Problem 16 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The standard deviation of a distribution is 30 and each item is raised by 3 , then new S.D. is (A) 32 (B) 28 (C) 27 (D) None of these

Step-by-Step Solution

Answer

The new standard deviation is 30, which means (D) None of these is correct.

Step 1: Understanding Standard Deviation

The standard deviation (SD) is a measure of how spread out the values in a dataset are. When each item in a dataset is increased by a constant, the distribution shifts but the spread (or variability) remains the same.

Step 2: Analyzing the Problem

We know the standard deviation of the original distribution is 30. According to the properties of standard deviation, adding a constant value to each data point does not change the standard deviation.

Step 3: Applying the Concept

Since increasing each item by 3 does not affect the spread or variability of the data, the new standard deviation remains unchanged from the original.

Step 4: Conclusion

The new standard deviation remains 30, as adding a constant to each data point does not change the standard deviation.

Key Concepts

Distribution

A distribution in statistics refers to the way values are spread or arranged within a dataset. It's like a blueprint that shows you where your data points sit across a spectrum. The kind of distribution you have is crucial as it helps you understand the typical behavior of your data.

The most common type of distribution is the normal distribution, often depicted as a bell curve. Here, most data points cluster around a central value, with fewer points appearing as you move away from the center.

Understanding distribution aids in identifying patterns and making predictions. In our exercise, the distribution remains the same, even when each item in the dataset is increased by a constant value. This is because adding a constant shifts all values equally, without altering the spread or shape of the distribution.

Variability

Variability tells us how much the values in a dataset differ from each other. It's essentially about the spread of the data. When we measure variability, we might be interested in finding out how tightly packed or widely spread our data points are.

Two primary measures of variability are standard deviation and variance. The standard deviation provides a clear score of how much individual data points deviate from the mean. In practical terms, it's like measuring how your data behaves relative to an average value.

• If your standard deviation is low, your data points are close to the mean.
• A higher standard deviation indicates data points are spread out widely.

In our original exercise, the variability of the dataset does not change, even when we add a constant value. This is because the standard deviation focuses only on the relative distances between data points, which remain constant if data points are just shifted together.

Dataset

A dataset refers to a collection of data points or values which are used for analysis. It can be as simple as a list of numbers or more complex data structures. Understanding your dataset is vital for statistical analysis as it sets the foundation for investigating data behavior.

Datasets can vary in size, quality, and type, and knowing these attributes helps direct which statistical methods to use.

• Size: How many data points?
• Type: Numerical or categorical data?
• Quality: Are there any errors or outliers?

In the context of our problem, the dataset initially had a standard deviation of 30. Raising each data point by a constant doesn’t alter the dataset's overall variability feature, such as the standard deviation. Rather, it merely shifts the values within the dataset uniformly.