Key Concepts and Formulas

• Standard Deviation ($\sigma$) and Variance ($\sigma^2$): The standard deviation is a measure of data dispersion, and its square is the variance. The relationship is fundamental: $\sigma^2 = (\text{Standard Deviation})^2$.
• Mean ($\mu$): For a set of $N$ observations $x_1, x_2, \dots, x_N$, the mean is the sum of all observations divided by the number of observations: $\mu = \frac{\sum x_i}{N}$
• Computational Formula for Variance: For efficiency in calculations, especially in problems involving unknown variables, the variance can be computed using the formula: $\sigma^2 = \frac{\sum x_i^2}{N} - \left(\frac{\sum x_i}{N}\right)^2$ Alternatively, using the mean $\mu$, this formula is often written as: $\sigma^2 = \frac{\sum x_i^2}{N} - \mu^2$ This formula helps avoid calculating individual deviations $(x_i - \mu)$, which can be cumbersome.

Step-by-Step Solution

We are given a set of numbers $\{-1, 0, 1, k\}$ and their standard deviation $\sigma = \sqrt{5}$, with the condition $k > 0$. Our goal is to find the value of $k$.

Step 1: Calculate the Variance ($\sigma^2$) from the Given Standard Deviation

• Why this step? The most convenient formula for variance directly uses $\sigma^2$. Since we are given the standard deviation ($\sigma$), the first logical step is to square it to obtain the variance, which can then be directly substituted into our chosen formula. Given standard deviation, $\sigma = \sqrt{5}$. $\text{Variance } (\sigma^2) = (\text{Standard Deviation})^2$ $\sigma^2 = (\sqrt{5})^2 = 5$ So, the variance of the given numbers is $5$.

Step 2: Calculate the Mean ($\mu$) of the Data Set in terms of $k$

• Why this step? The variance formula $\sigma^2 = \frac{\sum x_i^2}{N} - \mu^2$ requires the mean ($\mu$) of the data set. We need to express this mean algebraically in terms of $k$. The data set is $\{-1, 0, 1, k\}$. The number of observations, $N$, is $4$. First, find the sum of all observations, $\sum x_i$: $\sum x_i = -1 + 0 + 1 + k = k$ Now, calculate the mean ($\mu$): $\mu = \frac{\sum x_i}{N} = \frac{k}{4}$

Step 3: Calculate the Sum of Squares ($\sum x_i^2$) of the Data Set in terms of $k$}

• Why this step? The variance formula also requires the term $\sum x_i^2$, which is the sum of the squares of each individual data point. We need to express this sum algebraically in terms of $k$. Calculate the square of each observation and then sum them: $\sum x_i^2 = (-1)^2 + (0)^2 + (1)^2 + (k)^2$ $\sum x_i^2 = 1 + 0 + 1 + k^2$ $\sum x_i^2 = 2 + k^2$

Step 4: Formulate the Equation using the Variance Formula

• Why this step? This is the crucial step where we combine all the components we calculated into a single algebraic equation. By substituting the values of $\sigma^2$, $\mu$, $\sum x_i^2$, and $N$ into the variance formula, we create an equation that can be solved to find $k$. Substitute the calculated values into the variance formula $\sigma^2 = \frac{\sum x_i^2}{N} - \mu^2$:
• $\sigma^2 = 5$
• $\mu = \frac{k}{4}$
• $\sum x_i^2 = 2 + k^2$
• $N = 4$ The equation becomes: $5 = \frac{2 + k^2}{4} - \left(\frac{k}{4}\right)^2$

Step 5: Solve the Algebraic Equation for $k$

• Why this step? We now have an algebraic equation with $k$ as the only unknown. Our goal is to isolate $k$ by simplifying and solving this equation. This will involve basic algebraic manipulations, leading to a quadratic equation. Simplify the equation: $5 = \frac{2 + k^2}{4} - \frac{k^2}{16}$ To combine the terms on the right-hand side, find a common denominator, which is $16$. Multiply the first fraction by $\frac{4}{4}$: $5 = \frac{4(2 + k^2)}{16} - \frac{k^2}{16}$ $5 = \frac{8 + 4k^2 - k^2}{16}$ Combine the $k^2$ terms in the numerator: $5 = \frac{8 + 3k^2}{16}$ Multiply both sides by $16$ to eliminate the denominator: $5 \times 16 = 8 + 3k^2$ $80 = 8 + 3k^2$ Subtract $8$ from both sides: $80 - 8 = 3k^2$ $72 = 3k^2$ Divide both sides by $3$: $k^2 = \frac{72}{3}$ $k^2 = 24$ Take the square root of both sides to find $k$: $k = \pm\sqrt{24}$ Simplify the radical $\sqrt{24}$: $\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}$ So, we have two possible values for $k$: $k = 2\sqrt{6} \quad \text{or} \quad k = -2\sqrt{6}$

Step 6: Apply the Given Condition ($k > 0$)

• Why this step? The problem statement explicitly provides the condition $k > 0$. This condition is crucial for selecting the correct, unique value of $k$ from the two mathematical solutions obtained in the previous step. Without this condition, both values would be mathematically valid. From Step 5, we found $k = 2\sqrt{6}$ or $k = -2\sqrt{6}$. Since the problem states that $k > 0$, we must choose the positive value. Therefore, $k = 2\sqrt{6}$.

Common Mistakes & Tips

• Algebraic Precision: Be extremely careful with algebraic manipulations, especially when squaring terms, finding common denominators, and simplifying expressions. A small error can lead to an incorrect final answer.
• Formula Selection: Always opt for the computational variance formula ($\sigma^2 = \frac{\sum x_i^2}{N} - \mu^2$) when dealing with unknown variables, as it significantly reduces the chances of errors compared to the definitional formula involving $(x_i - \mu)^2$.
• Utilize All Conditions: Do not forget to apply all given conditions, such as $k > 0$, at the final stage of solving. These conditions are typically provided to help narrow down multiple mathematical solutions to a unique physical or problem-specific answer.
• Radical Simplification: Always simplify square roots completely. For example, $\sqrt{24}$ should be expressed as $2\sqrt{6}$ to match standard answer formats and options.

Summary

This problem required us to find an unknown value $k$ in a data set given its standard deviation. We systematically approached this by first converting the standard deviation to variance. Then, we calculated the mean and the sum of squares of the data points, both expressed in terms of $k$. These components were substituted into the computational formula for variance, leading to an algebraic equation. Solving this equation yielded two possible values for $k$. Finally, applying the given condition $k > 0$ allowed us to select the unique correct value.

The final answer is $\boxed{2\sqrt{6}}$, which corresponds to option (A).