Equality of which of the following quandtities in two data sets of the same size will ensure equality of their standard deviations?

Correct option is D

Solution:

The standard deviation of a data set is influenced by both the mean and the deviations from the mean, specifically how the values are spread around it. Standard deviation is computed using the variance, which takes into account the average of the squared differences between each data point and the mean.

Formula for Standard Deviation:

The standard deviation (σ) is calculated as:

​$$\text{Standard Deviation:} \quad \sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2}$$​​

where:

• xix_ixi​ are the individual data points,
• $\mu$ is the mean of the data set,
• $n$ is the number of data points.

The variance is the average of the squared differences from the mean.

Step 1: Understanding the Components

• The mean of a data set affects the squared deviations (the distance of data points from the mean).
• The average of the squares of all terms tells us about the spread or dispersion of the data points around the mean.

Thus, both the mean and the average of the squares of all terms (which is related to the variance) are needed to determine the standard deviation.

Step 2: Analyzing the Options

• Option (a): Their means
  If the means are equal, it does not guarantee that the standard deviations are equal. The spread of data could still be different.
  Incorrect.
• Option (b): The sums of positive and negative deviations from the respective means
  The sum of the deviations from the mean is always zero, so this does not affect the standard deviation.
  Incorrect.
• Option (c): The averages of squares of all terms
  The average of the squares of the terms is directly related to the variance. However, the mean is also a crucial factor in calculating the standard deviation.
  Incorrect by itself.
• Option (d): The averages of squares of all terms and their means
  This is correct because both the mean and the average of the squares of the terms (which is related to the variance) are necessary to determine the standard deviation. If these two quantities are equal between two data sets, their standard deviations will be equal.
  Correct.

Final Answer:

(d) The averages of squares of all terms and their means.