We are given a frequency distribution table and asked to calculate the geometric mean, harmonic mean, and first quartile. The distribution table shows the mass (in kg) and the corresponding frequency.

1. Problem Description

We are given a frequency distribution table and asked to calculate the geometric mean, harmonic mean, and first quartile. The distribution table shows the mass (in kg) and the corresponding frequency.

2. Solution Steps

First, we create a table with the midpoints of each class interval and the corresponding frequencies.

Mass (kg) | Frequency (f) | Midpoint (x) | f*x | f*ln(x) | f/x

---|---|---|---|---|---

41-45 | 5 | 43 | 215 | 19.026 | 0.1163

46-50 | 2 | 48 | 96 | 7.746 | 0.0417

51-55 | 6 | 53 | 318 | 24.462 | 0.1132

56-60 | 7 | 58 | 406 | 28.357 | 0.1207

61-65 | 5 | 63 | 315 | 21.892 | 0.0794

66-70 | 3 | 68 | 204 | 12.455 | 0.0441

Total | 28 | | 1554 | 113.938 | 0.5154

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6. Geometric Mean (G):

The geometric mean is given by:

$G = exp(\frac{\Sigma f \cdot ln(x)}{\Sigma f})$

$G = exp(\frac{113.938}{28})$

$G = exp(4.0692)$

$G = 58.504$

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7. Harmonic Mean (H):

The harmonic mean is given by:

$H = \frac{\Sigma f}{\Sigma (f/x)}$

$H = \frac{28}{0.5154}$

$H = 54.327$

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8. First Quartile ($Q_1$):

The first quartile corresponds to the 25th percentile. We need to find the value below which 25% of the data falls.

Total frequency = 28

25% of 28 = 0.25 * 28 = 7

Cumulative frequencies:

41-45: 5

46-50: 5 + 2 = 7

Since the cumulative frequency reaches 7 at the end of the 46-50 interval, the first quartile falls in this interval.

$Q_1 = L + (\frac{\frac{N}{4} - cf}{f}) * w$

Where:

L = Lower limit of the first quartile class (46)

N = Total frequency (28)

cf = Cumulative frequency of the class before the first quartile class (5)

f = Frequency of the first quartile class (2)

w = Class width (5)

$Q_1 = 46 + (\frac{7 - 5}{2}) * 5$

$Q_1 = 46 + (\frac{2}{2}) * 5$

$Q_1 = 46 + 1 * 5$

$Q_1 = 46 + 5$

$Q_1 = 51$

3. Final Answer

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6. The geometric mean, G is computed as 58.

5

0

4.

1

7. The harmonic mean, H is computed as 54.

3

2

7.

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8. The first quartile, $Q_1$ is estimated as

5

1.