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

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, so the answer is (D) None of these.

Step 1: Identify Given Information

We know the standard deviation of the original distribution, which is 30. This is the initial important figure to consider.

Step 2: Determine Impact of Adding a Constant

Understand that adding a constant value to each data point in a dataset does not affect the standard deviation of the dataset. This is because standard deviation measures how data points deviate from the mean, not their absolute values.

Step 3: Conclude the New Standard Deviation

Since adding 3 to each item in the data set does not change the standard deviation, the new standard deviation remains the same as the original standard deviation, which is 30.

Key Concepts

Impact of Adding a Constant

When you add a constant to every item in a dataset, you might assume many things could change, but that's not the case with standard deviation. The standard deviation is a measure of how spread out numbers are in a dataset. It looks at the differences from the mean. Adding a constant shifts all the data points up or down, but

• the distances between each point remain unchanged, and
• the spread of the data stays the same.

So, the standard deviation does not change.
If your original data has a standard deviation of 30 and you add 3 to each number, the standard deviation remains at 30. This concept can be understood better by imagining elevating or lowering an entire distribution without stretching or squeezing it. The relative positions between the data points remain identical.

Data Distribution

Data distribution refers to how data points are spread out. It could be concentrated around a central point or more spread out, forming different shapes such as bell curves in normal distributions or skewed distributions. Imagine your dataset as points on a number line.
If you have many points close together around a central number, the distribution is narrow. Points further apart indicate a wider distribution.

• Adding a constant to the data effectively shifts the entire distribution left or right,
• but maintains its original shape and spread.

If the original dataset had a mean, each data point would shift but the distances between them would remain, so the standard deviation stays the same. Understanding this can help us anticipate how distributions behave under transformation without altering their core features.

Statistical Measures

Statistical measures like mean, median, and standard deviation help us understand data in different ways. They provide insights into various aspects of data such as central tendency, variation, and spread.

• Mean: The average of all the data points.
• Median: The middle value in a dataset.
• Standard Deviation: Indicates how much each data point deviates from the average (mean).

When a constant is added to a dataset, the

• mean will increase by the constant amount,
• but the standard deviation doesn't change,
• because it's about the spread, not the center.

Keep in mind that although the position of the data will change with a constant addition, the relative spread remains the same. Exploring these measures allows us to better interpret and predict changes in datasets, making it crucial for statistical analysis.