13.3.3. Example 2

The marks distribution of 30 students in a mathematics examination are given in Table 13.3 of Example 1. Find the mode of this data. Also compare and interpret the mode and the mean.

In this example, we are considering the marks obtained by 30 students in a mathematics examination. We want to find the mode, which is the most frequently occurring mark range in the given data. Alongside that, we will compare the mode with the mean, which is the average of all marks.

Think of a school class where students frequently get multiple-choice questions wrong. If most students get 2 out of 5 questions right, but the average score (considering all questions) is 3, that highlights the difference between the mode (most common score) and the mean (average score).

Solution: Refer to Table 13.3 of Example 1. Since the maximum number of students (i.e., 7) have got marks in the interval 40 - 55, the modal class is 40 - 55. Therefore, the lower limit (l) of the modal class = 40, the class size (h) = 15, the frequency (f) of modal class = 7, the frequency (f) of the class preceding the modal class = 3, the frequency (f) of the class succeeding the modal class = 6.

To find the mode, we identify the range of marks where the highest number of students received their scores. In our case, 7 students scored in the range of 40 to 55. This is our modal class. We then note some key values: the lower limit of this range is 40, the class size (which is the range width) is 15, and the frequencies of the classes around this modal class to help in the calculation.

Consider a basketball game where you observe that most players score between 10 and 15 points. If 7 players scored in that range while very few scored lower or higher, that range (10-15 points) represents the 'mode' of scoring among the players.

Now, using the formula: Mode = l + ((f1 - f0) / (2f1 - f0 - f2)) * h, we get Mode = 40 + ((7 - 3) / (14 - 6 - 3)) * 15 = 52.

To calculate the mode mathematically, we apply a specific formula that uses the values we gathered: the lower limit, the frequencies of the modal class and its neighbouring classes, and the class size. Plugging our values into the formula gives us a calculated mode of 52 marks.

Imagine you're mixing paints. If you have different amounts of red, blue, and yellows, using the right formula helps you find the best mix. Here, calculating the mode is like finding the perfect mix of colors that highlight the most common score range among students.

So, the mode marks are 52. Now, from Example 1, you know that the mean marks is 62. So, the maximum number of students obtained 52 marks, while on an average, a student obtained 62 marks.

After finding that the mode marks are 52, we compare this with the mean marks of 62 from a previous example. This shows that while most students scored around 52 marks, the average score across all students is higher at 62.

Think of a marathon race where most runners finish in about 4 hours, but the overall average finishing time calculated (including some slower runners) is around 5 hours. This includes everyone’s performance in the calculation, showing a nuanced view of overall performance.

Remarks: 1. In Example 6, the mode is less than the mean. But for some other problems it may be equal or more than the mean also. 2. It depends upon the demand of the situation whether we are interested in finding the average marks obtained by the students or the average of the marks obtained by most of the students. In the first situation, the mean is required and in the second situation, the mode is required.

The remarks help us understand that the relationship between mode and mean can vary based on the data. In this case, the mode was less than the mean, indicating that while many students did well, some performed even better, pulling the average up. Depending on what we’re analyzing, we might want either the mean or the mode to get a clearer picture.

In a classroom, if the majority of students receive tests scores around 75%, but a few score 100%, the 'mode' reflects the common score. However, the 'mean' is skewed upwards due to those higher scores, which may not represent most students' performance. This illustrates how context matters when interpreting results.

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