Statistics · Lesson 2 of 4
Mode of Grouped Data
“Find the value that stands out most, even when the data arrives packed into intervals.”
• Understand the meaning of mode and modal class. • Identify the modal class from grouped data. • Use the grouped-data mode formula correctly. • Interpret l, h, f₀, f₁ and f₂. • Compare what mode and mean tell us.
Mode
When we look at a set of data, one useful question is: Which value appears most often? The value that occurs with the highest frequency is called the mode.
For example, if the shoe sizes of a group of students are:
6, 7, 7, 8, 7, 9, 8
then 7 is the mode because it appears more often than any other value.
Mode is especially useful when we want to identify the most common or most popular value in a data set. It can help us find things like the most common size, the most frequently chosen option, or the value that appears most often in a survey or observation.
In simple, ungrouped data, the mode can often be found just by looking at the frequencies. But in grouped data, individual values are combined into class intervals such as 10–20, 20–30, 30–40, and so on. This means we cannot directly see which exact value occurs most often.
Instead, we first identify the modal class, which is the class interval with the greatest frequency. Once the modal class is known, we use a formula to estimate the mode within that interval.
So, finding the mode in grouped data involves two main ideas: first, locate the class where the data is most concentrated, and then estimate the value inside that class that best represents the most frequent observation.
The class interval with the greatest frequency is called the modal class.
The modal class is an interval. The mode is a single estimated value inside that interval.
| Symbol | Meaning |
|---|---|
| l | Lower limit of the modal class |
| h | Class size |
| f₁ | Frequency of the modal class |
| f₀ | Frequency of the preceding class |
| f₂ | Frequency of the succeeding class |
Step-by-Step Method
Find the maximum frequency, identify its class, record the two neighbouring frequencies, then substitute only after all five quantities have been written clearly.
| Family size | 1–3 | 3–5 | 5–7 | 7–9 | 9–11 |
|---|---|---|---|---|---|
| Frequency | 7 | 8 | 2 | 2 | 1 |
Problem
Find the mode of the distribution above.
- 1.The highest frequency is 8, so modal class = 3–5.
- 2.l = 3, h = 2, f₁ = 8, f₀ = 7, f₂ = 2.
- 3.Mode = 3 + [(8 − 7)/(16 − 7 − 2)] × 2.
- 4.Mode = 3 + (1/7) × 2.
- 5.Mode ≈ 3.286.
Why the Neighbouring Frequencies Matter
The highest frequency tells us which class contains the mode, but the neighbouring frequencies indicate how the distribution rises into and falls away from the modal class. This is what lets the formula estimate a position inside the class.
| Class interval | 10–25 | 25–40 | 40–55 | 55–70 | 70–85 | 85–100 |
|---|---|---|---|---|---|---|
| Frequency | 2 | 3 | 7 | 6 | 6 | 6 |
Problem
Find the mode of the distribution above. Its mean is 62.
- 1.The largest frequency is 7, so the modal class is 40–55.
- 2.l = 40, h = 15, f₁ = 7, f₀ = 3, f₂ = 6.
- 3.Mode = 40 + [(7 − 3)/(14 − 3 − 6)] × 15.
- 4.Mode = 40 + (4/5) × 15 = 52.
- 5.The mean is 62 while the mode is 52.
- 6.Mean describes the overall average; mode estimates the most frequent value.
| Age interval | 5–15 | 15–25 | 25–35 | 35–45 | 45–55 | 55–65 |
|---|---|---|---|---|---|---|
| Frequency | 6 | 11 | 21 | 23 | 14 | 5 |
Problem
Find the mode of the age distribution above.
- 1.The greatest frequency is 23, so modal class = 35–45.
- 2.l = 35, h = 10, f₁ = 23, f₀ = 21, f₂ = 14.
- 3.Mode = 35 + [(23 − 21)/(46 − 21 − 14)] × 10.
- 4.Mode = 35 + (2/11) × 10.
- 5.Mode ≈ 36.82 years.
• Picking the class with the largest class mark instead of largest frequency, • Reporting the modal class itself as the mode, • Interchanging f₀ and f₂, • Using the upper limit instead of the lower limit l, • Applying the formula without continuous intervals where continuity is required,
Quiz
What is the modal class in a grouped frequency distribution?
The frequencies of the class intervals 0–10, 10–20, 20–30 and 30–40 are 4, 9, 15 and 7 respectively. Which is the modal class?
In the grouped-data mode formula, what does f₁ represent?
For a grouped distribution, the modal class is 20–30, with l = 20, h = 10, f₁ = 12, f₀ = 8 and f₂ = 6. What is the mode?
If the mean is 26 and the median is 27, what is the estimated mode using the empirical relationship?
Practice Problems
- Lifetimes 0–20, 20–40, 40–60, 60–80, 80–100, 100–120 have frequencies 10, 35, 52, 61, 38, 29. Find the mode.
- Monthly expenditure intervals 1000–1500, 1500–2000, 2000–2500, 2500–3000, 3000–3500, 3500–4000, 4000–4500, 4500–5000 have frequencies 24, 40, 33, 28, 30, 22, 16, 7. Find the mode.
- Classes 20–30, 30–40, 40–50, 50–60, 60–70 have frequencies 8, 15, 26, 18, 9. Find the mode.
Key Takeaways
• The modal class has the maximum frequency. • The mode is an estimated value inside the modal class. • The formula uses the modal frequency and both neighbouring frequencies. • Mode is useful when the most common value is the main interest.
Next, we use cumulative frequency to find the middle value of grouped data: the median.