Question

If the S.D. of \(\mathrm{y}_{1}, \mathrm{y}_{2}, \mathrm{y}_{3} \ldots \ldots \ldots, \mathrm{y}_{\mathrm{n}}\) is 6, then variance of \(\left(\mathrm{y}_{1}-3\right),\left(\mathrm{y}_{2}-3\right),\left(\mathrm{y}_{3}-3\right)\), \(\ldots\left(\mathrm{y}_{\mathrm{n}}-3\right)\) is (1) 6 (2) 36 (3) 3 (4) 27

Short answer

Answer: The variance of the adjusted data points is 36.

Step-by-step solution

Define given variables

The given standard deviation (S.D.) of \(\mathrm{y}_{1}, \mathrm{y}_{2}, \mathrm{y}_{3} \ldots, \mathrm{y}_{\mathrm{n}}\) is 6.

Relationship between variance and standard deviation

Recall the relationship between variance and standard deviation: Variance = (Standard Deviation)^2 In this case, the variance of \(\mathrm{y}_{1}, \mathrm{y}_{2}, \mathrm{y}_{3} \ldots, \mathrm{y}_{\mathrm{n}}\) is \((6)^2 = 36\).

Determine the effect of subtracting a constant on variance

When you add or subtract a constant value (in this case, 3) from each data point, it affects the mean, but does not affect the variance. The variance remains the same for the adjusted data points. So the variance of the adjusted data points, \(\left(\mathrm{y}_{1}-3\right),\left(\mathrm{y}_{2}-3\right),\left(\mathrm{y}_{3}-3\right), \ldots\left(\mathrm{y}_{\mathrm{n}}-3\right)\), is still 36. Thus, the correct answer is: (2) 36

Key concepts

Understanding Standard Deviation

Standard deviation is a measure that quantifies the amount of variation or dispersion in a set of data values. If the data points are all close together, the standard deviation will be low, but if they are spread out, the standard deviation will be higher.
Think of it like this: If you have a class of students and their heights, a high standard deviation in their heights means that some students are very tall and some are very short. In contrast, a low standard deviation means they are all around the same height.
In mathematical terms, the standard deviation is the square root of the variance. It gives us an insight into how much the data set deviates from the average. The formula for standard deviation \( \sigma \) is \[\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N}(x_i - \mu)^2}\]Where:

• \(\sigma\) is the standard deviation
• \(x_i\) is each data point
• \(\mu\) is the mean of the data points
• \(N\) is the number of data points

Understanding standard deviation helps in assessing variability and volatility of data, which is crucial in statistics and data analysis. It provides context to the mean and offers insights into the consistency of different data sets.

Exploring Variance

Variance is closely related to standard deviation and is a measure of how much the data points in a data set differ from the mean. It is the average of the squared differences from the mean, indicating the dispersion within the data.
For instance, if you have the heights of students, the variance will give you an idea of how much these heights differ from the average student height within that group.
The mathematical expression for variance \( \sigma^2 \) is:\[\sigma^2 = \frac{1}{N} \sum_{i=1}^{N}(x_i - \mu)^2\]Where:

• \(\sigma^2\) is the variance
• \(x_i\) is each data point
• \(\mu\) is the mean of the data points
• \(N\) is the number of data points

Variance plays an important role in indicating how much spread or variability there is in a dataset. The larger the variance, the greater the data spread. Conversely, a smaller variance denotes that the data points tend to be closer to the mean.

Effects of Data Transformation

Data transformation refers to the process of changing the scale or distribution of a data set. It is an essential technique used in statistics to adjust the values of data elements without changing their fundamental essence.
In simpler terms, it's like shifting or scaling the entire dataset to get a new perspective or to fit within a specific framework while maintaining the same general shape or structure of the data.
One interesting aspect of data transformation is how adding or subtracting a constant affects statistical measures:

• Adding or subtracting a constant influences the mean of the data set, but leaves the variance and standard deviation unchanged.
• This is because variance and standard deviation are measures of dispersion, which rely on how data points differ from each other, rather than their absolute positions.

This means that if you subtract or add a constant to every point in the data set, the spread remains the same. Understanding the impact of such transformations can be crucial, particularly when normalizing or comparing different data sets.