Problem 55 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\bar{X}\) and M.D. be the mean and the mean deviation about \(\overline{\mathrm{X}}\) of n observations \(\mathrm{x}_{\mathrm{i}}, \mathrm{i}=1,2, \ldots \ldots, \mathrm{n}\). If each of the observations is increased by 5 , then the new mean and the mean deviation about the new mean, respectively, are: [Online April 12, 2014] (a) \(\overline{\mathrm{X}}, \mathrm{M} . \mathrm{D}\). (b) \(\overline{\mathrm{X}}+5, \mathrm{M} . \mathrm{D}\). (c) \(\overline{\mathrm{X}}, \mathrm{M} . \mathrm{D} .+5\) (d) \(\overline{\mathrm{X}}+5, \mathrm{M} . \mathrm{D} .+5\)

Step-by-Step Solution

Answer

The answer is (b) \( \overline{\mathrm{X}}+5, \mathrm{M.D.} \).

Step 1: Understanding the Problem

We have the mean \( \bar{X} \) and the mean deviation \( M.D. \) of a set of observations. We need to find how these statistics change when each observation \( x_i \) is increased by 5.

Step 2: Calculating the New Mean

When each observation is increased by 5, the new observations become \( x_i + 5 \). The new mean is calculated by adding 5 to the original mean: \( \bar{X}_{new} = \bar{X} + 5 \).

Step 3: Calculating Mean Deviation with Shifted Observations

The mean deviation about the mean is the average of the absolute differences between each observation and the mean. Assuming the original mean deviation is \( M.D. = \frac{1}{n} \sum |x_i - \bar{X}| \). For the new observations \( x_i + 5 \) with mean \( \bar{X}_{new} = \bar{X} + 5 \), the mean deviation becomes \( M.D_{new} = \frac{1}{n} \sum |(x_i + 5) - (\bar{X} + 5)| = \frac{1}{n} \sum |x_i - \bar{X}| = M.D. \). Hence, the mean deviation remains the same.

Step 4: Conclusion and Choice

The new mean is \( \bar{X} + 5 \) and the new mean deviation is still \( M.D. \). Thus, the correct answer is option (b) \( \overline{\mathrm{X}}+5, \mathrm{M.D.} \).

Key Concepts

Mean Deviation

Mean deviation is a statistical measure that helps us understand how spread out the values are in a data set from its mean. It reflects the average distance between each data point and the mean. This measure gives insight into the variability within a set of data.

• The mean deviation is calculated by taking the absolute differences between each observation and the mean.
• Then, these absolute differences are averaged out to get the mean deviation.

The formula for mean deviation is given by: \[ M.D. = \frac{1}{n} \sum |x_i - \bar{X}| \] where:

• \( M.D. \) is the mean deviation.
• \( x_i \) represents each observation in the data set.
• \( \bar{X} \) is the mean of the data set.
• \( n \) is the total number of observations.

In our problem, the mean deviation does not change even if each data point is increased by the same value. This is because the relative distances between each observation and the mean stay consistent.

Mean Calculation

The mean, or average, is one of the most basic yet powerful descriptive statistics. It represents the central value of a data set. To calculate the mean, you sum up all the observations and divide by the number of observations, which gives you an idea of the "typical" value.
The formula for mean is:\[ \bar{X} = \frac{1}{n} \sum x_i \] where:

• \( \bar{X} \) is the mean.
• \( x_i \) are the individual data points.
• \( n \) is the number of data points.

In the context of the provided exercise, when all observations are increased by 5, the effect is directly mirrored on the mean, making it increase by the same amount. Thus, the new mean can be calculated as:\[ \bar{X}_{new} = \bar{X} + 5 \] This reflects how shifts in the data set impact the mean directly, showing the transformation in the data distribution.

Statistical Deviation

Statistical deviation is a term that refers to how much individual data points differ from a central measure, typically the mean. It helps to characterize the spread within your data set.
Mean deviation is one form of statistical deviation, focusing on the average distance of data points from the mean. Unlike variance or standard deviation, which square the differences (focusing more on larger deviations), mean deviation uses absolute values to maintain linearity without emphasizing outliers. This makes it a straightforward and intuitive measure of spread.
Understanding statistical deviation is crucial for analyzing the consistency and dispersion of data, which can influence decisions and predictions in data science and analytics.

Transformation of Data Sets

Transforming data sets involves applying certain operations or modifications to all components of your data. This can be adding, subtracting, multiplying, or dividing each element by a constant, which shifts the data without changing its inherent structure.
In the given exercise, each observation in the set is increased by 5. This type of transformation is an example of a linear shift. It affects the mean by the same amount (plus 5), but it does not affect the mean deviation or the statistical deviation, since these rely on the relative distances between data points and need the underlying distribution shape to change.
Understanding how transformations affect statistical metrics allows one to better manipulate the data for analysis and insight. It also simplifies complex calculations by reducing the impact of variations that do not provide meaningful information about relative spread.