Finding the mode for grouped data involves identifying the class interval with the highest frequency and then using a formula to estimate the mode within that interval. Here's a step-by-step guide:
1. Identify the Modal Class:
The modal class is the class interval with the highest frequency. Look through your frequency distribution table and locate the class interval that has the maximum frequency. This class is your modal class.
2. Determine the Size (h) of the Modal Class:
Calculate the class size (h) by subtracting the lower limit of the modal class from its upper limit.
h = Upper Limit - Lower Limit
3. Apply the Mode Formula:
Use the following formula to calculate the mode:
Mode = L + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] * h
Where:
* `L` = Lower limit of the modal class
* `f₁` = Frequency of the modal class
* `f₀` = Frequency of the class preceding the modal class
* `f₂` = Frequency of the class succeeding the modal class
* `h` = Class size of the modal classExample:
Let's say we have the following grouped data:
| Class Interval | Frequency |
|---|---|
| 0-10 | 5 |
| 10-20 | 12 |
| 20-30 | 15 |
| 30-40 | 8 |
| 40-50 | 3 |
-
Modal Class: The modal class is 20-30 because it has the highest frequency (15).
-
Class Size (h):
h = 30 - 20 = 10 -
Applying the Formula:
L = 20(Lower limit of the modal class)f₁ = 15(Frequency of the modal class)f₀ = 12(Frequency of the class preceding the modal class)f₂ = 8(Frequency of the class succeeding the modal class)h = 10
Mode = 20 + [(15 - 12) / (2 * 15 - 12 - 8)] * 10
Mode = 20 + [3 / (30 - 20)] * 10
Mode = 20 + [3 / 10] * 10
Mode = 20 + 3
Mode = 23
Therefore, the estimated mode for this grouped data is 23.
In summary, finding the mode for grouped data involves identifying the modal class (the class with the highest frequency) and then applying a specific formula that takes into account the lower limit of the modal class, the class size, and the frequencies of the preceding and succeeding classes to estimate the mode within that interval.
