Harmonic Mean Calculator – हार्मोनिक माध्य कैलकुलेटर | Speed, Rates Calculator

Harmonic Mean Calculator – हार्मोनिक माध्य कैलकुलेटर | Speed, Rates Calculator

Harmonic Mean Calculator

हार्मोनिक माध्य कैलकुलेटर

Calculate harmonic mean for speed, rates, ratios with frequency support

Simple Data
With Frequency

All values must be positive numbers greater than 0.

Calculation Result
Harmonic Mean: 3.43
\[ \text{Harmonic Mean} = \frac{n}{\sum \frac{1}{x}} \]

How to use the Harmonic 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 harmonic mean with detailed calculation
  5. Download the result for future reference

When to use Harmonic Mean:

  1. For calculating average speed when distances are equal
  2. For rates and ratios (like price-earnings ratio)
  3. When dealing with time and rate problems
  4. For data expressed as rates per unit
  5. With frequency data when values repeat multiple times

Comparison of Three Means:

Mean Type Formula (Simple) Formula (With Frequency) Use Case
Arithmetic Mean \(\frac{\sum x}{n}\) \(\frac{\sum f \cdot x}{\sum f}\) General average
Geometric Mean \(\sqrt[n]{\prod x}\) \(\exp\left(\frac{\sum f \cdot \ln(x)}{\sum f}\right)\) Growth rates, ratios
Harmonic Mean \(\frac{n}{\sum \frac{1}{x}}\) \(\frac{\sum f}{\sum \frac{f}{x}}\) Speed, rates
Simple Data
With Frequency

All values must be positive numbers greater than 0.

Calculation Result
Harmonic Mean: 3.43
\[ \text{Harmonic Mean} = \frac{n}{\sum \frac{1}{x}} \]

How to use the Harmonic 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 harmonic mean with detailed calculation
  5. Download the result for future reference

When to use Harmonic Mean:

  1. For calculating average speed when distances are equal
  2. For rates and ratios (like price-earnings ratio)
  3. When dealing with time and rate problems
  4. For data expressed as rates per unit
  5. With frequency data when values repeat multiple times

Comparison of Three Means:

Mean Type Formula (Simple) Formula (With Frequency) Use Case
Arithmetic Mean \(\frac{\sum x}{n}\) \(\frac{\sum f \cdot x}{\sum f}\) General average
Geometric Mean \(\sqrt[n]{\prod x}\) \(\exp\left(\frac{\sum f \cdot \ln(x)}{\sum f}\right)\) Growth rates, ratios
Harmonic Mean \(\frac{n}{\sum \frac{1}{x}}\) \(\frac{\sum f}{\sum \frac{f}{x}}\) Speed, rates
Harmonic Mean Calculator - हार्मोनिक माध्य कैलकुलेटर: पूरी गाइड | Ganit Calculator

Harmonic Mean Calculator - हार्मोनिक माध्य कैलकुलेटर: पूरी गाइड

Harmonic Mean (हार्मोनिक माध्य) statistics का एक specialized measure of central tendency है जो rates, speeds, और ratios के लिए specially designed है। इस comprehensive guide में हम harmonic mean के सभी aspects को detailed तरीके से cover करेंगे।

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

Harmonic Mean या हार्मोनिक माध्य n numbers का reciprocal of the arithmetic mean of their reciprocals होता है। यह rates और speeds के लिए ideal है।

\[ \text{Harmonic Mean} = \frac{n}{\sum \frac{1}{x}} = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \ldots + \frac{1}{x_n}} \]

Harmonic Mean के प्रकार और Calculation Methods

← Scroll horizontally to view full table →
Data Type Formula उपयोग
Simple Data \[ HM = \frac{n}{\sum \frac{1}{x}} \] Individual values, speeds, rates
With Frequency \[ HM = \frac{\sum f}{\sum \frac{f}{x}} \] Frequency distribution data
Weighted Harmonic Mean \[ HM = \frac{\sum w}{\sum \frac{w}{x}} \] Weighted data, different importance

Calculation Methods

1. Simple Harmonic Mean

जब data individual values के form में हो:

\[ \text{Harmonic Mean} = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \ldots + \frac{1}{x_n}} \]

2. Harmonic Mean with Frequency

जब values और उनकी frequencies दी गई हों:

\[ \text{Harmonic Mean} = \frac{\sum f}{\sum \frac{f}{x}} \]

3. Weighted Harmonic Mean

जब different weights के साथ values दी गई हों:

\[ \text{Harmonic Mean} = \frac{\sum w}{\sum \frac{w}{x}} \]

Key Properties of Harmonic Mean

Reciprocal Nature

Harmonic mean reciprocal relationships को handle करता है, जो rates और speeds के लिए perfect है।

Most Sensitive to Small Values

Harmonic mean small values के प्रति most sensitive है, क्योंकि reciprocals में small values large values बन जाते हैं।

Always ≤ Geometric Mean ≤ Arithmetic Mean

किसी भी positive data set के लिए, harmonic mean ≤ geometric mean ≤ arithmetic mean होता है।

Applications of Harmonic Mean

Average Speed Calculation

जब same distance different speeds से cover की गई हो, तो harmonic mean correct average speed देता है

Rates and Ratios

Price-earnings ratios, productivity rates, और अन्य rates के average के लिए perfect

Time and Work Problems

Different workers की different work rates के combined rate calculation के लिए accurate results देता है

Financial Analysis

Investment returns, interest rates, और financial ratios के analysis के लिए use होता है

Solved Examples

Example 1: Simple Harmonic Mean

Problem: 2, 4, 8 का harmonic mean निकालें

Solution:
Values: 2, 4, 8
\[ n = 3 \]
\[ \text{Sum of reciprocals} = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} = 0.5 + 0.25 + 0.125 = 0.875 \]
\[ \text{Harmonic Mean} = \frac{3}{0.875} = 3.4286 \]
\[ \text{Harmonic Mean} \approx 3.43 \]

Example 2: Harmonic Mean with Frequency

Problem: निम्नलिखित data का harmonic mean निकालें:

← Scroll horizontally to view full table →
Value (x)Frequency (f)
23
42
81
Solution:
← Scroll horizontally to view full table →
xff/x
231.5
420.5
810.125
Total62.125
\[ \sum f = 6 \]
\[ \sum \frac{f}{x} = 2.125 \]
\[ \text{Harmonic Mean} = \frac{6}{2.125} = 2.8235 \]
\[ \text{Harmonic Mean} \approx 2.82 \]

Example 3: Average Speed Calculation

Problem: एक car 60 km की distance 30 km/h की speed से और फिर same 60 km की distance 60 km/h की speed से cover करती है। Average speed निकालें।

Solution:
Speeds: 30 km/h, 60 km/h
\[ \text{Harmonic Mean} = \frac{2}{\frac{1}{30} + \frac{1}{60}} \]
\[ = \frac{2}{0.0333 + 0.0167} = \frac{2}{0.05} = 40 \text{ km/h} \]
Verification:
Total distance = 60 + 60 = 120 km
Total time = 60/30 + 60/60 = 2 + 1 = 3 hours
Average speed = 120/3 = 40 km/h ✓

Harmonic Mean vs Other Means

Aspect Harmonic Mean Geometric Mean Arithmetic Mean
Formula n / ∑(1/x) √[n](∏x) (∑x)/n
Use Case Rates, speeds Growth rates, ratios General average
Sensitivity Most sensitive to small values Moderately sensitive Sensitive to large values
Data Requirements All values > 0 All values > 0 Any real numbers
Relationship HM ≤ GM ≤ AM GM between HM and AM AM ≥ GM ≥ HM

Frequently Asked Questions (FAQ)

1. Harmonic mean कब use करना चाहिए?

Harmonic mean use करें जब:

  • Average speed calculate करना हो (जब distances equal हों)
  • Rates या ratios का average निकालना हो
  • Data rates per unit के form में हो
  • Time और work problems solve करने हों
  • Small values को more weightage देना हो

2. Harmonic mean में zero values क्यों नहीं use कर सकते?

Harmonic mean में zero values use नहीं कर सकते क्योंकि:

  • Zero का reciprocal (1/0) undefined होता है
  • Mathematical calculation possible नहीं है
  • Infinite values आते हैं जो meaningful results नहीं देते

3. Harmonic mean हमेशा सबसे छोटा क्यों होता है?

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

Key Points to Remember

  • Harmonic mean rates और speeds के लिए ideal है
  • All values must be positive (> 0)
  • Most sensitive to small values among the three means
  • Always less than or equal to geometric and arithmetic means
  • Perfect for average speed when distances are equal
  • Useful for time, work, and rate problems

Important Note

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

Real-World Applications

Transportation

Average vehicle speed calculation, fuel efficiency analysis, और transportation planning में harmonic mean का extensive use होता है।

Finance and Economics

Price-earnings ratios, investment returns, economic indicators, और financial ratios के analysis के लिए harmonic mean important है।

Science and Engineering

Electrical circuits में resistance calculation, physics में average velocity, और engineering में rate problems के लिए harmonic mean use होता है।

Quality Control

Manufacturing processes में production rates, efficiency ratios, और performance metrics के analysis के लिए harmonic mean valuable है।

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