10, 10, average = 10.
3. Adding or subtracting a constant from each number in a dataset changes the average by the same amount. Example: If the data are 5, 10, 15 (average = 10). Adding 2 to each → 7, 12, 17 → new average = 12.
4. Multiplying or dividing every number in the data by a constant also multiplies/divides the average. Example: For 4, 8, 12 (average = 8). Multiplying each by 2 → 8, 16, 24 → new average = 16.
5. The sum of the deviations of the data from the average is always zero. For 2, 4, 6 (average = 4): (2–4) + (4–4) + (6–4) = 0.

5.0Types of Averages 

In mathematics, the averages of a dataset are classified into three types, all of which have the same purpose of giving an idea of all the values of that data. These three types are: 

1. Arithmetic Mean (AM) 

The average is also known as the arithmetic mean; therefore, the mean and average have the same formula, that is: 

$\text{ Mean }=\frac{\text{ Sum of all the values }}{\text{ Number of Values }}$

2. Median: 

Median is the middle value of a certain dataset when all the values are arranged in increasing or decreasing order. It is considered the average because of this property itself. 

Also, in ungrouped data, if the number of values in a data set is odd, then the middle value is the median, while if it is even, then the median is the average of the two middle values of that dataset. 

3. Mode: 

Mode is the value which occurs the most number of times in a certain dataset. It is called an average because it indicates “the most common” or “the most typical value” in the data. 

In summary, all of these are known as the averages because they have a common goal, that is, to describe a dataset with a single representative value. 

6.0Solved Examples on Averages 

Problem 1: The average of 6 numbers is 42. If one number is missing and the sum of the remaining five numbers is 190, find the missing number.

Solution: Given that, the average of 6 numbers x=42

Let the missing number = a 

The sum of the remaining 5 numbers is 190, 

Now, 

$\begin{array}{rl}& \bar{x}=\frac{x_1+x_2+x_3+x_4+x_5+a}{n}\\ & 42=\frac{190+a}{6}\\ & 42\times 6-190=a\\ & a=62\end{array}$

Problem 2: The average marks of 20 students in a class are 56. Later, it was discovered that the marks of one student were wrongly entered as 72 instead of 42. Find the correct average.

Solution: Given that, the incorrect average marks of 20 students = 56

$\begin{array}{rl}& \text{ Incorrect Average }=\frac{\text{ Incorrect sum }}{\text{ Number of Values }}\\ & 56=\frac{\text{ Incorrect sum }}{20}\end{array}$

Incorrect sum = 5620 = 1120

According to the question, 

Correct Sum = 1120 – 72 + 42 = 1090

$\text{ Correct Average }=\frac{1090}{20}=54.5$

Problem 3: A cricketer has scored an average of 50 runs in his first 20 innings. In his next 5 innings, his average score falls by 2 runs. What is his new average score?

Solution: According to the question

Average in first 20 innings $=\frac{\text{ Total runs in first }20\text{ innings }}{20}$

Total runs in first 20 innings =1000

New Average = 50 – 2 = 48

Similarly, 

Total runs in 25 innings = 2548 = 1200

Therefore, 

Runs scored in the last 5 innings = 1200 – 1000 = 200