Answer :
The Harmonic Mean is a type of average used when finding averages of rates. It's particularly useful when the numbers in the dataset are defined in relation to some unit or rate, such as speed or density.
To find the Harmonic Mean of a data set, you use the formula:
[tex]\text{Harmonic Mean} = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}[/tex]
Where [tex]n[/tex] is the total number of observations and [tex]x_i[/tex] is each individual observation.
Here's how to calculate the Harmonic Mean for the given data set: 15, 250, 15.7, 157, 1.57, 105.7, 10.5, 1.06, 25.7, 0.257.
Identify each value and find the reciprocal:
- [tex]x_1 = 15 \rightarrow \frac{1}{15}[/tex]
- [tex]x_2 = 250 \rightarrow \frac{1}{250}[/tex]
- [tex]x_3 = 15.7 \rightarrow \frac{1}{15.7}[/tex]
- [tex]x_4 = 157 \rightarrow \frac{1}{157}[/tex]
- [tex]x_5 = 1.57 \rightarrow \frac{1}{1.57}[/tex]
- [tex]x_6 = 105.7 \rightarrow \frac{1}{105.7}[/tex]
- [tex]x_7 = 10.5 \rightarrow \frac{1}{10.5}[/tex]
- [tex]x_8 = 1.06 \rightarrow \frac{1}{1.06}[/tex]
- [tex]x_9 = 25.7 \rightarrow \frac{1}{25.7}[/tex]
- [tex]x_{10} = 0.257 \rightarrow \frac{1}{0.257}[/tex]
Sum all the reciprocals:
- [tex]\sum \frac{1}{x_i} = \frac{1}{15} + \frac{1}{250} + \frac{1}{15.7} + \frac{1}{157} + \frac{1}{1.57} + \frac{1}{105.7} + \frac{1}{10.5} + \frac{1}{1.06} + \frac{1}{25.7} + \frac{1}{0.257}[/tex]
Calculate the Harmonic Mean:
- Substitute the value of [tex]n = 10[/tex] and the sum of the reciprocals into the Harmonic Mean formula:
- [tex]\text{Harmonic Mean} = \frac{10}{\sum_{i=1}^{10} \frac{1}{x_i}}[/tex]
- Evaluating this will give the Harmonic Mean.
This entire process shows the step-by-step calculation to find the Harmonic Mean of the given data set. Make sure to calculate the sum of the reciprocals accurately, as this step is crucial for the correct result.
