Problem 25 - Algebra Form and Function

Question

Find the mean of each data set: (a) Five readings equaling (not totaling) \(120,\) three readings equaling 130 , two readings equaling 140 , four readings equaling 150 , and one reading equaling 160 . (b) Three readings equaling \(x_{1}\), six readings equaling \(x_{2}\), seven readings equaling \(x_{3}\), five readings equaling \(x_{4},\) and four readings equaling \(x_{5}\). (c) \(n_{1}\) readings equaling \(x_{1}, n_{2}\) readings equaling \(x_{2}\), and so on, up to \(n_{5}\) readings equaling \(x_{5}\).

Step-by-Step Solution

Answer

Question: Calculate the mean for each dataset. a) Dataset A: {[(120)x5], [(130) x 3], [(140)x2], [(150)x4], [(160)x1]} b) Dataset B: {[(x1)x3], [(x2) x 6], [(x3)x7], [(x4)x5], [(x5)x4]}, where x1, x2, x3, x4, and x5 are the numerical values. c) Dataset C: {[(x1)xn1], [(x2)xn2], [(x3)xn3], [(x4)xn4], [(x5)xn5]}, where x1, x2, x3, x4, x5 are values and n1, n2, n3, n4, n5 are the corresponding readings. Answer: a) Mean of Dataset A = \(\frac{(5 \times 120) + (3 \times 130) + (2 \times 140) + (4 \times 150) + (1 \times 160)}{15}\) b) Mean of Dataset B = \(\frac{(3x_1) + (6x_2) + (7x_3) + (5x_4) +(4x_5)}{25}\) c) Mean of Dataset C = \(\frac{(n_1x_1) + (n_2x_2) + (n_3x_3) + (n_4x_4) +(n_5x_5)}{n_1+n_2+n_3+n_4+n_5}\)

Step 1: (a) Determine the mean formula for the dataset#a# The mean of a dataset can be calculated using the formula: Mean = \(\frac{\text{Sum of values}}{\text{Total number of values}}\)

(a) Calculate the sum of values in the dataset#a# Remember that we have: - 5 readings equaling 120 - 3 readings equaling 130 - 2 readings equaling 140 - 4 readings equaling 150 - 1 reading equaling 160 Sum of values = (5 x 120) + (3 x 130) + (2 x 140) + (4 x 150) + (1 x 160)

Step 2: (a) Count the total number of values in the dataset#a# - Number of values = 5 + 3 + 2 + 4 + 1 = 15

(a) Calculate the mean of the dataset#a# Mean = \(\frac{Sum of values}{Total number of values} = \frac{(5 \times 120) + (3 \times 130) + (2 \times 140) + (4 \times 150) + (1 \times 160)}{15}\) The mean for dataset (a) is the result of this calculation.

Step 3: (b) Find the mean using the general formula for the dataset#b# For dataset (b), we need to find the mean as a function of \(x_1, x_2, x_3, x_4,\) and \(x_5\). We apply the same process as in (a), but using these general values. Mean = \(\frac{\text{Sum of values}}{\text{Total number of values}} = \frac{(3x_1) + (6x_2) + (7x_3) + (5x_4) +(4x_5)}{\text{Total number of values}}\) Total number of values = 3+6+7+5+4 = 25 Mean = \(\frac{(3x_1) + (6x_2) + (7x_3) + (5x_4) +(4x_5)}{25}\) The mean for dataset (b) is expressed as a function of these variables.

(c) Find the mean using the general formula for the dataset#c# For dataset (c), we need to find the mean as a function of \(x_1, x_2, x_3, x_4, x_5, n_1, n_2, n_3, n_4\) and \(n_5\). We apply the same process as in (a) and (b), but using these general values. Mean = \(\frac{\text{Sum of values}}{\text{Total number of values}} = \frac{(n_1x_1) + (n_2x_2) + (n_3x_3) + (n_4x_4) +(n_5x_5)}{\text{Total number of values}}\) Total number of values = \(n_1+n_2+n_3+n_4+n_5\) Mean = \(\frac{(n_1x_1) + (n_2x_2) + (n_3x_3) + (n_4x_4) +(n_5x_5)}{n_1+n_2+n_3+n_4+n_5}\) The mean for dataset (c) is expressed as a function of these variables.

Key Concepts

Dataset Analysis

Dataset analysis is the process of using data to discover meaningful insights and answer specific questions. When you're given a dataset, it's crucial to understand its composition. This helps in making accurate calculations and interpretations.

In our example, the datasets vary in complexity:

• One dataset involves concrete numbers, where each set of readings has specified values.
• The second dataset uses placeholders, hinting at flexible variables that might change based on context.
• The final set consists of a more generalized approach, using both placeholder values and the number of readings.

Each dataset requires understanding two key attributes:

• The value of individual entries, i.e., what each reading equals.
• The frequency of these readings, or how many times each value appears.

By grasping these elements, you're better positioned to perform further analyses, such as calculating the mean.

Arithmetic Mean

The arithmetic mean, often simply called the mean, is a fundamental concept in data analysis. It represents the central point of a data set by averaging all its numbers. To compute the mean, follow these straightforward steps:

• Add up all the values in your dataset. This sum provides the total value.
• Count the total number of values or entries that exist in your dataset.
• Divide the total sum by the number of values to obtain the mean.

For example, if your dataset consists of readings that are repeated multiple times as in dataset (a) from the exercise, compute the product of each reading with its frequency before summing them up. The mean of such a dataset can be represented as: \[\text{Mean} = \frac{(5 \times 120) + (3 \times 130) + (2 \times 140) + (4 \times 150) + (1 \times 160)}{15}\]By following this method, you can determine the mean of any given dataset, making it a powerful tool for summarizing data and understanding patterns.

Summation Formula

The summation formula is essential when working with multiple readings or large datasets. It simplifies the process of adding a series of numbers. The summation notation (Σ) helps compactly express these calculations.

In practice, to calculate the sum of the dataset's values, we use the formula:\[\sum_{i=1}^{n} x_i \cdot f_i \]Here, \(x_i\) represents each unique reading or value, and \(f_i\) stands for the frequency of that reading. The index \(i\) runs through all unique readings in the dataset from 1 to \(n\). This formula allows one to easily factor in each value's frequency without separately listing every instance. For exercises with variable values, as seen in dataset (b) and (c), the summation formula maintains its importance. It provides a concise way to compute total values: \[\sum (n_{i} \cdot x_{i})\]By using this approach, even large and complex datasets become manageable and clear, ensuring accurate and efficient analysis.