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Relations among AM, GM and HM - Measures of Central Tendency, Business Mathematics & Statistics - Business Mathematics and Statistics - B Com

Relations among A.M., G.M. and H.M. :

1. The Arithmetic Mean is never less than the Geometric Mean, again Geometric Mean is never less than the Harmonic Mean. i.e. A.M. ≥ G. M. ≥ H. M.

Uses of H.M. : Harmonic mean is useful in finding averages involving rate, time, price and ratio.

Example 23 : For the numbers 2, 4, 6, 8, 10, find GM & HM and show that AM > GM > HM

Relations among AM, GM and HM - Measures of Central Tendency, Business Mathematics & Statistics | Business Mathematics and Statistics - B Com

Relations among AM, GM and HM - Measures of Central Tendency, Business Mathematics & Statistics | Business Mathematics and Statistics - B Com

Relations among AM, GM and HM - Measures of Central Tendency, Business Mathematics & Statistics | Business Mathematics and Statistics - B Com

Relations among AM, GM and HM - Measures of Central Tendency, Business Mathematics & Statistics | Business Mathematics and Statistics - B Com

We get A.M. =6, G. M. = 5.211, H.M. = 4.379 i.e. A.M. ≥ G. M. ≥ H. M

Note : In only one case the above relation is not true. When all the variates are equal, we will find that AM = GM = HM

Example 24 : A.M. and G.M. of two observations are respectively 30 and 18. Find the observations. Also find H.M.

Relations among AM, GM and HM - Measures of Central Tendency, Business Mathematics & Statistics | Business Mathematics and Statistics - B Com

Or, xy = 324 or, (60 – y). y = 324, from (1)
Or, y2 – 60 y + 324 = 0 or, (y –54) (y –6) = 0, y = 54, 6
∴y = 54 , x = 6 or, y = 6, x = 54.
∴ Required observations are 6, 54.

Relations among AM, GM and HM - Measures of Central Tendency, Business Mathematics & Statistics | Business Mathematics and Statistics - B Com

The document Relations among AM, GM and HM - Measures of Central Tendency, Business Mathematics & Statistics | Business Mathematics and Statistics - B Com is a part of the B Com Course Business Mathematics and Statistics.
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FAQs on Relations among AM, GM and HM - Measures of Central Tendency, Business Mathematics & Statistics - Business Mathematics and Statistics - B Com

1. What are AM, GM, and HM in statistics?
Ans. In statistics, AM stands for Arithmetic Mean, GM stands for Geometric Mean, and HM stands for Harmonic Mean. These are measures of central tendency used to represent the average or typical value of a set of data.
2. How are AM, GM, and HM calculated?
Ans. The Arithmetic Mean (AM) is calculated by summing all the values in a dataset and dividing it by the number of values. The Geometric Mean (GM) is calculated by taking the nth root of the product of all values in a dataset, where n is the number of values. The Harmonic Mean (HM) is calculated by dividing the number of values by the sum of their reciprocals.
3. When should I use AM, GM, or HM?
Ans. The choice between AM, GM, and HM depends on the nature of the data and the purpose of analysis. AM is commonly used when dealing with continuous data and provides a representative value. GM is useful when analyzing exponential growth or rates of change. HM is often used in situations where the reciprocal of a value is of interest, such as average speed or average resistance.
4. What are the differences between AM, GM, and HM?
Ans. The main difference between AM, GM, and HM lies in their calculation methods and the types of data they represent. AM gives equal weight to each value, GM gives greater weight to larger values, and HM gives greater weight to smaller values. AM can be influenced by extreme values, while GM and HM are less affected. Additionally, AM and GM are always equal or greater than HM.
5. Can AM, GM, and HM be used together?
Ans. Yes, AM, GM, and HM can be used together to gain a more comprehensive understanding of a dataset. For example, when analyzing financial data, one can use AM to calculate the average return on investment, GM to calculate the compound annual growth rate, and HM to calculate the average cost of borrowing. Each measure provides valuable insights into different aspects of the data.
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