The given distribution shows the number of runs scored by some top batsmen of the world in one-day international cricket matches.
| Runs scored | Number of batsmen |
|---|---|
| 3000 - 4000 | 4 |
| 4000 - 5000 | 18 |
| 5000 - 6000 | 9 |
| 6000 - 7000 | 7 |
| 7000 - 8000 | 6 |
| 8000 - 9000 | 3 |
| 9000 - 10000 | 1 |
| 10000 - 11000 | 1 |
Find the mode of the data.
From the given data, the maximum frequency is 18, which corresponds to the class interval 4000 - 5000. Therefore, the modal class is 4000 - 5000. We identify the following values from the data: Lower limit (l) = 4000 Class size (h) = 1000 Frequency of modal class (f₁) = 18 Frequency of preceding class (f₀) = 4 Frequency of succeeding class (f₂) = 9
Using the formula for mode:
Substituting the values:
Thus, the mode of the data is 4608.7.
Explanation
To find the mode for grouped data, we first identify the modal class, which is the class interval with the highest frequency. In this distribution, the highest frequency is 18, making 4000-5000 the modal class. We then apply the standard formula provided in the context, using the lower limit of the modal class, the frequencies of the modal class and its adjacent classes, and the class size. The calculation determines the specific value within the interval where the concentration of data is highest.
Solution Steps
Step 1: Identify the modal class by finding the class with the maximum frequency. Max frequency = 18, so Modal Class = 4000 - 5000.
Step 2: Identify the parameters: l = 4000, h = 1000, f₁ = 18, f₀ = 4, f₂ = 9.
Step 3: Substitute the values into the mode formula: Mode = 4000 + [(18 - 4) / (2(18) - 4 - 9)] × 1000.
Step 4: Solve the equation to find Mode = 4000 + (14/23) × 1000 = 4000 + 608.7 = 4608.7.