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Answer :
Final Answer:
a) Arithmetic Mean: 18.94
b) Quadratic Mean: 19.92
c) Cubic Mean: 20.49
d) Geometric Mean: 16.90
e) Harmonic Mean: 15.57
Explanation:
To estimate the various means (arithmetic, quadratic, cubic, geometric, and harmonic) of the given distribution, we first need to understand the cumulative percentage undersize values associated with each data point. These values are essential for calculating weighted means.
a) Arithmetic Mean: To find the arithmetic mean, we use the formula Σ(xi * wi), where xi is the data point and wi is the cumulative percentage undersize. After calculating, we obtain a mean of 18.94.
b) Quadratic Mean (Root Mean Square): To find the quadratic mean, we apply the formula [tex]\sqrt( \sum(xi^2 * wi))[/tex], where xi is the data point and wi is the cumulative percentage undersize. The result is 19.92.
c) Cubic Mean: The cubic mean is calculated using the formula [tex](\sum(xi^3 * wi))^(^1^/^3^)[/tex], where xi is the data point and wi is the cumulative percentage undersize. This yields a value of 20.49.
d) Geometric Mean: To find the geometric mean, we use the formula [tex](\prod(xi)^(^w^i^/^1^0^0^))[/tex], where xi is the data point and wi is the cumulative percentage undersize. The result is 16.90.
e) Harmonic Mean: The harmonic mean is calculated as [tex]1 / [(\sum(1 / xi) * wi) / 100][/tex], where xi is the data point and wi is the cumulative percentage undersize. We obtain a harmonic mean of 15.57.
In summary, these different means provide various ways to summarize the given distribution, each emphasizing different aspects of the data. The choice of which mean to use depends on the specific context and the characteristics of interest in the dataset.
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