Mode of Grouped Data
The mode of grouped data is the value that occurs most frequently in a data set. For grouped (class-interval) data, it is calculated using a formula based on the modal class -- the class with the highest frequency. This topic is covered in CBSE Class 10 Mathematics, Chapter 14 (Statistics).
Unlike the mean (which uses all values) and the median (which depends on position), the mode identifies the most typical or popular value in the data.
The mode is particularly useful for categorical data and data where "most common" is more meaningful than "average" -- such as the most popular shoe size, the most frequent salary bracket, or the most common age group.
What is Mode of Grouped Data - Formula, Modal Class & Solved Examples?
Definition: The mode of a data set is the value (or class) that has the highest frequency. For grouped data, it is the value within the modal class that best represents the peak of the distribution.
Key terms:
- Modal class -- the class interval with the highest frequency.
- f_1 -- frequency of the modal class.
- f_0 -- frequency of the class just BEFORE the modal class.
- f_2 -- frequency of the class just AFTER the modal class.
- l -- lower limit of the modal class.
- h -- class size (width) of the modal class.
Special cases:
- If two non-adjacent classes have the same highest frequency, the data is bimodal.
- If no class has a distinctly higher frequency, the data has no mode.
- The formula works only when the modal class has a unique highest frequency.
Mode of Grouped Data Formula
Mode of Grouped Data:
Mode = l + [(f_1 - f_0) / (2f_1 - f_0 - f_2)] x h
Where:
- l = lower limit of the modal class
- f_1 = frequency of the modal class
- f_0 = frequency of the class preceding the modal class
- f_2 = frequency of the class succeeding the modal class
- h = class size
Steps to find the mode:
- Identify the modal class (highest frequency).
- Note f_1, f_0, f_2, l, and h.
- Substitute into the formula.
Empirical Relationship:
Mode = 3 x Median - 2 x Mean (approximately)
Derivation and Proof
Derivation of the Mode Formula (Interpolation Method):
- The modal class is the class with the highest frequency.
- The mode lies within this class, but where exactly?
- It should be closer to the side with the larger neighbouring frequency.
- The "excess" frequency on the lower side: Delta_1 = f_1 - f_0.
- The "excess" frequency on the upper side: Delta_2 = f_1 - f_2.
- The mode divides the class in the ratio Delta_1 : Delta_2.
- Distance from lower limit = [Delta_1/(Delta_1 + Delta_2)] x h.
- = [(f_1 - f_0)/((f_1 - f_0) + (f_1 - f_2))] x h
- = [(f_1 - f_0)/(2f_1 - f_0 - f_2)] x h
- Mode = l + [(f_1 - f_0)/(2f_1 - f_0 - f_2)] x h. Hence derived.
Note: The denominator (2f_1 - f_0 - f_2) must be positive. If it equals zero, the formula is undefined.
Types and Properties
Problems on mode of grouped data include:
Type 1: Direct Calculation
- Given a frequency distribution, identify the modal class and apply the formula.
Type 2: Finding Mode Using Empirical Relationship
- Given mean and median, find mode using Mode = 3 x Median - 2 x Mean.
Type 3: Finding Missing Frequency
- Given the mode and the rest of the data, find the missing frequency.
Type 4: Comparing Mean, Median, and Mode
- Calculate all three for the same distribution and verify the empirical relationship.
Type 5: Bimodal Data
- When two classes share the highest frequency.
Methods
Step-by-step method to find mode of grouped data:
- Identify the modal class -- the class with the highest frequency.
- Note the values:
- l = lower limit of the modal class
- f_1 = frequency of the modal class
- f_0 = frequency of the class just before the modal class
- f_2 = frequency of the class just after the modal class
- h = class size
- Apply the formula: Mode = l + [(f_1 - f_0)/(2f_1 - f_0 - f_2)] x h.
- Simplify to get the answer.
Common Mistakes:
- Confusing f_0 and f_2 -- f_0 is BEFORE the modal class, f_2 is AFTER.
- Choosing the wrong modal class (pick the class with the HIGHEST frequency).
- If the first class is the modal class, use f_0 = 0. If the last class, use f_2 = 0.
Solved Examples
Example 1: Finding Mode of Grouped Data
Problem: Find the mode of the following distribution:
| Class | Frequency |
|---|---|
| 0-10 | 5 |
| 10-20 | 8 |
| 20-30 | 15 |
| 30-40 | 12 |
| 40-50 | 10 |
Solution:
Modal class: 20-30 (highest frequency = 15).
- l = 20, f_1 = 15, f_0 = 8, f_2 = 12, h = 10
Mode = 20 + [(15 - 8)/(2 x 15 - 8 - 12)] x 10
- = 20 + [7/(30 - 20)] x 10
- = 20 + (7/10) x 10
- = 20 + 7 = 27
Answer: The mode is 27.
Example 2: Mode of Student Marks
Problem: Find the mode of marks scored by 50 students:
| Marks | Students |
|---|---|
| 0-20 | 6 |
| 20-40 | 8 |
| 40-60 | 16 |
| 60-80 | 12 |
| 80-100 | 8 |
Solution:
Modal class: 40-60 (f_1 = 16).
- l = 40, f_1 = 16, f_0 = 8, f_2 = 12, h = 20
Mode = 40 + [(16 - 8)/(32 - 20)] x 20 = 40 + (8/12) x 20 = 40 + 13.33 = 53.33.
Answer: The mode is 53.33.
Example 3: Mode Using Empirical Relationship
Problem: The mean of a data set is 24.5 and the median is 25.8. Find the mode.
Solution:
Using Mode = 3 x Median - 2 x Mean:
- Mode = 3(25.8) - 2(24.5)
- = 77.4 - 49.0
- = 28.4
Answer: The mode is 28.4.
Example 4: Mode of Family Size Data
Problem: Find the mode of the number of family members:
| Family Size | Families |
|---|---|
| 1-3 | 7 |
| 3-5 | 10 |
| 5-7 | 18 |
| 7-9 | 15 |
| 9-11 | 5 |
Solution:
Modal class: 5-7 (f_1 = 18).
- l = 5, f_1 = 18, f_0 = 10, f_2 = 15, h = 2
Mode = 5 + [(18 - 10)/(36 - 25)] x 2 = 5 + (8/11) x 2 = 5 + 1.45 = 6.45.
Answer: The modal family size is 6.45.
Example 5: Modal Class is the First Class
Problem: Find the mode of: 0-5 (20), 5-10 (14), 10-15 (8), 15-20 (5), 20-25 (3).
Solution:
Modal class: 0-5 (f_1 = 20). Since this is the first class, f_0 = 0.
- l = 0, f_1 = 20, f_0 = 0, f_2 = 14, h = 5
Mode = 0 + [(20 - 0)/(40 - 0 - 14)] x 5 = [20/26] x 5 = (10/13) x 5 = 3.85.
Answer: The mode is 3.85.
Example 6: Finding Missing Frequency Using Mode
Problem: The mode of the following data is 36. Find the missing frequency f.
| Class | Frequency |
|---|---|
| 0-10 | 8 |
| 10-20 | 10 |
| 20-30 | f |
| 30-40 | 16 |
| 40-50 | 12 |
| 50-60 | 6 |
Solution:
Since mode = 36 lies in 30-40, modal class is 30-40.
- l = 30, f_1 = 16, f_0 = f, f_2 = 12, h = 10
36 = 30 + [(16 - f)/(32 - f - 12)] x 10
- 6 = [(16 - f)/(20 - f)] x 10
- 6(20 - f) = 10(16 - f)
- 120 - 6f = 160 - 10f
- 4f = 40
- f = 10
Answer: The missing frequency f = 10.
Example 7: Comparing Mean, Median, and Mode
Problem: For the data: 10-20 (3), 20-30 (5), 30-40 (7), 40-50 (5), 50-60 (5), find mean, median, and mode. Verify the empirical relationship.
Mean:
- Sigma(f_i x_i) = 3(15) + 5(25) + 7(35) + 5(45) + 5(55) = 45 + 125 + 245 + 225 + 275 = 915
- Mean = 915/25 = 36.6
Median:
- n/2 = 12.5. cf: 3, 8, 15, 20, 25. Median class: 30-40 (cf = 15 first exceeds 12.5).
- Median = 30 + [(12.5 - 8)/7] x 10 = 30 + 6.43 = 36.43
Mode:
- Modal class: 30-40 (f_1 = 7). f_0 = 5, f_2 = 5.
- Mode = 30 + [(7-5)/(14-10)] x 10 = 30 + (2/4) x 10 = 35
Verifying:
- 3 x Median - 2 x Mean = 3(36.43) - 2(36.6) = 109.29 - 73.2 = 36.09
- Actual mode = 35. Close to 36.09 (relationship holds approximately).
Answer: Mean = 36.6, Median = 36.43, Mode = 35.
Example 8: Mode of Production Data
Problem: Daily production (units) over 60 days:
| Production | Days |
|---|---|
| 20-30 | 5 |
| 30-40 | 12 |
| 40-50 | 22 |
| 50-60 | 14 |
| 60-70 | 7 |
Solution:
Modal class: 40-50 (f_1 = 22).
- l = 40, f_1 = 22, f_0 = 12, f_2 = 14, h = 10
Mode = 40 + [(22 - 12)/(44 - 26)] x 10 = 40 + (10/18) x 10 = 40 + 5.56 = 45.56.
Answer: The modal production is 45.56 units.
Real-World Applications
Retail and Manufacturing:
- The most frequently sold shoe size, clothing size, or product variant is the mode. It helps decide production quantities.
Elections:
- The modal age group of voters or the most common income bracket of supporters helps in targeted campaigning.
Weather:
- The most common temperature range or rainfall bracket in a region is found using mode.
Transportation:
- The modal speed on a highway or the most common commute time is used for traffic planning.
Education:
- The most common marks range identifies the typical performance level of students.
Healthcare:
- The most common blood pressure range or BMI category is used for health screening.
Key Points to Remember
- Mode is the value that occurs most frequently in a data set.
- For grouped data: Mode = l + [(f_1 - f_0)/(2f_1 - f_0 - f_2)] x h.
- The modal class is the class with the highest frequency.
- f_0 is the frequency BEFORE the modal class; f_2 is the frequency AFTER.
- If the first class is the modal class, f_0 = 0. If the last class, f_2 = 0.
- Mode is the only measure of central tendency for categorical data.
- A data set can be unimodal, bimodal, or multimodal.
- Empirical relationship: Mode = 3 x Median - 2 x Mean (approximate).
- The denominator (2f_1 - f_0 - f_2) must be positive for the formula to work.
- The mode always lies within the modal class interval.
Practice Problems
- Find the mode of: 0-10 (4), 10-20 (7), 20-30 (12), 30-40 (8), 40-50 (4).
- Find the mode of: 100-200 (8), 200-300 (15), 300-400 (20), 400-500 (12), 500-600 (5).
- If the mode is 67, find the missing frequency in: 40-50 (5), 50-60 (x), 60-70 (15), 70-80 (12), 80-90 (7).
- The mean of a data set is 50 and the mode is 44. Find the median using the empirical relationship.
- Find the mode of: 2-4 (3), 4-6 (5), 6-8 (11), 8-10 (7), 10-12 (4).
- Compare mean, median, and mode for: 0-20 (10), 20-40 (15), 40-60 (25), 60-80 (20), 80-100 (5).
Frequently Asked Questions
Q1. What is the mode of grouped data?
The mode is the most frequently occurring value. For grouped data: Mode = l + [(f_1 - f_0)/(2f_1 - f_0 - f_2)] x h, where the modal class has the highest frequency.
Q2. How do you identify the modal class?
The modal class is the class interval with the highest frequency. Look at all the frequencies and pick the largest.
Q3. What is f_0 and f_2 in the mode formula?
f_0 is the frequency of the class immediately BEFORE the modal class. f_2 is the frequency of the class immediately AFTER the modal class.
Q4. What if two classes have the same highest frequency?
If they are adjacent, a more advanced method may be needed. If non-adjacent, the data is bimodal.
Q5. What is the empirical relationship between mean, median, and mode?
Mode = 3 x Median - 2 x Mean (approximately). This holds for moderately skewed distributions.
Q6. Can mode be used for non-numerical data?
Yes. Mode is the only measure of central tendency for categorical data (e.g., most popular colour, most common blood group).
Q7. What if the modal class is the first or last class?
If first class, set f_0 = 0. If last class, set f_2 = 0. The formula still works.
Q8. How is mode different from mean and median?
Mean uses all values (sum/count). Median is the middle value. Mode is the most frequent value. Mode is least affected by extremes and is the only one for categorical data.
Related Topics
- Mean of Grouped Data
- Median of Grouped Data
- Mode of Data
- Empirical Relationship Between Mean, Median, Mode
- Statistics - Collection and Presentation
- Frequency Distribution Table
- Histogram of Grouped Data
- Frequency Polygon
- Mean of Ungrouped Data
- Median of Ungrouped Data
- Cumulative Frequency Distribution
- Ogive (Cumulative Frequency Curve)
- Mean by Assumed Mean Method
- Mean by Step-Deviation Method
