Frequency Distribution Calculating Mean Mode And Median

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In the realm of statistics, understanding frequency distribution is paramount for analyzing data sets effectively. Frequency distribution illustrates how often each value occurs in a dataset, providing valuable insights into the data's central tendencies and spread. When dealing with numerical data, measures like mean, mode, and median are crucial for summarizing and interpreting the information. This article delves into the concepts of mean, mode, and median, demonstrating their calculation using a given frequency distribution table. We will explore how to determine these measures and interpret their significance in understanding the data's characteristics.

Problem Statement: Deciphering Marks Distribution

Let's consider a scenario where we have the frequency distribution of marks obtained in a test. This distribution provides a snapshot of how the scores are spread across different marks. The table below summarizes the data:

Marks 3 4 5 6 7 8
Frequency 5 X - 1 X 9 4 1

In this table, the 'Marks' row represents the scores obtained by students, and the 'Frequency' row indicates the number of students who achieved those scores. Notice that the frequencies for marks 4 and 5 are expressed in terms of an unknown variable, X. This adds an interesting twist to the problem, requiring us to first determine the value of X before we can proceed with further analysis.

Our task is twofold:

  1. Given that the mean mark is 5, we need to calculate the value of X.
  2. Once we have determined X, we need to find the mode and median of the distribution.

This problem combines fundamental statistical concepts and algebraic problem-solving, making it a valuable exercise in understanding data analysis.

Calculating the Value of X: Harnessing the Power of the Mean

To calculate the value of X, we need to understand the concept of the mean. The mean, often referred to as the average, is a measure of central tendency that represents the sum of all values in a dataset divided by the number of values. In the context of a frequency distribution, the mean is calculated by multiplying each value by its frequency, summing these products, and then dividing by the total frequency.

Understanding the Mean Formula

The formula for the mean (xˉ{\bar{x}}) in a frequency distribution is:

xˉ=∑(fi∗xi)∑fi{\bar{x} = \frac{\sum(f_i * x_i)}{\sum f_i}}

Where:

  • xi{x_i} represents each individual mark.
  • fi{f_i} represents the corresponding frequency of that mark.
  • ∑(fi∗xi){\sum(f_i * x_i)} represents the sum of the products of each mark and its frequency.
  • ∑fi{\sum f_i} represents the total frequency, which is the sum of all frequencies.

In our problem, we are given that the mean is 5. Therefore, we can set up the equation as follows:

5=(3∗5)+(4∗(X−1))+(5∗X)+(6∗9)+(7∗4)+(8∗1)5+(X−1)+X+9+4+1{5 = \frac{(3 * 5) + (4 * (X - 1)) + (5 * X) + (6 * 9) + (7 * 4) + (8 * 1)}{5 + (X - 1) + X + 9 + 4 + 1}}

Solving for X: A Step-by-Step Approach

Now, let's solve this equation step-by-step to find the value of X:

  1. Simplify the numerator and the denominator:

    Numerator:

    (3 * 5) + (4 * (X - 1)) + (5 * X) + (6 * 9) + (7 * 4) + (8 * 1)

    = 15 + 4X - 4 + 5X + 54 + 28 + 8

    = 9X + 101

    Denominator:

    5 + (X - 1) + X + 9 + 4 + 1

    = 2X + 18

  2. Substitute the simplified expressions back into the equation:

    5=9X+1012X+18{5 = \frac{9X + 101}{2X + 18}}

  3. Multiply both sides by (2X + 18) to eliminate the fraction:

    5∗(2X+18)=9X+101{5 * (2X + 18) = 9X + 101}

  4. Distribute and simplify:

    10X+90=9X+101{10X + 90 = 9X + 101}

  5. Isolate X by subtracting 9X from both sides:

    X+90=101{X + 90 = 101}

  6. Subtract 90 from both sides to solve for X:

    X=101−90{X = 101 - 90}

    X=11{X = 11}

Therefore, the value of X is 11. This means that the frequency of students scoring 5 marks is 11, and the frequency of students scoring 4 marks is 11 - 1 = 10.

Determining Mode and Median: Unveiling Data's Central Values

Now that we have calculated the value of X, we can complete the frequency distribution table and proceed to find the mode and median. These are two other measures of central tendency that provide different perspectives on the data's typical value.

The Updated Frequency Distribution Table

Substituting X = 11 into the table, we get the complete frequency distribution:

Marks 3 4 5 6 7 8
Frequency 5 10 11 9 4 1

Finding the Mode: Identifying the Most Frequent Score

The mode is the value that appears most frequently in a dataset. In a frequency distribution, the mode is the value with the highest frequency. Looking at our updated table, we can see that the highest frequency is 11, which corresponds to a mark of 5.

Therefore, the mode of this distribution is 5. This indicates that the most common score obtained by students in the test was 5.

Calculating the Median: Pinpointing the Middle Value

The median is the middle value in a dataset when the values are arranged in ascending order. In a frequency distribution, the median is the value that divides the distribution into two equal halves. To find the median, we need to determine the middle position in the dataset.

  1. Calculate the Total Frequency:

    The total frequency is the sum of all frequencies: 5 + 10 + 11 + 9 + 4 + 1 = 40. This means there are 40 students in total.

  2. Determine the Median Position:

    The median position is given by (Total Frequency + 1) / 2. In this case, it is (40 + 1) / 2 = 20.5. Since we have a decimal, the median lies between the 20th and 21st values.

  3. Find the Cumulative Frequencies:

    To locate the 20th and 21st values, we need to calculate the cumulative frequencies:

    • Marks 3: Cumulative frequency = 5
    • Marks 4: Cumulative frequency = 5 + 10 = 15
    • Marks 5: Cumulative frequency = 15 + 11 = 26

  4. Identify the Median Value:

    The cumulative frequency of 15 tells us that the first 15 values fall within marks 3 and 4. The 20th and 21st values fall within the cumulative frequency of 26, which corresponds to a mark of 5.

Therefore, the median of this distribution is 5. This means that half of the students scored 5 or below, and half scored 5 or above.

Conclusion: Interpreting Mean, Mode, and Median Together

In this article, we have successfully calculated the value of X in a frequency distribution table, and we have determined the mode and median of the distribution. Let's summarize our findings:

  • The value of X is 11.
  • The mode of the distribution is 5.
  • The median of the distribution is 5.

It's interesting to note that in this case, the mean (given as 5), mode (5), and median (5) are all equal. This indicates that the distribution is symmetrical and that the data is clustered around the central value of 5. However, this is not always the case. In skewed distributions, the mean, mode, and median can differ, providing valuable insights into the data's skewness and central tendencies.

Understanding these measures of central tendency is crucial for interpreting data and making informed decisions. The mean provides an average value, the mode identifies the most frequent value, and the median pinpoints the middle value. By analyzing these measures together, we can gain a comprehensive understanding of the data's characteristics and draw meaningful conclusions.

This exercise demonstrates how statistical concepts can be applied to real-world scenarios, helping us to analyze data, understand patterns, and make informed decisions. From calculating the value of an unknown variable to determining the mode and median, each step provides valuable insights into the distribution of marks and the overall performance of the students in the test.