Geometric Mean Calculator – ज्यामितीय माध्य कैलकुलेटर | Ganit Calculator

Geometric Mean Calculator with Frequency – ज्यामितीय माध्य कैलकुलेटर | Growth Rate Calculator

Geometric Mean Calculator

ज्यामितीय माध्य कैलकुलेटर

Calculate geometric mean for growth rates, ratios, and percentage increases with frequency support

Simple Data
With Frequency

All values must be positive numbers greater than 0.

Calculation Result
Geometric Mean: 21.45
\[ \text{Geometric Mean} = \sqrt[n]{x_1 \times x_2 \times x_3 \times \ldots \times x_n} \]

How to use the Geometric Mean Calculator:

  1. Select data type: Simple Data or With Frequency
  2. Enter your data values according to the selected type
  3. For frequency data, enter both values and their frequencies
  4. Click Calculate to see the geometric mean with detailed calculation
  5. Download the result for future reference

When to use Geometric Mean:

  1. For calculating average growth rates or compound interest
  2. For ratios and percentage increases
  3. When values have different ranges or units
  4. For data that follows exponential patterns
  5. With frequency data when values repeat multiple times
Simple Data
With Frequency

All values must be positive numbers greater than 0.

Calculation Result
Geometric Mean: 21.45
\[ \text{Geometric Mean} = \sqrt[n]{x_1 \times x_2 \times x_3 \times \ldots \times x_n} \]

How to use the Geometric Mean Calculator:

  1. Select data type: Simple Data or With Frequency
  2. Enter your data values according to the selected type
  3. For frequency data, enter both values and their frequencies
  4. Click Calculate to see the geometric mean with detailed calculation
  5. Download the result for future reference

When to use Geometric Mean:

  1. For calculating average growth rates or compound interest
  2. For ratios and percentage increases
  3. When values have different ranges or units
  4. For data that follows exponential patterns
  5. With frequency data when values repeat multiple times
Geometric Mean Calculator - ज्यामितीय माध्य कैलकुलेटर: पूरी गाइड | Ganit Calculator

Geometric Mean Calculator - ज्यामितीय माध्य कैलकुलेटर: पूरी गाइड

Geometric Mean (ज्यामितीय माध्य) statistics का एक powerful measure of central tendency है जो growth rates, ratios, और percentage increases के लिए specially designed है। इस comprehensive guide में हम geometric mean के सभी aspects को detailed तरीके से cover करेंगे।

Geometric Mean क्या है? (What is Geometric Mean?)

Geometric Mean या ज्यामितीय माध्य n numbers का nth root of their product होता है। यह multiplicative relationships और growth rates के लिए ideal है।

\[ \text{Geometric Mean} = \sqrt[n]{x_1 \times x_2 \times x_3 \times \ldots \times x_n} \]

Geometric Mean के प्रकार और Calculation Methods

← Scroll horizontally to view full table →
Data Type Formula उपयोग
Simple Data \[ GM = \sqrt[n]{x_1 \times x_2 \times \ldots \times x_n} \] Individual values, growth rates
With Frequency \[ GM = \exp\left(\frac{\sum f \cdot \ln(x)}{\sum f}\right) \] Frequency distribution data
Logarithmic Form \[ GM = e^{\frac{\sum \ln(x)}{n}} \] Computational efficiency

Calculation Methods

1. Simple Geometric Mean

जब data individual values के form में हो:

\[ \text{Geometric Mean} = \sqrt[n]{x_1 \times x_2 \times x_3 \times \ldots \times x_n} \]

2. Geometric Mean with Frequency

जब values और उनकी frequencies दी गई हों:

\[ \text{Geometric Mean} = \exp\left(\frac{\sum f \cdot \ln(x)}{\sum f}\right) \]

3. Alternative Logarithmic Form

Computational efficiency के लिए logarithmic form:

\[ \text{Geometric Mean} = e^{\frac{\ln(x_1) + \ln(x_2) + \ldots + \ln(x_n)}{n}} \]

Key Properties of Geometric Mean

Multiplicative Nature

Geometric mean multiplicative relationships को handle करता है, जबकि arithmetic mean additive relationships के लिए better है।

Scale Invariant

Geometric mean scale changes से affect नहीं होता। अगर सभी values को same factor से multiply करें, तो GM भी उसी factor से multiply होगा।

Always ≤ Arithmetic Mean

किसी भी positive data set के लिए, geometric mean हमेशा arithmetic mean से कम या equal होता है।

Applications of Geometric Mean

Growth Rates Calculation

Compound annual growth rates (CAGR), investment returns, population growth rates की calculation के लिए ideal

Ratio Analysis

Financial ratios, performance ratios, और अन्य ratios के average के लिए perfect

Percentage Changes

Successive percentage increases/decreases के average calculation के लिए accurate results देता है

Quality Control

Manufacturing processes में product quality ratios और performance indices के लिए use होता है

Solved Examples

Example 1: Simple Geometric Mean

Problem: 2, 4, 8 का geometric mean निकालें

Solution:
Values: 2, 4, 8
\[ n = 3 \]
\[ \text{Product} = 2 \times 4 \times 8 = 64 \]
\[ \text{Geometric Mean} = \sqrt[3]{64} = 4 \]
\[ \text{Alternative method: } GM = e^{\frac{\ln(2) + \ln(4) + \ln(8)}{3}} = e^{\frac{0.6931 + 1.3863 + 2.0794}{3}} = e^{1.3863} = 4 \]

Example 2: Geometric Mean with Frequency

Problem: निम्नलिखित data का geometric mean निकालें:

← Scroll horizontally to view full table →
Value (x)Frequency (f)
23
42
81
Solution:
← Scroll horizontally to view full table →
xfln(x)f × ln(x)
230.69312.0794
421.38632.7726
812.07942.0794
Total66.9314
\[ \sum f = 6 \]
\[ \sum f \cdot \ln(x) = 6.9314 \]
\[ \text{Geometric Mean} = \exp\left(\frac{6.9314}{6}\right) = \exp(1.1552) = 3.1748 \]

Example 3: Growth Rate Calculation

Problem: एक company के revenue 3 years में क्रमशः 10%, 20%, और 15% बढ़े। Average annual growth rate निकालें।

Solution:
Growth factors: 1.10, 1.20, 1.15
\[ \text{Geometric Mean} = \sqrt[3]{1.10 \times 1.20 \times 1.15} \]
\[ = \sqrt[3]{1.518} = 1.149 \]
\[ \text{Average Annual Growth Rate} = 1.149 - 1 = 0.149 = 14.9\% \]

Geometric Mean vs Arithmetic Mean

Aspect Geometric Mean Arithmetic Mean
Formula √[n](x₁ × x₂ × ... × xₙ) (x₁ + x₂ + ... + xₙ)/n
Use Case Growth rates, ratios, percentages Additive data, normal distributions
Extreme Values Less affected by extremes Highly affected by extremes
Data Requirements All values must be positive Any real numbers
Relationship GM ≤ AM (always) AM ≥ GM (always)

Frequently Asked Questions (FAQ)

1. Geometric mean कब use करना चाहिए?

Geometric mean use करें जब:

  • Growth rates या compound interest calculate करना हो
  • Ratios या percentages का average निकालना हो
  • Data multiplicative pattern follow करता हो
  • Extreme values के effect को minimize करना हो

2. Geometric mean में negative numbers क्यों नहीं use कर सकते?

Geometric mean में negative numbers use नहीं कर सकते क्योंकि:

  • Negative numbers का logarithm undefined होता है
  • Even number of negative numbers का product positive होता है, जो misleading results दे सकता है
  • Mathematically inconsistent results आते हैं

3. Geometric mean हमेशा arithmetic mean से छोटा क्यों होता है?

AM-GM inequality के according, किसी भी set of positive numbers के लिए arithmetic mean हमेशा geometric mean से बड़ा या equal होता है। Equality केवल तब होती है जब सभी numbers equal हों।

Key Points to Remember

  • Geometric mean multiplicative relationships के लिए ideal है
  • All values must be positive (> 0)
  • Growth rates और ratios के लिए arithmetic mean से better है
  • Less sensitive to extreme values compared to arithmetic mean
  • Can be calculated using product method or logarithmic method
  • Always less than or equal to arithmetic mean (AM ≥ GM)

Important Note

Geometric mean केवल positive numbers के लिए defined है। अगर आपके data में zero या negative numbers हैं, तो geometric mean calculate नहीं कर सकते। ऐसे cases में arithmetic mean या other measures का use करें।

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