Science

A Detailed Overview of The Average Concept

Everyone is not good at mathematics as it is not a simple subject at all. Due to horrible looking formulas and equations, many students hate this subject. But gone are the days when it used to be a tension as the free calculate average by calculator-online.net helps you to analyse various maths concepts. So let’s move on and discuss how this particular term and its relevant tool aids in minimising the complexity involved during calculations.

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What Is The Average?

In layman’s words, an average is a single number that reflects the whole collection of uneven numbers. The total of provided observations divided by the number of observations “n” yields an average. The letter A stands for it. Arithmetic mean is another name for average. The central tendency of a group of data observations is called the arithmetic mean. To calculate the average amount, you can use an online calculate average that is specially designed to determine the average value of any data set or a number of values.

Average Formula:

If you wish to calculate the average of a group of numbers, simply add them all together and divide by the number of numbers you have. For a better understanding, consider the following formula for average:

The average equation is as follows:

Average of a set of numbers = Sum of the terms / Number of the terms

Remember that when computing the average for a bunch of integers, the easy to use average calculator by calculator-online.net employs the same algorithm.

Properties of Average:

Being a mathematical term, average also exhibits certain properties that you must have a look upon. So let’s get ahead towards it now:

(i)

The supplied data’s average is smaller than the biggest observation and bigger than the smallest observation.

Consider the following scenario: 

Calculate the average of 4, 8, 10, and 14

Here we have:

Average = 4+8+10+14/4

Average = 36/4

Average = 9

The average here is 9, which is lower than the highest observation (14) but higher than the smallest observation (nine).

(ii)

If all of the data’s observations are the same, the average will be the same as the observations.

Consider the scenario given below:

Compute the average of  2, 2, 2, 2, and 2

Here we have:

Average = 2+2+2+2+2+/5

Average = 10/5

Average = 2

The average is the same as the observations in this case. The free calculate average also 

depicts the same results but immediately and saving you a lot of time. 

(iii)

If 0 is one of the observations in the supplied collection of data, it should be included in the average calculation as well.

Consider the following scenario: 

Calculate the average of 3, 10, and 8

Now we have:

Average = 3+10+8/3

Average = 21/3

Average = 7

Use the best online calculating average in case you feel it is difficult to figure out.

(iv)

If the value of each measurement is raised or lowered by a certain amount, the average will be raised or lowered by the same amount.

Let’s consider the example below:

We have the following data values:

2, 5, 6, 8, and 9

No doubt their average will be:

Average = 2+5+6+8+9/5

Average = 30/5

Average = 6

Now if we add 3  to each value given, then the average is:

5, 8, 9, 11, and 12

Average = 5+8+9+11+12/5

Average = 45/5

Average = 9

The free calculate average by calculator-online.net also aids you to determine the same increment or decrement in the values immediately. 

The Types of Average Problems:

In this section below, we will be discussing various methodologies in which the average concept is broadly taken into consideration. Let’s go!

Taking The Average of A Set of Data:

This is one of the most straightforward sorts of questions found in competitive examinations. We can use the formula right away in this case. In this style of inquiry, the total number of values is divided by the sum of all supplied values.

Average = sum of values given/total values in number

Example:

Determine the average of the numbers given below:

2, 8, 5, 64, and 1

Solution:

Sum of given values = 2+8+5+64+1

Sum of given values = 80

Total number of values = 5

As we know:

Average = sum of values given/total values in number

Average = 80/5

Average = 16

You can get the same results in no time by subjecting yourself to the free online average calculator. 

Finding The Average of A Set of Numbers:

These questions are similar to type 1 questions, except they can be answered using straight equations. Rather than adding the provided set of numbers one by one, it is better to know and memorise the direct equations shown below. Applying the technique will take significantly less time, and we all know how important time management is in any competitive test.

 

Average of first “n” natural numbers (n+1)/2
Average of first “n” even numbers n+1
Average of first “n” odd numbers n
Average of consecutive numbers First number + Last number/2
Average of 1 to “n” odd numbers Last odd number + 1 /2
Average of 1 to “n” even numbers Last even number + 2/2
Average of squares of first “n” natural numbers (n+1)(2n+1)/6
Average of cubes of first “n” natural numbers n(n+)^2/2
Average of “n” multiples of any numbers Number * (n+1)/2

Items May Be Added or Removed:

Certain elements must be added to these questions, and then the impact of these additions on the final average value must be determined. The value of added observations is n(b-a)+b if the average of “n” observations is “a” but becomes “b” when one observation is added.

Certain things are to be Avoided in these questions, and then the influence of these showdowns on the final average value is determined. If the average of “n” observations is “a,” but when one is removed, the average becomes “b,” the value of the deleted observations is n(a-b)+b.

Example:

The mean weight of the 21 boys was 64 kg. When the teacher’s weight was included in, the average rose by 1 kilogram. What was the weight of the teacher?

Solution:

We must utilise the formula n(b-a)+b in this case.

n=21, a=64, and b=(64+1)=65

We obtain as a result of substituting values,

⇒21(65-64)+65

⇒21(1)+65

⇒21+65=86

As a result, the teacher’s weight is 86 kg. You can also verify this result by using a free averaging calculator.

Example:

The average height of 100 large trees in a garden is 3 metres, while the height of a Banyan tree is also 3 metres. If the height of the Banyan tree is removed, the average falls by 2 metres. What is the height of a Banyan tree?

Solution:

We must utilise the formula n(a-b)+b in this case.

n=100, a=3, and b=(3-2)=1. (the old average + the new observation)

We obtain the following as a result of substitution of values:

⇒101(3-1)+1

⇒101(2)+1

⇒203

As a result, the banyan tree’s height is 203 metres.

Calculating The Average After Data Restoration:

Some of the data in these questions will be replaced. There can sometimes be a discrepancy between the real and substituted values. Consider a set of “n” observations, some of which are replaced by fresh observations (a1, a2, a3, etc.). And if the mean goes up or down by “b,” then we have:

Value of new observations= a ± nb

Example:

When one of the females, who is 126 cm tall, is replaced by another girl, the average height of the three girls increases by 4 cm. What is the new girl’s height?

Solution:

We must use the formula in this case,

a + nb = new observation value

(As the average grew, we added a + symbol.)

n=3, a=100cm, and b=4cm

new girl’s height=100+34=112 cm

You can also verify the results by using the best average calculator absolutely for free. 

Using A Weighted Average:

Two or more separate sets of data are combined in these questions to generate a new group. The weighted average is the name for this form of average. It is vital to highlight that the particular group’s average is known.

Weighted average = sum of data from all groups/number of data values of all the groups

If there are Z sets containing averages A1, A2, A3, A4, Ax and n1, n2, n3, n4,….nx members, the weighted average is calculated as follows:

Weighted Average = n1A1+n2A2+n3A3+…+nxAx/n1+n2+n3+…+nx

Example: 

A man travels to work at 60 km/hr and comes home at 30 km/hr along the same route. Calculate average speed.

Solution:

Here we have the formula as follows:

Weighted Average = n1A1+n2A2/n1+n2

Weighted Average = 60*65+40*45/60+40

Weighted Average = 51

You can also take into consideration the use of the free calculate average to resolve such complicated queries.

Statistical Types of Average:

In statistical analysis, the concept of average differs a little. It is not only restricted to the calculated the most common value among all. Let’s have a look at various average kinds that take the concept of average in another domain!

Mean:

It is also considered as the normal average. Yes, if you look at its formula below, you will understand yourself:

 Mean = sum of values in a data set/number of values

It is also known as the arithmetic mean. Just suppose you have the following generic data set:

a_1, a_2, a_3, …, a_n-1, a_n 

For this set of values, we have the average as follows: 

Mean = a_1, a_2, a_3, …, a_n-1, a_n/n 

This expression can be displayed by the following notation as well: 

Mean Types: 

Following are a couple of mean types that can also be simplified by using free average calculators.

Population Mean:

A particular mean of the whole population sample is known as the population mean.

Sample mean:

It is considered the mean of the small sample of the population given.

Median:

It is the second type of the average. It is defined as the number that is present right in the middle after the data set is arranged from least to greatest. 

Median displays the relation of the entire set of the numbers that is given. In the light of statistics, the mean is used to describe the measure of dispersion of the data set values.

Mode:

In a data set,  the number or the value that repeats itself most of the time is referred to as the mode.

You can also determine the mode with the use of the free to use average calculator proposed by calculator online.

The interesting fact here to consider is that there  can be more than a single mode for one data set. It depends upon the frequency of the numbers as long as the numbers have the same frequency and frequency is the highest as well,  you have to consider all those numbers as the modals of the data set.

Mode For Grouped Data:

You can determine the mode for the grouped data by subjecting to either calculate average or the formula as follows:

 Mode = L+(f_1-f_o/2f_1-f_o-f_2)*h

Where: 

L = lower class limit of model class

F_1 = model class’ frequency

F_o = frequency of class before modal class

F_2 = frequency of class after modal class

H = class interval

Range:

In a data set of values, the range is defined as the difference between the highest and the lowest value. 

Range = Highest Value – Lowest Value 

Wrapping It Up:

So far, we had a detailed discussion of the topic under  consideration in this read. Average plays an important role in our  lives. When you have less number of packages to distribute among a large audience, there comes the average that helps you to estimate how much quantity of food each person will be getting. Moreover, the free average calculator also aids in this regard a lot to know how to find average instantly. We hope this guidepost is going to be a matter of discussion for you.

 

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