Arithmetic Mean Formula: Definition, Limitations, Solved Examples

Arithmetic Mean in statistics is used to find central tendencies of skewed distribution. However, apart from being used in mathematics, its application is evident in almost every academic field like economics, anthropology and history. It is also applicable in finding per capita income of a demographic or nation’s population, measuring average temperature of earth to measure global warming or calculate average score in sports such as cricket. 

Key Terms: Arithmetic Mean, average, statics, Direct Method, Short cut method, Deviation Method

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What is Arithmetic Mean?

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Arithmetic mean is generally used for calculating a mean or average of a grouped data. It is calculated by dividing the sum of a collection of integers by the number of numbers in the set. It is used when all of the values in a set of data have the same measuring unit, such as heights, miles, hours, and so on.

Consider if you are asked to find the arithmetic mean of 13, 6, 9. Then you first find the sum of the data and divide it with the count of data. 13+ 6+ 8 = 27, so the arithmetic mean is 9. Arithmetic mean is sometimes called just ‘mean’ or ‘average’. In the next section we will see the formula to find the mean of cumulative data.

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Formula for Arithmetic Mean

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In a discrete frequency distribution the arithmetic mean may be computed by any one of the following methods:

1. Direct Method
2. Short cut method
3. Step deviation method

Direct Method

If x1, x2, ... , xn has corresponding f1, f2, … fn frequencies, then the arithmetic mean is given by,

X = (f1 x1 + f2 x2 + …. +fn xn) / (f1 + f2 +....+ fn)

Or, X= (\(\sum_{i=1}^{n}\)fi xi) / (\(\sum_{i=1}^{n}\)fi)

Example 1. Find the arithmetic mean of the following distribution:

Solution: 

Using the formula,

X= (\(\sum_{i=1}^{n}\)fi  xi) / (\(\sum_{i=1}^{n}\)fi)

X = 360 / 40 = 90

Therefore the arithmetic mean of the distribution is 90.

Short Cut Method

If values of the frequencies are large then the calculation of arithmetic mean becomes very difficult and time consuming. So, a better approach is the shortcut method.

Let x1, x2, ... ,xn has f1, f2, ... ,fn respective frequencies, taking deviation about a random point ‘A’ we get,

di = xi - A

fi di = fi(xi - A)

\(\sum_{i=1}^{n}\) fi di = \(\sum_{i=1}^{n}\) fi (xi - A)

Or, \(\sum_{i=1}^{n}\)fi di =\(\sum_{i=1}^{n}\) fi xi -\(\sum_{i=1}^{n}\)fi A

Or, \(\sum_{i=1}^{n}\)fi di =\(\sum_{i=1}^{n}\) fi xi -AN [here \(\sum_{i=1}^{n}\)fi = N ]

Or, (1/N)(\(\sum_{i=1}^{n}\)fi di) = (1/N)(\(\sum_{i=1}^{n}\) fi xi) -A

Or, X = A + (1/N)(\(\sum_{i=1}^{n}\) fi di)

Example: Find the Arithmetic mean of the following distribution table:

Weight: 67 70 72 73 75

No. of students: 4 3 2 2 1

Solution:

Let assume A = 72

Using the formula, we get

 X = A + (1/N)(\(\sum_{i=1}^{n}\)fi di)

X = 72 + (1/12)(-21) = 70.25kg 

Therefore the mean weight is 70.25 kg.

Also read Difference Between Variance and Standard Deviation

Step deviation Method

If there is a visible common number in the data distribution then the step deviation method shortens the calculation of the mean.

Consider h is the visible common number

ui = (xi - A)/h

Or, xi = A + h ui

Or, fi xi = fi A +hf ui

Or, \(\sum_{i=1}^{n}\)fi xi =A\(\sum_{i=1}^{n}\) fi + h\(\sum_{i=1}^{n}\)fi ui

Or, (1/N)(\(\sum_{i=1}^{n}\)fi xi) = (A/N)(\(\sum_{i=1}^{n}\) fi) + (h/N) \(\sum_{i=1}^{n}\)fi ui

Or, X = A +h(1/N) \(\sum_{i=1}^{n}\)fi ui

Example: Find the average of the following distribution table:

Solution:

Let A = 900 and h = 20

Using the formula, we get,

X = A +h(1/N) \(\sum_{i=1}^{n}\)fi ui

X = 900 + (20/100) (-44) = 900 -8.80 =891.2

Therefore, the average of the given data distribution is 891.2.

Read Also: Degree of Polynomial

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Limitations of Arithmetic Mean

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Following are the Arithmetic mean limitations.

• Extreme values have a big impact on the arithmetic mean.
• It is unable to properly average percentages & ratios.
• For strong data distribution, it is not a suitable average.
• If any piece is missing, it is impossible to calculate precisely.
• The mean does not always agree with any of the observations.

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Things to Remember

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• Arithmetic mean is generally used for calculating a mean or average of a grouped data. 
• Arithmetic mean is calculated by dividing the sum of a collection of integers by the number of numbers in the set.
• In a discrete frequency distribution the arithmetic mean may be computed by Direct Method, Short Shortcut method, & Step deviation method.
• Direct method:If x1, x2, ... , xn has corresponding f1, f2, … fn frequencies, then the arithmetic mean is given by, X= (\(\sum_{i=1}^{n}\)fi xi) / (\(\sum_{i=1}^{n}\)nfi).
• Shortcut Method: X = A + (1/N)(\(\sum_{i=1}^{n}\)fi di), where A is an arbitrary point and di = xi - A.
• Step deviation method: X = A +h(1/N) \(\sum_{i=1}^{n}\)fi ui , ui = (xi - A)/h

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