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. 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 observation. The arithmetic mean is the most popularly used measure of central tendency.

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 series of figures in a given data

n = number of figures or elements

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

Example 1

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

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4. ORGANIC MANURING
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  4. FARM YARD MANURE
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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

         N

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

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 countsFrequency (f)
8 10 12 16 18 20 24II I IIII III III I IIII I2 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) + (24 x 6)

20

=          16+10+48+48+54+20+144

                                    20

=          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

                                        Ʃf

Example of how to calculate arithmetic mean

Calculate the mean of the following marks scored by students in an 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.

dopportunity cost

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 examination.

Scores (Groupings)X Class MarkTally or EventsF Freq.(fx)
0 – 9 10 – 19 20 – 29 30 – 39 40 – 49 50 – 59 60 – 69 4.5 14.5 24.5 34.5 44.5 54.5 64.5III IIII IIII IIII I IIII IIII II I3 9 6 5 4 2 113.5 130.5 147.0 172.5 178.0 109.0 64.5
   Sf=30Sfx=815

Arithmetic mean x =

=815

    30    = 27.2

Advantages of arithmetic mean

  • 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 arithmetic mean may not be revealed.
  • It may be difficult to obtain without calculations.
  • Arithmetic mean can lead to distorted result.
  • Arithmetic mean may be badly affected by extreme values in a distribution.
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