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In this article we will discuss about the calculation of simple and weighted arithmetic mean with the help of formulas.
Arithmetic mean is a commonly used average to represent a data. It is obtained by simply adding all the values and dividing them by the number of items. Arithmetic mean can be a simple arithmetic mean or weighted arithmetic mean.
1. Simple Arithmetic Mean:
Simple arithmetic mean is calculated differently for different sets of data, that is, the calculation of arithmetic mean differs for individual observations, for discrete series and for continuous series.
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Let us have a look at each of them:
(i) Individual Observations:
(a) Direct Method:
In simple arithmetic mean, there are no frequencies. To calculate simple arithmetic mean under direct method all the observations are added and divided by the total number of items.
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When a variable X takes the values x1, x2, x3, x4, ….xn, the average value of X is given by the formula,
Where, X is the mean, and
N is the number of values.
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The formula can also be written as:
(b) Short-Cut Method:
In short-cut method, an arbitrary origin is taken and deviations are calculated from this arbitrary origin.
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Then, the mean is calculated using the following formula:
Where, A is the assumed mean,
and d is the deviation of the values from the assumed mean.
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(ii) Discrete Series:
In discrete series, the arithmetic mean is calculated by the following methods:
(a) Direct Method:
In direct method, the arithmetic mean is calculated by the following formula:
The above formula shows that the sum of product of frequencies with their respective variables (Σfx) is to be divided by the sum of the frequencies (Σf) to derive arithmetic mean.
(b) Short-Cut Method:
The following formula is used to calculate the mean by this method:
Where, A = Assumed mean,
d = X – A,
Σf = Sum of the frequencies, and
F = Individual frequency.
Under this method, the AM is calculated by multiplying respective frequencies (f) with the deviations (d) of the variables from the assumed mean. Then, this total of the product of deviation and respective frequencies (Σfd) is divided by the sum of the frequencies (Σf) and added to assumed mean (A).
(iii) Continuous Series:
When the data is very large, it may be difficult to add every item and divide it by the number of values to obtain the arithmetic mean; therefore, the data has to be grouped. For instance, if there are 50 students in a class, rather than adding the marks of all the 50 students they can be grouped into different classes such as the number of students who have scored between 0 to 10, 10 to 20, 20 to 30, 30 to 40, and 40 to 50 and so on.
Here, the upper limit of one class is the lower limit of the next class. A student who has scored exactly 10 marks can be included in the 10 to 20 class interval. This method is known as exclusive method. In an inclusive method, the class interval may be taken as 0 to 10, 11 to 20, and 21 to 30 and so on. Mean is, then, calculated by taking the middle value of each class and applying the formula used in discrete series,
(a) Direct Method:
Example:
Calculate the arithmetic mean from the following data:
Solution:
Here, the mid-point for each class is calculated by adding the lower limit and the upper limit and dividing it by 2. For the first class 15-18, it is calculated as (15+18)/2 = 16.5. Then, the midpoints (m) are multiplied by frequencies of the respective classes and the product is divided by sum of frequencies (Σf) to derive AM.
(b) Short-Cut Method or Step Deviation Method:
The average can also be calculated by assuming one of the values from the given figures as the assumed mean.
The mean is then calculated using the following formula:
Where, A is the assumed mean,
f is the frequency of each class,
d = deviations from the mid-point (m – A), and Σf is the total frequency.
The method of calculating the mean taking deviations from the assumed mean is also called as the step deviation method.
Calculation of Arithmetic Mean in Open-End Class Intervals:
Open-end classes are those that do not have a lower or an upper boundary. For example, in a data on income distribution, when the last income class is written as 30 lakhs and above, it is an open end class. And, when the lowest income class is written as less than one lakh, it is also an open-end class.
In such cases, an assumption has to be made about the upper or lower limits. The lower limit could be assumed as zero for the income ‘less than one lakh’ and the upper class limit for the income class ’30 lakhs and above’, could be assumed based on the other class intervals.
For example- observe the following data:
In this example, the appropriate assumption for first class would be 0 – 20 and since the class interval is 20, the appropriate assumption for the last class would be 80 – 100.
2. Weighted Arithmetic Mean:
Simple arithmetic mean gives equal importance to each item in the series. But in practice, the importance of each item in the series may be different. This factor is taken into consideration by weighted arithmetic mean which takes into account the weights (importance) assigned to each and every value. The weights represent the relative importance of each item.
When weights are provided, the arithmetic mean is calculated using the following formula:
Arithmetic mean is a widely used measure of central value due to the following advantages:
1. It is easy to understand.
2. It is simple to compute.
3. It takes each and every item into consideration.
4. It is a reliable measure as the value does not change when computed at different points of time.
1. It is affected by extreme values.
2. It is not an appropriate measure when the distribution is skewed.
3. It is not accurate when items are missing.
4. It cannot be applied when the data is qualitative in nature like honesty, level of satisfaction etc.
5. Assumptions regarding class intervals in case of open end classes may be inaccurate.
Geometric mean is a special type of average.
The geometric mean is the nth root of the product of n values and is symbolically expressed as follows:
Geometric mean is generally used to compare things with different properties. It is a better measure than the arithmetic mean for describing proportional growth or exponential growth. It is applied in the calculation of the Human Development Index (HDI) which is based on three dimensions, namely, life expectancy, education and income. Usage of geometric mean in the calculation of HDI decreases the level of substitutability between dimensions. Geometric mean is also applied in computing financial indices as it is more reliable and a better measure than arithmetic mean.
Harmonic Mean (HM):
Harmonic mean is calculated as the average of the reciprocals of the values given.
It is symbolically expressed as follows:
Harmonic mean is an appropriate measure when average of rates or ratios has to be computed. It is widely applied in physics in calculating quantities such as speed.
Relationship between Arithmetic Mean, Geometric Mean and Harmonic Mean:
Relationship between arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM) can be expressed as:
AM x HM = GM2
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