What Is Mode- Definition, Formula, Practice Questions & Answers

The mode is basically the value that appears most frequently in a set of data. Knowing the mode helps us get a clear picture of how the data is spread out and what values are most common.

By identifying the mode, we can see not only the central tendency of the data but also understand which values are repeated more often than others.

Definition Of Mode

Mode is the value that appears most frequently in a given data set. It represents the peak occurrence among all values, making it a crucial measure in statistical analysis.

Calculating Mode For Ungrouped Data

Process

To calculate the mode for ungrouped data, you identify the value that appears most frequently. This value represents the mode of the dataset.

For instance, in a set of numbers like 2, 4, 5, 3, 4, and 7, the mode is 4 since it occurs twice.

Applicability

In scenarios where you have a list of individual data points without any grouping or categorization, finding the mode can help determine the most common observation.

For example, in a survey collecting respondents' favourite colours without grouping them by age or gender, identifying the mode colour gives insight into the prevalent choice among all participants.

Importance

Identifying the mode in ungrouped datasets is crucial for various reasons. It provides a simple measure of central tendency that highlights the most frequent observation.

Calculating Mode For Grouped Data

In grouped data, the modal class is the class interval with the highest frequency. It represents the range with the maximum occurrences.

To find the mode in grouped data, you first identify the modal class. Next, within this modal class, determine the modal value using the formula:

Mode=𝐿+(𝑓_𝑚−𝑓_𝑚−1) / 2𝑓_𝑚−𝑓_𝑚−1−𝑓_𝑚+1×𝑐

• The modal class, which is the one with the most occurrences, has a lower boundary denoted as L

• fm is the frequency of the modal class.

• fm−1​ is the frequency of the class preceding the modal class.

• fm+1​ is the frequency of the class succeeding the modal class.

• c is the class width (the difference between the upper and lower boundaries of the modal class).

This formula estimates the mode for grouped data by interpolating between the modal class and its neighbouring classes, considering their frequencies and class widths.

Bimodal, Trimodal & Multimodal Distributions

Bimodal

A bimodal distribution is characterized by having two peaks or modes in the data. This means that two values occur most frequently in the dataset.

For example, a bimodal distribution could represent the distribution of heights in a population where there are peaks for both tall and short individuals.

Trimodal

In a trimodal distribution, three distinct peaks or modes are present in the dataset. This suggests that there are three values with high frequencies.

An example of a trimodal distribution could be the distribution of exam scores in a classroom where students fall into three distinct performance groups - high, medium, and low achievers.

Multimodal

A multimodal distribution goes beyond three modes and can have multiple peaks in the data. This indicates that several values occur frequently.

An illustration of a multimodal distribution could be the distribution of temperatures throughout the year in a region with distinct seasons like spring, summer, fall, and winter.

Comparing Mean, Median & Mode

Mean vs. Median

Mean represents the average value of a dataset, calculated by adding all values and dividing by the total number. Median, on the other hand, is the middle value when data is arranged in ascending order.

When data has extreme outliers, the median is more reliable as it is not influenced by these outliers. In scenarios where there are no outliers, and a balanced distribution exists, the mean provides a better representation of the central tendency.

Mode In Data Analysis

While mean and median focus on typical values, mode highlights the most frequent data point in a set. It is especially useful in categorical data analysis, where identifying the most common category is essential.

Complementary Roles

In statistical analysis, mean, median, and mode each serve unique purposes. They complement each other by providing different perspectives on central tendencies within datasets and data values.

Click here to learn more about various topics related to quantitative aptitude, including a detailed explanation of what mode is!

Selected Practice Questions With Solution 

Practice not only helps sharpen your skills but also boosts your confidence in handling diverse datasets. By engaging with different levels of difficulty, you can enhance your ability to recognize patterns and find solutions effectively.

Question 1: What is the mode of the following dataset: 2, 3, 5, 5, 7, 7, 7, 9?

a) 2 

b) 5 

c) 7 

d) 9

Correct answer: c) 7

Explanation: In the given dataset: 2, 3, 5, 5, 7, 7, 7, 9, the number 7 appears three times, which is more frequent than any other number. Therefore, the mode of this dataset is 7.

Question 2: Find the mode of the following dataset: 4, 7, 7, 8, 9, 9, 10.

a) 9

b) 7

c) 10 

d) 7 & 9

Correct answer: d) 7 & 9

Explanation: The dataset is multimodal if multiple numbers appear most frequently in the dataset. Therefore, the mode of this dataset is 7 & 9. 

Question 3: What is the mode of the following dataset: 3, 5, 7, 7, 9, 9, 9, 11?

a) 3

b) 5

c) 7

d) 9

Correct answer: d) 9

Explanation: In the given dataset, the number 9 appears three times, which is more frequent than any other number. Therefore, the mode of this dataset is 9.

Question 4: Determine the mode of the following dataset: 2, 4, 4, 6, 6, 6, 8, 8, 8, 8.

a) 8

b) 4

c) 6

d) 2

Correct answer: a) 8

Explanation: Here, 8 appears most frequently. Hence, it's the correct answer. 

Question 5: If a dataset's mode is 10, which of these could be the dataset?

a) 5, 8, 10, 10, 12 

b) 10, 10, 10, 12, 15 

c) 6, 8, 10, 11, 11 

d) 7, 8, 9, 10, 11

Correct answer: b) 10, 10, 10, 12, 15

Explanation: In the given options, option b) contains the dataset where 10 appears most frequently, making it the mode.

Question 6: If a dataset's mode is 10, which of these could be the dataset?

a) 5, 6, 7, 8, 10, 10, 11

b) 10, 10, 11, 11, 12, 13, 14

c) 8, 9, 10, 10, 10, 11, 12

d) 9, 10, 10, 13, 11, 11, 12

Correct answer: c) 8, 9, 10, 10, 10, 11, 12

Explanation: In a dataset, the mode is the value that appears most frequently. In option c), the value 10 appears three times, making it the mode of the dataset.

Question 7: The following table shows the frequency distribution of scores in a class:

What is the mode of the scores?

a) 25

b) 35

c) 45

d) 55

Correct answer: c) 45

Explanation: The mode for grouped data is estimated as the midpoint of the class interval with the highest frequency. The class interval with the highest frequency is 41-50, with a frequency of 15. Therefore, the mode is estimated to be the midpoint of this class interval, which is (41 + 50) / 2 = 45.

Question 8: The following table represents the distribution of ages (in years) of participants in a marathon:

What is the mode of the ages of the participants?

a) 35

b) 45

c) 55

d) 65

Correct answer: c) 55

Explanation: The mode for grouped data is estimated as the midpoint of the class interval with the highest frequency. Therefore, the mode is estimated to be the midpoint of this class interval, which is (50 + 59) / 2 = 54.5. So, the correct answer is c) 55. 

Conclusion

You now have a solid grasp of what mode is, how to calculate it for different types of data, and its significance in statistics. By understanding bimodal, trimodal, and multimodal distributions, you can effectively identify patterns in your data.

Practice problems will further enhance your skills in finding the mode efficiently. Keep honing your mode calculation skills by practising with different datasets.

Frequently Asked Questions (FAQs)

1. What is the mode in statistics?

In statistics, the mode is the number that shows up the most in a set of data. It's one way to figure out the middle of the numbers, along with the mean and median.

2. How do you calculate the mode for ungrouped data?

To calculate the mode for ungrouped data, simply identify the value that occurs most frequently in the dataset. If multiple values have the same highest frequency, then the data set is considered multimodal.

3. What is the significance of identifying bimodal, trimodal, or multimodal distributions?

Identifying bimodal, trimodal, or multimodal distributions helps to understand the complexity of data patterns. It provides insights into multiple peaks or clusters within the dataset, indicating diverse trends or subgroups.

4. How does finding the mode effectively benefit statistical analysis?

Finding the mode effectively helps in understanding common patterns or predominant values within a dataset. It simplifies data interpretation by highlighting the most frequent observations, aiding decision-making processes based on popular trends.

5. Can you explain how to compare mean, median, and mode in data analysis?

In data analysis, comparing mean, median, and mode allows for a comprehensive evaluation of central tendencies. While the mean reflects the average value and the median represents the middle point, mode indicates the most recurring value, offering a holistic view of data distribution.

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