Arithmetic Mean Vs Geometric Mean
The main difference between the arithmetic mean and geometric mean is that the arithmetic mean is related to the sum and the geometric mean is related to the product of data values. Let us see what is the arithmetic mean vs geometric mean in several perspectives.
| 1. | What is Arithmetic Mean Vs Geometric Mean? |
| 2. | Difference Between Arithmetic Mean and Geometric Mean Table |
| 3. | FAQs on Arithmetic Mean Vs Geometric Mean |
What is Arithmetic Mean Vs Geometric Mean?
Here is arithmetic mean vs geometric mean in terms of their formulas. For a set of data values, say, x₁, x₂, x₃, ..., xₙ,
- The arithmetic mean (AM) = (x₁ + x₂ + x₃ + ... + xₙ) / n.
- The geometric mean (GM) = (x₁ · x₂ · x₃ · ... · xₙ)1/n.
Example: For the values 1, 3, 5, 7, and 9:
Arithmetic mean = (1 + 3 + 5 + 7 + 9) / 5 = 5.
Geometric mean = (1 × 3 × 5 × 7 × 9)1/5 ≈ 3.93.
Thus, arithmetic mean is the sum of the values divided by the total number of values. In other words, the arithmetic mean is nothing but the average of the values. On the other hand, the geometric mean is the product of the values raised to the multiplicative inverse of the total number of values. This is the difference between AM and GM in terms of meaning and formula. Here are the differences between arithmetic mean and geometric mean in several ways.
Difference in Terms of Results
The geometric mean for a set of data values is always less than (or equal to) that of the arithmetic mean. We can see in the above example that 3.93 (GM) < 5 (AM).
Difference in Terms of Data Values
The geometric mean applies only to positive values whereas the arithmetic mean applies to both positive and negative values.
Difference in Terms of Effect of Outliers
The geometric mean doesn't get affected much by the outlier whereas the arithmetic mean does. For example, consider a set of data values with an outlier, say, 10, 12, 14, and 99. Let us calculate AM and GM.
AM = (10 + 12 + 14 + 99) / 4 = 33.75
GM = (10 × 12 × 14 × 99)1/4 ≈ 20.19.
We can see that most of the data values are very far from AM whereas GM is not that much affected.
Difference in Terms of Ease of Use
Arithmetic mean is easy to use as it involves the sum whereas the geometric mean is difficult to use as it involves the product and taking roots.
Difference in Terms of Accuracy
AM is accurate when the data values are not skewed and are independent of each other. GM is more accurate when there is volatility in the data.
Difference in Terms of Application
The arithmetic mean (which is nothing but average) is widely used in the fields of statistics, economics, history, and sociology. The geometric mean (which is nothing but compounded growth) is used to calculate the average growth rates in finance.
Difference Between Arithmetic Mean and Geometric Mean Table
The differences between AM and GM that are mentioned in the previous section are summarized in the table below.
| Arithmetic Mean (AM) | Geometric Mean (GM) |
|---|---|
| 1. It is calculated by dividing the sum of data values by the number of data values. | 1. It is calculated by raising the product of data values by reciprocal of the number of data values. |
| 2. For a given set of data values x₁, x₂, x₃, ..., xₙ, the arithmetic mean = (x₁ + x₂ + x₃ + ... + xₙ) / n. | 2. For a given set of data values x₁, x₂, x₃, ..., xₙ, the geometric mean = (x₁ · x₂ · x₃ · ... · xₙ)1/n. |
| 3. AM ≥ GM always for any set of data values. | 3. GM ≤ AM always for any set of data values. |
| 4. It applies to both positive and negative values. | 4. It applies only to positive values. |
| 5. It is affected by outliers. | 5. It is not much affected by outliers. |
| 6. It is easy to use. | 6. It is harder to use compared to AM. |
| 7. It gives a good and accurate approximation when there is not much variation in the data. | 7. It gives a good and accurate approximation when there is much variation in the data. |
| 8. It is mostly used in the fields of maths and statistics. | 9. It is used mostly in the field of finance. |
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FAQs on Arithmetic Mean Vs Geometric Mean
What is the Difference Between Arithmetic Mean and Geometric Mean Formulas?
For any set of values a₁, a₂, a₃, ..., aₙ, the formula for arithmetic mean is (a₁ + a₂ + a₃ + ... + aₙ) / n. The formula for geometric mean for the same set of data values is (a₁ a₂ a₃ ... aₙ)1/n.
What is Arithmetic Mean Vs Geometric Mean Inequality?
The arithmetic mean (AM) is always greater than or equal to the geometric mean (GM). i.e., AM ≥ GM.
What is the Difference Between Arithmetic Mean and Geometric Mean Applications?
Arithmetic mean is not only used in the fields of maths and statistics but also in the fields like history, sociology, etc. Geometric mean as it is compounded growth is used in the field of finance.
What is Arithmetic Mean Vs Geometric Mean Vs Harmonic Mean Relation?
The two important relationships between arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM) are:
- AM ≥ GM ≥ HM
- AM × HM = GM2