what is arithmetic mean and its importance

THE ARITHMETIC MEAN, Definition of arithmetic mean, also popularly referred to as the ‘mean’ is the average of a series of figures or values.

mean is widely used in various fields, including mathematics, statistics, economics, and everyday life. It provides a way to summarize a set of values into a single representative value

It is obtained by dividing the sum of these figures by the total number of the figures or values. It is also the average of a collection of observations. The arithmetic mean is the most popularly used measure of central tendency.

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The arithmetic mean, also known as the average, is a common measure of central tendency used in statistics and mathematics. It is calculated by summing up a set of values and then dividing the sum by the total number of values.

To find the arithmetic mean, follow these steps:

Add up all the values in the set.
Count the total number of values in the set.
Divide the sum by the total number of values.

Mathematically, the arithmetic mean (M) of a set of values (x1, x2, x3, …, xn) is given by the formula:

M = (x1 + x2 + x3 + … + xn) / n

Where: M = Arithmetic mean x1, x2, x3, …, xn = Individual values in the set n = Total number of values in the set

For example, let\’s say we have a set of numbers: 5, 8, 10, 12, and 15. To find the arithmetic mean, we add up all the values and divide by the total number of values:

Arithmetic mean = (5 + 8 + 10 + 12 + 15) / 5 Arithmetic mean = 50 / 5 the mean is = 10

Therefore, the mean of the given set of numbers is 10.

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Formula for calculating arithmetic mean

Arithmetic mean, x =

Where

x = arithmetic mean

Ʃ = represents a Greek letter denoting “sum of”

x = series of figures in a given data

Ʃx = the total of the values of a series of figures in a given data

n = number of figures or elements

Note: This formula is used especially when the figures are small and ungrouped.

Example 1

Calculate the arithmetic mean for the scores of eight students in the NECO economics examination in the year 1999. The scores are: 14, 18, 24, 16, 30 12, 20 and 10.

Solution

Step I: Add up the numbers or scores

Ex = 14 + 18 + 24 + 16+ 30 + 12 + 20 + 10 = 144

Step II: N = no. of figures or scores which is = 8

Step III: x = Ex =144

Note:

When the figures given in a data are large and in most cases repeat themselves, then frequencies are used. Frequency is the number of times a particular event or information occurs.

Frequency distribution is usually used when data presented are large and most of the numbers appear more than once.

In this case the formula for calculating the arithmetic mean will change slightly to:

Arithmetic mean, x = >>>>>>>>>>>>>>>>>

Where f = no. of times a particular number occurs (frequency). Other symbols remain the same.

Example 2

Calculate the mean of the following sets of numbers:

8, 16, 24, 8, 12, 12, 16, 18, 24, 10, 16, 20, 24, 24, 12, 24, 12, 16, 24, 18, 18.

Solution $\"soil$

Step I: Identify the numbers that occur in the set, i.e. 8, 10, 12, 16, 18, 20 and 24. Arrange these numbers in a frequency distribution table (table 2.11).

Step II: Arrange the numbers starting from the smallest number, which is 8, to the highest number, which is 24, as shown in table 2.11.

Step III: Arrange the figures or numbers in a frequency distribution table as shown in table 2.11.

Table 2.11: Frequency distribution

Numbers (X)	Tally or counts	Frequency (f)
8 10 12 16 18 20 24	II I IIII III III I ~~IIII~~ I	2 1 4 3 3 1 6

N = 20

Step IV: Apply the formula

Arithmetic mean, x =

= (8 x 2) + (10 x 1) + (12 x 4) + (16 x 3) + (18 x 3) + (20 x 1) + 16+10+48+48+54+20+144

= 340

20 =17.0

Mean of group data

The arithmetic mean can also be prepared for grouped data. In this case, the class mark (mid-points) of the individual class interval is used for the X- column.

Formula used is

Arithmetic mean x = Ʃfx

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Example of how to calculate the arithmetic mean

Calculate the mean of the following marks scored by students in an economics examination:

8, 31, 45, 38, 22, 28, 16, 51, 65, 48, 6, 24,

18, 12, 16, 48, 38, 50, 44, 6, 18, 16, 24, 32,

36, 26, 14, 20, 18.

Solution

Use the steps as in example 2
Use a class interval of 0 – 9, 10 – 19, 20 – 29, etc. as shown in table 2.12.

Prepare a frequency table as in Table 2.12

Table 2.12: Frequency table for marks scored by students in economics examination.

Arithmetic mean x =

=815

30 = 27.2

Advantages of arithmetic mean

The arithmetic mean is very easy to calculate.

It gives an exact value

Arithmetic mean is the best known average.

It is very easy to understand.

Arithmetic mean provides a good method of comparing values

It makes use of all available information in data

Disadvantages of arithmetic mean

Arithmetic means cannot be obtained graphically.

Certain facts in mean may not be revealed.

It may be difficult to obtain without calculations.

It can lead to distorted results.

mean may be badly affected by extreme values in a distribution.