Tenth Grade Mathematics: Mode of Grouped Data and Modal Class
In statistical analysis, while the mean represents the mathematical balance point of a dataset, the mode represents the most common, popular, or frequently occurring value. In business inventory, manufacturing, and shoe sizing, the mode is the primary measure of central tendency because retailers must stock the sizes that the largest number of consumers purchase. In grouped continuous data, individual values are grouped inside class intervals, meaning we can no longer identify a single most frequent number by simple inspection; instead, we locate the modal class and interpolate the mode using a specialized formula.
Definition of Mode and the Modal Class
For ungrouped data, the mode is simply the observation having the maximum frequency. For continuous grouped frequency distributions:
Formal Definition:
The Modal Class is defined as the class interval that possesses the HIGHEST
frequency in the entire frequency distribution.
Once the modal class is identified, the actual mode lies somewhere inside that interval. Its precise location is skewed toward whichever adjacent class interval has the higher neighboring frequency.
Derivation of the Grouped Mode Formula
Consider the histogram representation of the modal class and its two immediate adjacent intervals:
Histogram Anatomy around the Modal Class:
Frequency
^
| +-------------+ f1 (Modal Class Frequency)
| | |
| +-------+ |
| | f0 | +-------+
| | | | f2 |
+--+-------+-------------+-------+---> Variable
<--h---><------h-----><---h--->
Preceding Modal Succeeding
Class Class Class
l l + h
Let the modal class have lower limit l and class width h. Let f1 be the frequency of the modal class. Let f0 be the frequency of the class preceding the modal class. Let f2 be the frequency of the class succeeding the modal class.
The difference in height on the left side is Delta_1 = f1 - f0. The difference in height on the right side is Delta_2 = f1 - f2. The mode divides the interval width h in proportion to these frequency differences:
Mode = l + [ Delta_1 / (Delta_1 + Delta_2) ] * h
Substituting Delta_1 = f1 - f0 and Delta_2 = f1 - f2: Delta_1 + Delta_2 = (f1 - f0) + (f1 - f2) = 2*f1 - f0 - f2.
This yields the universal Mode Formula for grouped data:
Theorem (Mode Formula for Grouped Data):
f1 - f0
Mode = l + [ ------------------- ] * h
2*f1 - f0 - f2
Component Summary:
- l = Lower limit of the modal class
- h = Size of the modal class interval
- f1 = Frequency of the modal class (the maximum frequency)
- f0 = Frequency of the class preceding the modal class
- f2 = Frequency of the class succeeding the modal class
Detailed Step-by-Step Example
Find the mode of the following data showing the ages of patients admitted to a hospital during a year:
Age (in years): 5-15 | 15-25 | 25-35 | 35-45 | 45-55 | 55-65
Number of Patients: 6 | 11 | 21 | 23 | 14 | 5
Step 1: Locate the maximum frequency:
The highest frequency is 23.
Therefore, the Modal Class is 35-45.
Step 2: Extract formula components:
Lower limit l = 35
Class size h = 45 - 35 = 10
f1 (frequency of modal class 35-45) = 23
f0 (frequency of preceding class 25-35) = 21
f2 (frequency of succeeding class 45-55) = 14
Step 3: Calculate numerator and denominator:
Numerator: f1 - f0 = 23 - 21 = 2
Denominator: 2*f1 - f0 - f2 = 2*(23) - 21 - 14 = 46 - 35 = 11
Step 4: Substitute into the formula:
Mode = l + [ (f1 - f0) / (2*f1 - f0 - f2) ] * h
Mode = 35 + [ 2 / 11 ] * 10
Mode = 35 + 20 / 11 = 35 + 1.818... = 36.82 years.
The mode of the dataset is approximately 36.82 years, indicating that the most common age of admitted patients is around 36.8 years.
Chapter Exercises
Exercise 1: The following table shows the marks distribution of 100 students: Marks 0-10 (4 students), 10-20 (8 students), 20-30 (10 students), 30-40 (12 students), 40-50 (16 students), 50-60 (20 students), 60-70 (14 students), 70-80 (10 students), 80-90 (6 students). Find the mode of the distribution.
Exercise 2: A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes: Cars 0-10 (7), 10-20 (14), 20-30 (13), 30-40 (12), 40-50 (20), 50-60 (11), 60-70 (15), 70-80 (8). Find the mode of the data.
Exercise 3: Explain why the mode of a grouped frequency distribution is sensitive only to the frequencies in and immediately adjacent to the modal class, whereas the arithmetic mean incorporates every observation in the distribution.
Solutions and Explanations
Solution 1: We inspect the frequencies to locate the modal class. The frequencies are: 4, 8, 10, 12, 16, 20, 14, 10, 6. The maximum frequency is 20, which belongs to the class interval 50-60. Therefore, the modal class is 50-60. Parameters: lower limit l = 50, class width h = 10, modal frequency f1 = 20, preceding frequency f0 = 16, succeeding frequency f2 = 14. Compute: f1 - f0 = 20 - 16 = 4. 2f1 - f0 - f2 = 2(20) - 16 - 14 = 40 - 30 = 10. Applying the mode formula: Mode = l + [(f1 - f0) / (2*f1 - f0 - f2)] * h = 50 + [4 / 10] * 10 = 50 + 4 = 54. Thus, the mode of the marks distribution is 54.
Solution 2: Inspecting the frequencies: 7, 14, 13, 12, 20, 11, 15, 8. The highest frequency is 20, which occurs in the class interval 40-50. Thus, the modal class is 40-50. Parameters: lower limit l = 40, class size h = 10, f1 = 20, f0 = 12, f2 = 11. Compute numerator: f1 - f0 = 20 - 12 = 8. Compute denominator: 2f1 - f0 - f2 = 2(20) - 12 - 11 = 40 - 23 = 17. Applying the mode formula: Mode = l + [(f1 - f0) / (2*f1 - f0 - f2)] * h = 40 + [8 / 17] * 10 = 40 + 80 / 17 = 40 + 4.706 = 44.71 cars. Thus, the modal number of passing cars is approximately 44.7.
Solution 3: The arithmetic mean is calculated by summing the products of all class marks and their frequencies across the entire dataset: Sum(fi * xi) / N. Consequently, changing any single frequency or class limit in any interval shifts the computed balance point. In contrast, the mode represents the single point of greatest concentration (the peak of the frequency curve). In the grouped mode formula, the computation depends strictly on the lower limit of the peak interval l, the peak height f1, and the rates of ascent and descent from the immediate neighbors f0 and f2. Intervals located farther away do not alter the local maximum of the curve; hence, the mode depends solely on the immediate vicinity of the modal class.