Finding The Value Of K In A Frequency Distribution With A Given Arithmetic Mean

by ADMIN 80 views

Introduction to Arithmetic Mean and Frequency Distribution

In statistics, the arithmetic mean, often referred to as the average, is a fundamental measure of central tendency. It represents the sum of a set of values divided by the number of values. Understanding the arithmetic mean is crucial in various fields, from data analysis to everyday decision-making. When dealing with frequency distributions, which group data into intervals or classes, calculating the mean requires a slightly different approach, incorporating the frequencies associated with each class.

A frequency distribution is a tabular representation of data that shows the number of items in each category or class. These distributions are essential for summarizing and interpreting large datasets. Each class is defined by a range, and the frequency indicates how many data points fall within that range. Frequency distributions are used extensively in statistical analysis to understand patterns and trends within data. Calculating the arithmetic mean of a frequency distribution involves using the midpoints of the class intervals and their corresponding frequencies.

The problem at hand presents a frequency distribution with a missing frequency, denoted as k, for one of the classes. The arithmetic mean of the entire distribution is given as 53. Our goal is to determine the value of k. This involves applying the formula for the arithmetic mean of a frequency distribution and solving for the unknown variable. This process not only reinforces the understanding of statistical calculations but also highlights the practical application of these concepts in real-world scenarios. The ability to find missing frequencies in a distribution is a valuable skill in statistical analysis, allowing for a more complete understanding of the dataset even when some data points are initially unknown.

Problem Statement

We are given a frequency distribution with the following classes and frequencies:

Class 0-20 20-40 40-60 60-80 80-100
Frequency 12 15 32 k 13

The arithmetic mean of this distribution is 53. Our task is to find the value of k, which represents the frequency of the class 60-80. To solve this, we will use the formula for the arithmetic mean of a frequency distribution.

Methodology: Calculating the Arithmetic Mean for Grouped Data

To find the value of k, we will use the formula for the arithmetic mean of a frequency distribution. The formula is:

Mean = (Σ(fᵢ * xᵢ)) / Σfᵢ

Where:

  • fáµ¢ is the frequency of the i-th class,
  • xáµ¢ is the midpoint of the i-th class interval, and
  • Σ represents the summation.

Step-by-Step Calculation

  1. Determine the Midpoints (xáµ¢) of Each Class Interval:

    • For the class 0-20, the midpoint is (0 + 20) / 2 = 10.
    • For the class 20-40, the midpoint is (20 + 40) / 2 = 30.
    • For the class 40-60, the midpoint is (40 + 60) / 2 = 50.
    • For the class 60-80, the midpoint is (60 + 80) / 2 = 70.
    • For the class 80-100, the midpoint is (80 + 100) / 2 = 90.

  2. Multiply Each Frequency (fáµ¢) by Its Corresponding Midpoint (xáµ¢):

    • For the class 0-20: 12 * 10 = 120
    • For the class 20-40: 15 * 30 = 450
    • For the class 40-60: 32 * 50 = 1600
    • For the class 60-80: k * 70 = 70k
    • For the class 80-100: 13 * 90 = 1170

  3. Sum the Products (fáµ¢ * xáµ¢):

    • Σ(fáµ¢ * xáµ¢) = 120 + 450 + 1600 + 70k + 1170 = 3340 + 70k

  4. Sum the Frequencies (fáµ¢):

    • Σfáµ¢ = 12 + 15 + 32 + k + 13 = 72 + k

  5. Apply the Arithmetic Mean Formula:

    • Mean = (Σ(fáµ¢ * xáµ¢)) / Σfáµ¢
    • 53 = (3340 + 70k) / (72 + k)

Solving for k

Now that we have set up the equation, we can solve for k. This involves algebraic manipulation to isolate k on one side of the equation.

  1. Multiply Both Sides by (72 + k):

    • 53 * (72 + k) = 3340 + 70k

  2. Expand the Left Side:

    • 3816 + 53k = 3340 + 70k

  3. Rearrange the Equation to Group Like Terms:

    • 70k - 53k = 3816 - 3340

  4. Simplify:

    • 17k = 476

  5. Divide Both Sides by 17 to Solve for k:

    • k = 476 / 17
    • k = 28

Thus, the value of k is 28.

Verification of the Result

To ensure the correctness of our solution, we can substitute the value of k back into the arithmetic mean formula and verify that it equals 53.

  1. Substitute k = 28 into the Formula:

    • Mean = (3340 + 70*28) / (72 + 28)
    • Mean = (3340 + 1960) / 100
    • Mean = 5300 / 100
    • Mean = 53

The calculated mean is indeed 53, which matches the given arithmetic mean. This confirms that our solution for k is correct.

Conclusion

In this problem, we successfully found the missing frequency, k, in a frequency distribution given the arithmetic mean. By applying the formula for the arithmetic mean of grouped data and using algebraic techniques, we determined that k = 28. The verification step further validated our solution. This exercise demonstrates the importance of understanding statistical measures and their applications in data analysis. The ability to calculate and interpret statistical measures like the arithmetic mean is crucial in many fields, allowing for informed decision-making based on data.

The process involved several key steps: identifying the midpoints of class intervals, calculating the sum of the product of frequencies and midpoints, summing the frequencies, and solving the resulting equation. Each step is essential in accurately determining the missing frequency. This problem not only reinforces mathematical skills but also enhances the understanding of statistical concepts, which are fundamental in data analysis and interpretation. By mastering these concepts, one can effectively analyze datasets and draw meaningful conclusions, contributing to better decision-making in various contexts. The combination of statistical knowledge and algebraic skills is a powerful tool in addressing real-world problems and gaining insights from data.