To determine the correct assertions, we need to understand how the mean and standard deviation of a list of numerical entries change when certain transformations are applied. We will analyze the impact of adding a constant to each entry and doubling each entry in a list. Let's assume we have a list of n entries denoted as $x_1, x_2, \dots, x_n$.

Mean of the List

The mean, often denoted as $\bar{x}$, is a measure of central tendency. It is calculated by summing all the entries and dividing by the number of entries:

$$\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$$

Standard Deviation of the List

The standard deviation, often denoted as $\sigma_x$, is a measure of the spread or dispersion of the entries around the mean. It quantifies how much the entries typically deviate from the mean:

$$\sigma_x = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}}$$

Now, let's evaluate each assertion based on these definitions.

Analyzing Assertion P: Adding to Mean

Assertion P: Adding 7 to each entry in a list adds 7 to the mean of the list.

• Let's consider a new list where each original entry $x_i$ has 7 added to it, so the new entries are $y_i = x_i + 7$.
• The new mean, $\bar{y}$, is calculated as:
• $$\bar{y} = \frac{\sum_{i=1}^{n} (x_i + 7)}{n}$$
• Using the property of summation that $\sum(a+b) = \sum a + \sum b$:
• $$\bar{y} = \frac{\sum_{i=1}^{n} x_i + \sum_{i=1}^{n} 7}{n}$$
• Since $\sum_{i=1}^{n} 7$ means adding 7 for $n$ times, it equals $7n$:
• $$\bar{y} = \frac{\sum_{i=1}^{n} x_i + 7n}{n}$$
• We can separate this into two fractions:
• $$\bar{y} = \frac{\sum_{i=1}^{n} x_i}{n} + \frac{7n}{n}$$
• We know that $\frac{\sum_{i=1}^{n} x_i}{n}$ is the original mean $\bar{x}$:
• $$\bar{y} = \bar{x} + 7$$
• This derivation shows that if you add a constant to every entry in a list, the mean of the list increases by that same constant.

Therefore, Assertion P is CORRECT.

Analyzing Assertion Q: Adding to Standard Deviation

Assertion Q: Adding 7 to each entry in a list adds 7 to the standard deviation of the list.

• Consider the new entries $y_i = x_i + 7$, with the new mean $\bar{y} = \bar{x} + 7$.
• The deviation of each new entry from the new mean is $(y_i - \bar{y})$. Let's substitute the expressions:
• $$y_i - \bar{y} = (x_i + 7) - (\bar{x} + 7)$$
• $$y_i - \bar{y} = x_i + 7 - \bar{x} - 7$$
• $$y_i - \bar{y} = x_i - \bar{x}$$
• This means the difference between each data point and its mean remains unchanged when a constant is added to all entries. The entire distribution just shifts.
• The new standard deviation, $\sigma_y$, is given by:
• $$\sigma_y = \sqrt{\frac{\sum_{i=1}^{n} (y_i - \bar{y})^2}{n}}$$
• Substituting $(y_i - \bar{y}) = (x_i - \bar{x})$:
• $$\sigma_y = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}}$$
• This is precisely the formula for the original standard deviation, $\sigma_x$.
• So, $\sigma_y = \sigma_x$. Adding a constant to each entry in a list does not change its standard deviation.

Therefore, Assertion Q is INCORRECT.

Analyzing Assertion R: Doubling the Mean

Assertion R: Doubling each entry in a list doubles the mean of the list.

• Let's consider a new list where each original entry $x_i$ is doubled, so the new entries are $z_i = 2x_i$.
• The new mean, $\bar{z}$, is calculated as:
• $$\bar{z} = \frac{\sum_{i=1}^{n} (2x_i)}{n}$$
• Using the property of summation that a constant factor can be pulled out:
• $$\bar{z} = \frac{2 \sum_{i=1}^{n} x_i}{n}$$
• We know that $\frac{\sum_{i=1}^{n} x_i}{n}$ is the original mean $\bar{x}$:
• $$\bar{z} = 2\bar{x}$$
• This shows that if you multiply every entry in a list by a constant, the mean of the list is also multiplied by that same constant.

Therefore, Assertion R is CORRECT.

Analyzing Assertion S: Doubling Standard Deviation

Assertion S: Doubling each entry in a list leaves the standard deviation of the list unchanged.

• Consider the new entries $z_i = 2x_i$, with the new mean $\bar{z} = 2\bar{x}$.
• The deviation of each new entry from the new mean is $(z_i - \bar{z})$. Let's substitute the expressions:
• $$z_i - \bar{z} = (2x_i) - (2\bar{x})$$
• $$z_i - \bar{z} = 2(x_i - \bar{x})$$
• The new standard deviation, $\sigma_z$, is given by:
• $$\sigma_z = \sqrt{\frac{\sum_{i=1}^{n} (z_i - \bar{z})^2}{n}}$$
• Substituting $(z_i - \bar{z}) = 2(x_i - \bar{x})$:
• $$\sigma_z = \sqrt{\frac{\sum_{i=1}^{n} (2(x_i - \bar{x}))^2}{n}}$$
• $$ \sigma_z = \sqrt{\frac{\sum_{i=1}^{n} 4(x_i - \bar{x})^2}{n}}$$
• Pulling the constant 4 out of the summation:
• $$ \sigma_z = \sqrt{4 \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}}$$
• Recognizing the term under the square root as $\sigma_x^2$:
• $$ \sigma_z = \sqrt{4 \sigma_x^2}$$
• $$ \sigma_z = 2\sigma_x$$
• This shows that if you multiply every entry in a list by a constant, the standard deviation is multiplied by the absolute value of that constant (in this case, 2). It does not remain unchanged.

Therefore, Assertion S is INCORRECT.

Summary of Correct Assertions

Based on our detailed analysis of each assertion:

• Assertion P is CORRECT: Adding a constant to each entry adds the same constant to the mean.
• Assertion Q is INCORRECT: Adding a constant to each entry does not change the standard deviation.
• Assertion R is CORRECT: Doubling each entry doubles the mean.
• Assertion S is INCORRECT: Doubling each entry doubles the standard deviation.

The assertions that are correct are P and R.

This table summarizes how basic linear transformations affect these fundamental statistical measures. Assertion P and Assertion R are consistent with these principles.