1. **Problem Statement:** Calculate the harmonic mean $H$ using the formula $$H = \frac{\sum f}{\sum f \left(\frac{1}{x_i}\right)}$$ where $f$ is the frequency and $x_i$ is the midpoint or observation. 2. **Given Data:** - Frequencies $f$: 9, 10, 17, 10, 5, 4, 5 - Midpoints $x_i$: 74.5, 94.5, 104.5, 134.5, 154.5, 174.5, 194.5 - Reciprocals $\frac{1}{x_i}$: 0.0134, 0.0105, 0.0087, 0.0074, 0.0064, 0.0057, 0.0051 - Products $f \times \frac{1}{x_i}$: 0.1206, 0.1050, 0.1479, 0.074, 0.032, 0.0226, 0.0255 3. **Summations:** - Total frequency $\sum f = 60$ - Sum of products $\sum f \left(\frac{1}{x_i}\right) = 0.5278$ 4. **Calculation:** Substitute values into the harmonic mean formula: $$H = \frac{60}{0.5278}$$ 5. **Result:** $$H \approx 113.6194$$ This means the harmonic mean of the given data is approximately 113.6194. The harmonic mean is useful when averaging rates or ratios, especially when the data involves quantities like speeds or densities.