Properties of Standard Deviation: Change of Origin and Scale

Standard deviation (SD) is a fundamental measure used in statistics to quantify the amount of variation or dispersion of a set of values. It tells us how spread out the numbers are from their average value (the mean).

Let's explore how standard deviation behaves when the data undergoes certain transformations:

• Change of Origin: This means adding or subtracting a constant value from every data point. For example, if we have data $x_1, x_2, ..., x_n$ and we transform it into $y_1, y_2, ..., y_n$ where $y_i = x_i + a$ (for some constant 'a'), the mean also shifts by 'a'. However, the difference between each data point and the new mean remains the same as the difference between the original data point and the original mean. Therefore, the standard deviation does not change. We write this as $SD(X+a) = SD(X)$.
• Change of Scale: This involves multiplying or dividing every data point by a constant value. If we transform the data $x_1, x_2, ..., x_n$ into $y_1, y_2, ..., y_n$ where $y_i = b \times x_i$ (for some constant 'b'), the mean also gets multiplied by 'b'. The deviations from the mean are also multiplied by 'b'. Consequently, the standard deviation is multiplied by the absolute value of the constant 'b'. We write this as $SD(bX) = |b| \times SD(X)$.

Considering these points for the given statements:

• Statement A suggests that the standard deviation is unaffected by both change of origin and change of scale. This is only partially true; it's independent of the origin but *not* independent of the scale. Thus, Statement A is incorrect.
• Statement B correctly states that the standard deviation is independent of the change of origin but is affected by a change of scale. This aligns with our analysis. Thus, Statement B is correct.

Calculating Standard Deviation for First n Natural Numbers

The set of the first 'n' natural numbers is $\{1, 2, 3, ..., n\}$. There is a well-established formula for the variance and standard deviation of this sequence.

The variance ($\sigma^2$) of the first 'n' natural numbers is given by:

$ \sigma^2 = \frac{\sum_{i=1}^{n} (i - \mu)^2}{n} = \frac{n^2 - 1}{12} $

where $\mu$ is the mean, $\mu = \frac{n+1}{2}$.

The standard deviation ($\sigma$) is the square root of the variance:

$ \sigma = \sqrt{\sigma^2} = \sqrt{\frac{n^2 - 1}{12}} $

Now let's look at the statements related to this formula:

• Statement C correctly states the formula for the standard deviation of the first 'n' natural numbers as $\sqrt{\frac{n^2-1}{12}}$. Thus, Statement C is correct.
• Statement D provides a different formula, $\sqrt{\frac{n-1}{6}}$, which is not the correct standard deviation for the first 'n' natural numbers. Thus, Statement D is incorrect.

Standard Deviation and Mean Deviation Relationship

Mean Deviation (MD) is another measure of dispersion, calculated as the average of the absolute differences from the mean. The relationship between Standard Deviation (SD) and Mean Deviation (MD) can help describe the shape of a distribution.

For distributions that exhibit moderate skewness (meaning they are slightly asymmetric), there's an approximate empirical relationship often used:

• The Mean Deviation is roughly 80% of the Standard Deviation. This can be written as: $MD \approx 0.8 \times SD$.
• Conversely, this implies that the Standard Deviation is about 1.25 times the Mean Deviation: $SD \approx \frac{1}{0.8} \times MD = 1.25 \times MD$.

Let's analyze Statement E based on this understanding:

• Statement E claims that for moderately skewed distributions, the standard deviation is 0.8 of the mean deviation. Mathematically, this means $SD = 0.8 \times MD$. This is the inverse of the commonly cited empirical rule ($MD \approx 0.8 \times SD$ or $SD \approx 1.25 \times MD$). Therefore, Statement E is incorrect as stated.

Summary of Correct Statements

After evaluating each statement based on the mathematical properties and formulas related to standard deviation:

• Statement A: Incorrect (dependent on scale).
• Statement B: Correct (independent of origin, dependent on scale).
• Statement C: Correct (standard formula for first n natural numbers).
• Statement D: Incorrect (wrong formula).
• Statement E: Incorrect (relationship stated is inverse of the common approximation).

The statements identified as correct are B and C.