Answer :
The Harmonic Mean (HM) is a measure of central tendency that is particularly useful when dealing with rates or ratios. It is calculated using the formula:
[tex]\text{HM} = \frac{n}{\sum \frac{1}{x_i}}[/tex]
where [tex]n[/tex] is the number of observations, and [tex]x_i[/tex] are the individual data points.
Let's calculate the Harmonic Mean for each given data set.
(a) Discrete Series
Data Set:
- Values [tex]X = \{10, 20, 30, 40, 50, 60\}[/tex]
- Frequency [tex]f = \{5, 4, 20, 18, 7, 6\}[/tex]
First, find the sum of the reciprocals of [tex]X[/tex] weighted by their frequencies:
[tex]\text{Sum} = \sum \left( \frac{f_i}{x_i} \right) = \frac{5}{10} + \frac{4}{20} + \frac{20}{30} + \frac{18}{40} + \frac{7}{50} + \frac{6}{60}[/tex]
Calculate the above, then apply the Harmonic Mean formula:
[tex]\text{HM} = \frac{\sum f}{\sum \left( \frac{f_i}{x_i} \right)}[/tex]
(b) Discrete Series
Data Set:
- Values [tex]X = \{2, 3, 4, 5, 6\}[/tex]
- Frequency [tex]f = \{2, 6, 12, 20, 5\}[/tex]
Calculate the sum of reciprocals weighted by frequencies:
[tex]\text{Sum} = \left( \frac{2}{2} + \frac{6}{3} + \frac{12}{4} + \frac{20}{5} + \frac{5}{6} \right)[/tex]
Then, apply the Harmonic Mean formula:
[tex]\text{HM} = \frac{\sum f}{\sum \left( \frac{f_i}{x_i} \right)}[/tex]
(c) Discrete Series
Data Set:
- Values [tex]X = \{5, 16, 25, 30, 40\}[/tex]
- Weights [tex]W = \{2, 4, 8, 6, 10\}[/tex]
Find the sum of the reciprocals weighted by their corresponding weights:
[tex]\text{Sum} = \left( \frac{2}{5} + \frac{4}{16} + \frac{8}{25} + \frac{6}{30} + \frac{10}{40} \right)[/tex]
Calculate the Harmonic Mean:
[tex]\text{HM} = \frac{\sum W}{\sum \left( \frac{W_i}{x_i} \right)}[/tex]
(d) Data Set
Data Set:
- Commodity: {A, B, C, D}
- Index No.: {180, 150, 200, 160}
- Weight: {3, 1, 4, 2}
Sum of the reciprocals weighted by weights:
[tex]\text{Sum} = \left( \frac{3}{180} + \frac{1}{150} + \frac{4}{200} + \frac{2}{160} \right)[/tex]
Harmonic Mean:
[tex]\text{HM} = \frac{\sum W}{\sum \left( \frac{W_i}{\text{Index No.}_i} \right)}[/tex]
(3) Continuous Series
In continuous series, the class interval midpoints are used as values [tex]X[/tex].
(a) Continuous Interval
- Class Interval (CI): 50-100, 100-150, 150-200, 200-250
- Frequencies [tex]f = \{15, 66, 35, 20\}[/tex]
Calculate the midpoints: [tex]75, 125, 175, 225[/tex].
Sum of reciprocals weighted by frequencies:
[tex]\text{Sum} = \left( \frac{15}{75} + \frac{66}{125} + \frac{35}{175} + \frac{20}{225} \right)[/tex]
Harmonic Mean:
[tex]\text{HM} = \frac{\sum f}{\sum \left( \frac{f_i}{ ext{Midpoint}_i} \right)}[/tex]
(b) Continuous Interval
- CI: 10-14, 15-19, 20-24, 25-29, 30-34
- Frequencies [tex]f = \{3, 17, 11, 9, 8\}[/tex]
Calculate the midpoints: [tex]12, 17, 22, 27, 32[/tex].
Use the same steps as above to find the Harmonic Mean.
To get an accurate Harmonic Mean for each set, carefully compute each step using the respective formulas.
