Problem 31 - A Complete Resource Book in Mathematics for JEE Main

Question

The arithmetic mean of a set of observation is. If each observation is divided by \(\alpha\) and then is increased by 10 , the means of the new series is (A) \(\frac{\bar{x}}{\alpha}\) (B) \(\frac{\bar{x}+10}{\alpha}\) (C) \(\frac{\bar{x}+10 \alpha}{\alpha}\) (D) \(\alpha \bar{x}+10\)

Step-by-Step Solution

Answer

The correct option is (C) \( \frac{\bar{x} + 10\alpha}{\alpha} \).

Step 1: Understanding the Arithmetic Mean

The arithmetic mean \( \bar{x} \) of a set of observations is given by \( \bar{x} = \frac{x_1 + x_2 + ... + x_n}{n} \). This is the average value of the observations, where \( x_1, x_2, \, ..., \, x_n \) are the individual observations and \( n \) is the number of observations.

Step 2: Apply the Transformation to Observations

Each observation \( x_i \) in the dataset is first divided by \( \alpha \) and then increased by 10. The new value of each \( x_i \) becomes \( \frac{x_i}{\alpha} + 10 \). Thus, the entire set of observations is transformed in this manner.

Step 3: Compute the New Mean

The mean for the new series of observations can be calculated by taking the mean of these transformed observations. Substitute \( \frac{x_i}{\alpha} + 10 \) for each observation in the original mean formula. The new arithmetic mean \( \bar{y} \) is given by: \[ \bar{y} = \frac{1}{n} \left( \frac{x_1}{\alpha} + 10 + \frac{x_2}{\alpha} + 10 + ... + \frac{x_n}{\alpha} + 10 \right) \] This can be simplified to: \[ \bar{y} = \frac{1}{n} \left( \frac{x_1 + x_2 + ... + x_n}{\alpha} + 10n \right) \]

Step 4: Simplify the New Mean Expression

Since \( \frac{x_1 + x_2 + ... + x_n}{n} \) is \( \bar{x} \), the mean of the new series becomes: \[ \bar{y} = \frac{\bar{x}}{\alpha} + 10 \] This new mean represents the transformed arithmetic mean of the data after adjustments.

Step 5: Conclusion and Correct Option

By considering the final expression \( \frac{\bar{x}}{\alpha} + 10 \), we see that option C, \( \frac{\bar{x} + 10\alpha}{\alpha} \), simplifies directly to \( \frac{\bar{x}}{\alpha} + 10 \), therefore, it is the correct answer.

Key Concepts

Transformation of Data

Data transformation refers to the process of changing or modifying data points in a dataset. Such transformations are often performed to achieve more meaningful insights or to adjust the data to a specific scale or format. In this exercise, each observation was divided by a constant \( \alpha \) and then increased by 10. This systematic change in data alters the overall dataset configuration.Here's what happens during such a transformation:

• Each original observation \( x_i \) is modified by first dividing it by \( \alpha \), thereby shrinking it in proportion to \( \alpha \).
• Next, every transformed observation is further increased by adding a constant of 10. This results in a shift of the entire dataset upward by 10 units.

Such operations help in studying the effects of scaling and shifting data, which are common techniques in data analysis and predictive modeling.

Arithmetic Mean Formula

The arithmetic mean, commonly known as the average, is a measure of central tendency. It is calculated by dividing the sum of values in a dataset by the number of observations. Mathematically, the arithmetic mean \( \bar{x} \) is given by:\[ \bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n} \]where \( x_1, x_2, \ldots, x_n \) are individual data points, and \( n \) is the total number of observations.The arithmetic mean is helpful because:

• It provides a simple summary of a large amount of data with a single value.
• It is useful in comparing differences between datasets.
• It helps in detecting trends and patterns over data points.

However, it is important to remember that the arithmetic mean can be affected by extreme values or outliers, which makes it essential to analyze data comprehensively before drawing conclusions.

Mean of Transformed Observations

When observations in a dataset are transformed, as in this exercise, it is essential to know how these changes affect the arithmetic mean. The transformations applied here involve dividing each observation by \( \alpha \) and then adding 10.To find the new mean \( \bar{y} \) of the transformed observations, we adapt the original mean formula:\[ \bar{y} = \frac{1}{n} \left( \frac{x_1}{\alpha} + 10 + \frac{x_2}{\alpha} + 10 + \ldots + \frac{x_n}{\alpha} + 10 \right) \]Here’s how it simplifies:

• Divide the sum of individual observations by \( \alpha \).
• Add \( 10n \) (since each observation contributes a 10 after transformation).
• This results in: \[ \bar{y} = \frac{\bar{x}}{\alpha} + 10 \]

The new mean expresses how the arithmetic mean shifts due to the scaling by \( \alpha \) and the upward adjustment by 10. Understanding such transformations is crucial for correctly interpreting transformed datasets, particularly when predicting or analyzing trends.