Understanding Average Ratios and Central Tendency
The question asks for the most appropriate measure of central tendency to calculate an average of ratios or rates, such as 'price per unit', 'work done per hour', or 'kilometres per hour'. These are examples of rates where we want to find an average value under certain conditions.
Measures of Central Tendency Explained
Central tendency measures provide a single value that represents the center or typical value of a dataset. Let's look at the options:
- Arithmetic Mean: This is the most common average, calculated by summing all values and dividing by the number of values. It's suitable for data where the values are directly additive, like heights or scores. Formula: \(\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}\).
- Geometric Mean: Used primarily for averaging rates of change or ratios where values are multiplied together, like average investment returns over several years. Formula: \(G = \sqrt[n]{x_1 \times x_2 \times \dots \times x_n}\).
- Harmonic Mean: This is specifically useful for averaging rates or ratios, particularly when the numerator represents a fixed quantity or when dealing with inverse relationships. It's calculated as the reciprocal of the arithmetic mean of the reciprocals of the observations. Formula: \(H = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}\).
- Mode: The mode is the value that appears most frequently in a dataset. It's used for categorical data or to find the most common occurrence, not for calculating an average rate.
Why Harmonic Mean is Suitable for Average Ratios
Ratios like 'price/unit' or 'kilometres/hour' represent rates. When calculating an average of such rates, the harmonic mean is often the correct choice because it appropriately accounts for the underlying structure of these rates.
Consider the example of calculating the average speed for a journey:
Suppose you travel 100 kilometres at 50 km/hr and then another 100 kilometres at 100 km/hr.
- Time for the first part: \(\frac{100 \text{ km}}{50 \text{ km/hr}} = 2 \text{ hours}\).
- Time for the second part: \(\frac{100 \text{ km}}{100 \text{ km/hr}} = 1 \text{ hour}\).
- Total distance = 200 km.
- Total time = 2 + 1 = 3 hours.
- Actual average speed = \(\frac{\text{Total Distance}}{\text{Total Time}} = \frac{200 \text{ km}}{3 \text{ hours}} \approx 66.67 \text{ km/hr}\).
If we incorrectly used the arithmetic mean:
Arithmetic mean of speeds = \(\frac{50 \text{ km/hr} + 100 \text{ km/hr}}{2} = 75 \text{ km/hr}\). This is incorrect because it doesn't account for the fact that you spent more time travelling at the slower speed.
Now, let's use the harmonic mean for the speeds:
Harmonic Mean (H) = \(\frac{2}{\frac{1}{50} + \frac{1}{100}}\)
H = \(\frac{2}{\frac{2}{100} + \frac{1}{100}}\)
H = \(\frac{2}{\frac{3}{100}}\)
H = \(\frac{2 \times 100}{3} = \frac{200}{3} \approx 66.67 \text{ km/hr}\).
This matches the actual average speed. Similarly, for 'price/unit', if you buy different quantities at different prices, the harmonic mean helps find the average cost per unit correctly, especially if considering the total amount spent. For 'work done/hour', it helps find the average rate of work.
Conclusion
Therefore, when the task is to find the average of rates or ratios like price per unit, work done per hour, or kilometres per hour, the harmonic mean is the suitable measure of central tendency.
Similar Questions
What is the median of the distribution ?
- What is the mode of the distribution?
- Consider the following data :
110, 41, 43, 95, 127, 99, 61, 92, 71, 93, 110, 36.
If 93 is replaced by 94, then consider the following statements:
1. The difference between new median and old median is 1.
2. The difference between new mean and old mean is less than 0·1.
3. The difference between new mode and old mode is zero.
Which of the statements given above are correct? - Which measure of central tendency is least affected by the presence of extreme observations in the data?
- The frequency distribution of marks of 100 candidates in a particular examination is as follows:What are the average marks of the candidates?
Marks Number of Candidates More than 10 100 More than 20 75 More than 30 60 More than 40 40 - The arithmetic mean of 200 observations is 60. If 5 is multiplied to each observation, then what will be the new arithmetic mean?
- A distribution consists of 3 components with frequencies 45, 40 and 55 having their means 2, 2.5 and 2 respectively. What is the mean of the combined distribution?
- Which one of the following is a positional average?
- What is the value of \(u\)?
- What is the value of \(v\)?
Important Questions from Elementary Statistics
In a colony 5 families have 1 child, 7 families have 2 children, 8 families have 3 children and 3 families have 4 children.What is the mode of the number of children.
For which set of numbers do the mean, median and mode all have the same value?
The median of 5, 8, 25, 22, 34, 18 is
Find the mean of first 5 two-digit multiples of 4.
The difference between the upper limit and the lower limit of a class is called :
