How to Find the Mode in Grouped and Ungrouped Data (With Formula)
Measures of central tendency mode is a vital concept in statistics. It helps us understand the most common value in a dataset, which is useful for analyzing trends in business, economics, and school or competitive exams. Knowing how to find the mode can support smarter decision-making and exam preparation.
| Measure | Definition | Formula (if applicable) | Usage |
|---|---|---|---|
| Mean | Average of all data values | \(\bar{x} = \frac{\sum{x_i}}{n}\) | Numerical, continuous, and discrete data |
| Median | Middle value when data is sorted | N/A (odd: middle value, even: mean of two middles) | When data has outliers or is skewed |
| Mode | Most frequent value | Ungrouped: Most repeated value Grouped: \( \text{Mode} = l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h \) |
Categorical, discrete, and grouped data |
Mode in Statistics
Mode in statistics means the value that appears most often in a dataset. This simple measure of central tendency quickly shows what is most typical in a set of numbers or categories. Mode is often used in business, economics, and daily situations to find popular products or common choices.
Mode Formula (Ungrouped and Grouped Data)
The formula to find mode depends on the type of data available. Knowing which formula to use saves time during exams and simplifies calculations.
| Data Type | Mode Formula | Variables |
|---|---|---|
| Ungrouped Data | Most repeated value in the dataset | None needed |
| Grouped Data | Mode = l + ( (f₁ - f₀) / (2f₁ - f₀ - f₂) ) × h |
l: lower limit of modal class f₁: frequency of modal class f₀: frequency before modal class f₂: frequency after modal class h: class interval |
How to Calculate Mode: Solved Examples
Solved examples help clarify the steps to find the mode. These are useful for Class 11/12 students and for last-minute revision.
Mode for Ungrouped Data
Example: Find the mode for the data set: 13, 16, 12, 14, 19, 12, 14, 13, 14.
Solution: Arrange and count frequencies:
12 (2), 13 (2), 14 (3), 16 (1), 19 (1)
Mode = 14 (appears 3 times)
Mode for Grouped Data
Example: Find mode for following frequency table:
| Class Interval | Frequency |
|---|---|
| 10 – 20 | 5 |
| 20 – 30 | 8 |
| 30 – 40 | 12 |
| 40 – 50 | 4 |
Mode class: 30–40 (highest frequency = 12)
l = 30, f₁ = 12, f₀ = 8, f₂ = 4, h = 10
Mode = 30 + ( (12 – 8) / (2 × 12 – 8 – 4) ) × 10
= 30 + (4 / (24 – 12)) × 10
= 30 + (4 / 12) × 10
= 30 + (0.333) × 10
= 30 + 3.33 = 33.33
Applications of Mode
Mode is widely used in economics and business to identify the most popular product, service, or choice among consumers. It helps companies decide which item to stock more. Mode is also useful in exam data, survey analysis, and quality control. At Vedantu, we make sure students see such real-world uses clearly in each topic.
Difference Between Mean, Median and Mode
Mean, median, and mode are all measures of central tendency, but they serve different purposes.
| Feature | Mean | Median | Mode |
|---|---|---|---|
| Definition | Sum of all values ÷ Number of values | Middle value when sorted | Most frequent value |
| Usefulness | General average; affected by outliers | Position-based; not affected by outliers | Shows most typical observation |
| Best For | Continuous, normal data | Skewed distributions | Categorical/discrete data, exam MCQ analysis |
| Empirical Relation | Mode ≈ 3 × Median – 2 × Mean (for moderate skew) | ||
When and Why to Use Mode
Use mode when you want to know what is most common. It is ideal for multiple-choice questions, preference surveys, sales data, and situations where outliers may affect the mean. Mode is also helpful for qualitative (non-numeric) data.
For stepwise guidance on mode calculations for all data types, visit Calculation of Mean, Median, and Mode.
To understand the process of organizing and presenting data before calculating mode, see Tabulation and Textual and Tabular Presentation Of Data.
Relation of Mode to Syllabus and Competitive Exams
Mode is directly covered in Class 11 and 12 Economics/Statistics syllabus. Questions appear in CBSE, ISC, and state boards, as well as in competitive exams like SSC, UPSC basics, and accountancy papers. Knowing mode can improve your accuracy in data handling and data analysis questions.
For more applications of mode in economics data, visit Statistics in Economics and Index Numbers.
At Vedantu, we explain measures of central tendency mode with clear steps, simple language, and real-life examples to ensure strong student understanding and exam confidence.
In summary, measures of central tendency mode points to the most frequent value in a set, helping students and business professionals identify trends. Understanding its calculation for ungrouped and grouped data is essential for exams and practical data analysis. Practice using mode alongside mean and median for best results in school and beyond.
FAQs on Measures of Central Tendency: Mode Explained with Examples
1. What is the mode in central tendency?
The mode is the value that appears most frequently in a dataset. It's one of the three main measures of central tendency, alongside the mean and median. Understanding the mode is crucial for analyzing data in statistics and commerce.
2. What is mode and its formula?
The mode represents the most frequent value in a dataset. For ungrouped data, it's the value that appears most often. There isn't a single formula for the mode, as its calculation depends on data type. For grouped data, a slightly more complex formula is used to estimate the modal class.
3. What is the mode of 13, 16, 12, 14, 19, 12, 14, 13, 14?
The mode is 14. This is because the number 14 appears three times, which is more frequent than any other value in the dataset.
4. What are the 4 measures of central tendency?
While mean, median, and mode are the three most common measures of central tendency, sometimes the midrange (average of the highest and lowest values) is also considered. The choice of measure depends on the dataset's characteristics and the type of analysis.
5. What is the mode in grouped data?
In grouped data, the mode is the midpoint of the modal class (the class with the highest frequency). The formula to calculate it is: Mode = l + [(f₁-f₀)/(2f₁-f₀-f₂)] × h, where 'l' is the lower class boundary of the modal class, f₁ is the modal frequency, f₀ is the frequency of the class before, f₂ is the frequency of the class after, and 'h' is the class interval.
6. Measures of central tendency mode formula?
There isn't one single formula for the mode. For ungrouped data, you identify the most frequent value. For grouped data, you use the formula: Mode = l + [(f₁-f₀)/(2f₁-f₀-f₂)] × h. This formula helps estimate the mode within the modal class.
7. Measures of central tendency mode example?
Consider the dataset: 2, 3, 3, 4, 4, 4, 5, 5. The mode is 4 because it appears most frequently. For grouped data, calculating the mode involves identifying the modal class (highest frequency) and using the formula to estimate its midpoint.
8. Measures of central tendency mode definition?
The mode is a measure of central tendency that identifies the most frequent value or observation in a dataset. It's a useful measure when dealing with categorical data or when you want to know the most typical value. The mode is one of three main measures of central tendency; the other two are the mean and the median.
9. Measures of central tendency mode grouped data?
For grouped data, you find the modal class (the class with the highest frequency) and then use a specific formula to estimate the mode within that class. The formula uses the lower boundary of the modal class, the frequencies of the modal class and neighboring classes, and the class width. This provides a more precise estimate of the mode than simply identifying the modal class itself.
10. What is the difference between mean, median and mode?
The mean is the average (sum of values divided by the count). The median is the middle value when data is ordered. The mode is the most frequent value. They offer different perspectives on central tendency; the mode is especially useful for categorical data or when outliers heavily influence the mean.
11. How is mode used in business decision-making?
In business, the mode helps identify popular products or services (most frequently purchased), understand consumer preferences, predict trends, and optimize inventory management. For example, a clothing store might use the mode to determine the most popular clothing size.
12. Why is mode sometimes preferred over mean or median in data analysis?
The mode is preferred when dealing with categorical data or when the most frequent value is important, irrespective of other values. It is also less sensitive to extreme outliers that could skew the mean.
