E.L.C.T NORTHERN DIOCESE MASAMA GIRLS HIGH SCHOOL Form Four Mathematics Problem Solution

Understanding the Frequency Distribution and the Problem

In this mathematics problem from E.L.C.T NORTHERN DIOCESE MASAMA GIRLS HIGH SCHOOL Form Four Home Package, we are presented with a frequency distribution table showing scores and the number of students who achieved those scores. Our task is to analyze this data and calculate three key statistical measures: the mean, the median, and the modal class. These measures will help us understand the central tendency and distribution of the scores. The mean provides the average score, the median indicates the middle score, and the modal class represents the score range with the highest frequency. To accurately determine these measures, we will utilize specific formulas and techniques tailored for grouped data. Understanding these concepts is crucial for students as they build a solid foundation in statistics, which is a fundamental branch of mathematics with applications across numerous fields, from science and engineering to social sciences and economics. This exercise not only enhances their computational skills but also fosters their ability to interpret and draw meaningful conclusions from data, a skill highly valued in both academic and professional settings. The application of an assumed mean in calculating the actual mean adds a layer of complexity that hones students' understanding of different statistical methods and their efficiency in different scenarios.

Calculating the Mean Using the Assumed Mean Method

The first part of the problem requires us to calculate the mean of the scores, given an assumed mean (A) of 50.5. The mean, often referred to as the average, is a fundamental measure of central tendency in statistics. It provides a single value that represents the center of a dataset. When dealing with grouped data, such as the frequency distribution table provided, we use a specific formula that takes into account the frequency of each class interval. The assumed mean method is a technique used to simplify the calculation of the mean, especially when dealing with large numbers. This method involves choosing a value within the data range as the assumed mean and then calculating deviations from this value. The formula for calculating the mean using the assumed mean method is:

Mean = A + (∑fd / ∑f)

Where:

• A is the assumed mean
• f is the frequency of each class
• d is the deviation of the midpoint of each class from the assumed mean (d = midpoint - A)
• ∑fd is the sum of the products of the frequencies and deviations
• ∑f is the sum of the frequencies (total number of students)

To apply this formula, we first need to determine the midpoint of each class interval. The midpoint is calculated by adding the upper and lower limits of the class and dividing by 2. For example, for the class interval 1-20, the midpoint is (1+20)/2 = 10.5. We then calculate the deviation (d) for each class by subtracting the assumed mean (50.5) from the midpoint. Next, we multiply the frequency (f) of each class by its corresponding deviation (d) to get fd. We sum all the fd values (∑fd) and the frequencies (∑f). Finally, we plug these values into the formula to calculate the mean. This method not only simplifies the calculations but also provides a deeper understanding of how deviations from a central point contribute to the overall average. By mastering this technique, students can efficiently calculate the mean for various grouped datasets, a skill essential in statistical analysis and data interpretation.

Determining the Median from the Frequency Distribution

The second part of the problem asks us to find the median of the scores. The median is another crucial measure of central tendency, representing the middle value in a dataset when it is arranged in ascending order. Unlike the mean, which is affected by extreme values, the median provides a more robust measure of the center, particularly in skewed distributions. For grouped data presented in a frequency distribution table, the median is the value that divides the distribution into two equal halves. To calculate the median, we first need to determine the median class, which is the class interval that contains the median. This is done by finding the cumulative frequency. The median class is the class where the cumulative frequency is greater than or equal to half the total frequency (∑f/2). Once we have identified the median class, we use the following formula to calculate the median:

Median = L + [(N/2 - CF) / f] * h

Where:

• L is the lower boundary of the median class
• N is the total number of observations (∑f)
• CF is the cumulative frequency of the class preceding the median class
• f is the frequency of the median class
• h is the class width (the difference between the upper and lower boundaries of the class)

This formula interpolates within the median class to estimate the exact median value. It takes into account the proportion of the data that falls within the median class and adjusts the lower boundary accordingly. Calculating the median for grouped data requires a careful understanding of cumulative frequencies and class boundaries, reinforcing students' skills in data organization and interpretation. The median is a valuable statistical tool, especially in situations where the dataset may contain outliers or extreme values that could distort the mean. By understanding how to calculate and interpret the median, students gain a more complete picture of the distribution of data and can make more informed decisions based on their analysis.

Identifying the Modal Class

The final part of the problem requires us to identify the modal class. The modal class is the class interval with the highest frequency. It represents the most common range of scores in the dataset. Unlike the mean and median, which provide a single central value, the modal class identifies the range where most of the data points are concentrated. Determining the modal class is straightforward: we simply look for the class interval with the highest frequency in the frequency distribution table. In some cases, a dataset may have more than one modal class (bimodal or multimodal), indicating multiple peaks in the distribution. The modal class is a useful measure for understanding the most typical or frequently occurring values in a dataset. It is particularly relevant in situations where we are interested in the most common category or range, rather than the average or middle value. For example, in market research, the modal class might represent the most popular price range for a product. In educational assessments, it could indicate the score range achieved by the majority of students. Understanding the modal class provides valuable insights into the distribution's shape and characteristics. It helps in identifying patterns and trends within the data, contributing to a more comprehensive analysis. By being able to identify the modal class, students can quickly grasp the most prevalent values in a dataset, enhancing their ability to interpret and communicate statistical information effectively. This skill is particularly useful in real-world applications where quick identification of trends and common occurrences is essential for decision-making.

Detailed Solution and Discussion of the Problem

To provide a comprehensive solution, let's assume the frequency distribution table is as follows:

i) Mean Calculation:

Using the assumed mean formula:

Mean = A + (∑fd / ∑f) = 50.5 + (-60 / 40) = 50.5 - 1.5 = 49

Therefore, the mean score is 49.

ii) Median Calculation:

Total number of students (N) = 40

N/2 = 40/2 = 20

The median class is the class where the cumulative frequency is greater than or equal to 20, which is the 41-60 class.

• L (lower boundary of median class) = 40.5
• CF (cumulative frequency of the class preceding the median class) = 14
• f (frequency of the median class) = 15
• h (class width) = 20

Median = L + [(N/2 - CF) / f] * h = 40.5 + [(20 - 14) / 15] * 20 = 40.5 + (6/15) * 20 = 40.5 + 8 = 48.5

Thus, the median score is 48.5.

iii) Modal Class:

The modal class is the class with the highest frequency, which is the 41-60 class (frequency = 15).

Conclusion

This comprehensive solution demonstrates the calculation of the mean, median, and modal class from a frequency distribution table. By using the assumed mean method, we efficiently calculated the mean score. The median was determined by identifying the median class and applying the appropriate formula. The modal class was simply the class with the highest frequency. These measures provide a clear understanding of the central tendency and distribution of the scores. This exercise reinforces the importance of statistical analysis in understanding and interpreting data, a crucial skill for students in various academic and professional fields. By mastering these concepts, students can confidently tackle statistical problems and draw meaningful conclusions from data, empowering them to make informed decisions based on evidence and analysis.