by Geometric Mean
Geometric Mean (GM) is a measure of central tendency that represents the average rate of change or growth in a set of positive numerical values, especially when values are multiplicative or vary exponentially. Unlike the arithmetic mean, which adds values, the geometric mean multiplies them and then takes the nth root, where n is the number of values.
Geometric Mean (GM) Formula
1. For Individual Data (n values)
GM = ⁿ√(x₁ × x₂ × x₃ × … × xₙ)
Or
GM = antilog [ (Σ log xᵢ) / n ]
2. For Frequency Distribution
GM = antilog [ (Σ f log x) / Σ f ]
Where:
xᵢ = Individual values
f = Frequency of each value
n = Total number of values
log = Logarithm base 10
Σ = Summation symbol
It is particularly useful in areas like finance (e.g., calculating average investment returns), population studies, and compound interest scenarios. GM is also applied when comparing items with different ranges or scales and is best suited for positive values only.
Geometric mean provides a more accurate average when data includes percentages, ratios, or growth rates, making it essential in economic, scientific, and financial analysis.
Methods of Calculation of Geometric Mean:
1. Individual Series
GM = ⁿ√(x₁ × x₂ × x₃ × … × xₙ)
Or,
GM = antilog [ (Σ log x) / n ]
Examples:
Data: 4, 8, 16
Step 1: Multiply values → 4 × 8 × 16 = 512
Step 2: n = 3 (number of values)
Step 3: GM = ³√512 = 8
Geometric Mean = 8
2. Discrete Series
GM = antilog [ (Σ f log x) / Σ f ]
Where:
x = Observed values
f = Frequencies
n = Number of values
Examples:
| Value (x) | Frequency (f) |
|————-|——————-|
| 2 | 3 |
| 4 | 2 |
| 8 | 1 |
Step 1: log₁₀(2) = 0.3010
log₁₀(4) = 0.6020
log₁₀(8) = 0.9030
Step 2: f × log x
- (3 × 0.3010) + (2 × 0.6020) + (1 × 0.9030)
- 0.9030 + 1.2040 + 0.9030 = 3.0100
Step 3: Σf = 6
GM = antilog(3.010 / 6) = antilog(0.5017) ≈ 3.17
Geometric Mean ≈ 3.17
3. Continuous Series
Step 1: Find midpoints (m) of each class
Step 2: Compute log m for each class
Step 3: Multiply log m by corresponding frequency (f)
Step 4: Use formula:
GM = antilog [ (Σ f log m) / Σ f ]
Examples:
| Class Interval | Frequency (f) |
|—————- —|——————-|
| 0–10 | 2 |
| 10–20 | 3 |
| 20–30 | 5 |
Step 1: Midpoints (m) → 5, 15, 25
Step 2: log m → log(5)=0.6990, log(15)=1.1761, log(25)=1.3979
Step 3: f × log m
- 2×0.6990 = 1.398
- 3×1.1761 = 3.5283
- 5×1.3979 = 6.9895
- Σf log m = 11.9158
Step 4: Σf = 10
GM = antilog(11.9158 / 10) = antilog(1.1916) ≈ 15.56
Geometric Mean ≈ 15.56
Harmonic Mean
Harmonic Mean (HM) is a measure of central tendency used to find the average of rates, ratios, or speeds. It is especially useful when values are expressed in terms like “per unit” (e.g., km/hr, price/unit). Unlike the arithmetic mean, the harmonic mean gives more weight to smaller values, making it suitable for situations where reciprocals of values are more relevant.
Harmonic mean is typically used in problems involving average speed (e.g., if a car travels the same distance at different speeds), cost per unit, or productivity rates. It is highly useful in fields like finance, engineering, and physics. Since it is sensitive to very small values, it gives a more accurate representation of averages involving ratios or fractions.
Methods of Calculation of Harmonic Mean:
1. Individual Series
HM = n / (Σ (1/x))
Where:
- n = Number of observations
- x = Each individual value
Examples:
Data: 2, 4, 5
Step 1: Find reciprocals
- 1/2 = 0.5
- 1/4 = 0.25
- 1/5 = 0.20
Step 2: Sum of reciprocals = 0.5 + 0.25 + 0.20 = 0.95
n = 3
Step 3: HM = n / Σ(1/x) = 3 / 0.95 ≈ **3.16**
Harmonic Mean = 3.16
2. Discrete Series
HM = Σf / (Σ (f/x))
Where:
- f = Frequency of each value
- x = Observed value
Examples:
| Value (x) | Frequency (f) |
|————-|——————–|
| 2 | 3 |
| 4 | 2 |
| 5 | 1 |
Step 1: Compute f/x
→ 3/2 = 1.5
→ 2/4 = 0.5
→ 1/5 = 0.2
Step 2: Σf = 6
Σ(f/x) = 1.5 + 0.5 + 0.2 = 2.2
Step 3: HM = Σf / Σ(f/x) = 6 / 2.2 ≈ **2.73**
Harmonic Mean ≈ 2.73
3. Continuous Series
Step 1: Find class midpoints (m)
Step 2: Compute f/m for each class
Step 3: Use formula:
HM = Σf / (Σ (f/m))
Where:
- f = Frequency of each class
- m = Midpoint of each class = (Lower limit + Upper limit) / 2
Examples:
| Class Interval | Frequency (f) |
|——————–|——————-|
| 0–10 | 2 |
| 10–20 | 3 |
| 20–30 | 5 |
Step 1: Midpoints (m): 5, 15, 25
Step 2: f/m
2/5 = 0.40
3/15 = 0.2
5/25 = 0.20
Step 3: Σf = 10
Σ(f/m) = 0.40 + 0.20 + 0.20 = 0.80
Step 4: HM = Σf / Σ(f/m) = 10 / 0.80 = **12.5**
Harmonic Mean = 12.5
