Key Concepts and Formulas

This problem requires a solid understanding of two fundamental statistical measures: the mean and the standard deviation. These concepts are crucial for analyzing and interpreting datasets.

1. Mean ($\overline{x}$): The mean is a measure of central tendency, representing the average value of a dataset. It is calculated by summing all observations and dividing by the total number of observations. For a set of $N$ observations $x_1, x_2, ..., x_N$, the formula is: $\overline{x} = \frac{\sum_{i=1}^{N} x_i}{N}$ The mean provides a single value that gives us a sense of the "center" or "typical" value of the data.

2. Standard Deviation ($\sigma$): The standard deviation is a measure of dispersion or variability, indicating how much individual data points typically deviate or spread out from the mean. It is the square root of the variance ($\sigma^2$). A smaller standard deviation implies data points are clustered closely around the mean, while a larger standard deviation suggests data points are more spread out. The formula for standard deviation is: $\sigma = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \overline{x})^2}{N}}$ Here, $(x_i - \overline{x})$ represents the deviation of each individual score from the mean. The sum of these squared deviations, divided by $N$, gives us the variance, and its square root is the standard deviation.

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Step-by-Step Solution

Step 1: Understand the Given Information and Identify the Goal

• We are given the marks for five tests: $45, 54, 41, 57, 43$.
• The score for the sixth test is unknown; let's represent it by $x$.
• The total number of tests is $N = 6$.
• The mean score for all six tests is given as $\overline{x} = 48$.
• Our ultimate goal is to calculate the standard deviation ($\sigma$) of the marks in these six tests.

Step 2: Calculate the Unknown Score ($x$) for the Sixth Test

• Why this step? To calculate the standard deviation, we need a complete dataset of all individual scores. Since one score is unknown, our first essential task is to determine its value using the provided mean.

• We use the formula for the mean: $\overline{x} = \frac{\text{Sum of all scores}}{\text{Number of scores}}$

• Substitute the known values into the mean formula: $48 = \frac{45 + 54 + 41 + 57 + 43 + x}{6}$

• First, sum the known scores from the five tests: $45 + 54 + 41 + 57 + 43 = 240$

• Now, substitute this sum back into the mean equation: $48 = \frac{240 + x}{6}$

• To isolate $x$, multiply both sides of the equation by 6: $48 \times 6 = 240 + x$ $288 = 240 + x$

• Finally, solve for $x$ by subtracting 240 from both sides: $x = 288 - 240$ $x = 48$ The score in the sixth test is $48$.

Step 3: List All Scores and the Mean

• Now that we have found the unknown score, we have the complete dataset for the six tests: $x_1 = 45, x_2 = 54, x_3 = 41, x_4 = 57, x_5 = 43, x_6 = 48$
• The mean score is $\overline{x} = 48$.
• The number of observations is $N=6$.

Step 4: Calculate the Deviations from the Mean ($x_i - \overline{x}$)

• Why this step? The standard deviation formula requires us to quantify how much each data point differs from the mean. These differences, known as "deviations," are the fundamental components for measuring dispersion.

• We calculate $(x_i - \overline{x})$ for each score:

  • $x_1 - \overline{x} = 45 - 48 = -3$
  • $x_2 - \overline{x} = 54 - 48 = 6$
  • $x_3 - \overline{x} = 41 - 48 = -7$
  • $x_4 - \overline{x} = 57 - 48 = 9$
  • $x_5 - \overline{x} = 43 - 48 = -5$
  • $x_6 - \overline{x} = 48 - 48 = 0$

Step 5: Calculate the Squared Deviations ($(x_i - \overline{x})^2$)

• Why this step? We square the deviations for two important reasons:

  1. Eliminate Negative Signs: If we simply summed the deviations, positive and negative values would cancel each other out (the sum of deviations from the mean is always zero), making the sum ineffective as a measure of spread. Squaring ensures all values are non-negative.
  2. Emphasize Larger Deviations: Squaring gives proportionally more weight to larger deviations. This means that data points further from the mean contribute more significantly to the overall measure of dispersion.

• We square each deviation calculated in the previous step:

  • $(-3)^2 = 9$
  • $(6)^2 = 36$
  • $(-7)^2 = 49$
  • $(9)^2 = 81$
  • $(-5)^2 = 25$
  • $(0)^2 = 0$

Step 6: Sum the Squared Deviations ($\sum (x_i - \overline{x})^2$)

• Why this step? This sum represents the total squared dispersion across all data points. It forms the numerator of the variance and standard deviation formula, providing a cumulative measure of how much the data points collectively vary from the mean.

• Sum all the squared deviations: $\sum (x_i - \overline{x})^2 = 9 + 36 + 49 + 81 + 25 + 0 = 200$

Step 7: Calculate the Standard Deviation ($\sigma$)

• Why this step? This is the final calculation to arrive at the standard deviation, which is the required answer. We substitute the sum of squared deviations and the number of observations ($N$) into the formula and simplify.

• Using the standard deviation formula: $\sigma = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \overline{x})^2}{N}}$

• Substitute the calculated sum of squared deviations (200) and $N=6$: $\sigma = \sqrt{\frac{200}{6}}$

• Simplify the fraction inside the square root by dividing both the numerator and denominator by their greatest common divisor (2): $\sigma = \sqrt{\frac{100}{3}}$

• Take the square root of the numerator and denominator separately: $\sigma = \frac{\sqrt{100}}{\sqrt{3}}$ $\sigma = \frac{10}{\sqrt{3}}$

The standard deviation of the marks in the six tests is $\frac{10}{\sqrt{3}}$.

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Common Mistakes & Tips

1. Arithmetic Errors: Be extremely careful with all calculations (addition, subtraction, squaring). A single arithmetic mistake in any step will propagate and lead to an incorrect final answer. Always double-check your calculations.
2. Sign Errors in Squaring: Remember that squaring any real number (positive or negative) always results in a positive number. For example, $(-7)^2 = 49$, not $-49$. This is a very common error.
3. Variance vs. Standard Deviation: The quantity $\frac{\sum (x_i - \overline{x})^2}{N}$ is the variance ($\sigma^2$). Don't forget the final crucial step of taking the square root to obtain the standard deviation ($\sigma$).
4. Missing Data Points: Ensure you have all data points before proceeding to calculate dispersion measures. If any are missing, as in this problem, use other given information (like the mean) to find them first.

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Summary

This problem effectively tests your ability to apply concepts of both central tendency (mean) and dispersion (standard deviation). We began by utilizing the definition of the mean to determine the missing score for the sixth test, a critical initial step to complete our dataset. With all six scores identified, we then systematically calculated the standard deviation: first, by finding the deviation of each score from the mean; second, by squaring these deviations to eliminate negative values and give more weight to larger spreads; third, by summing these squared deviations; and finally, by taking the square root of the average squared deviation (variance). The calculated standard deviation, $\frac{10}{\sqrt{3}}$, quantifies how much the students' scores typically vary around their average score of 48.

The final answer is $\boxed{\frac{10}{\sqrt{3}}}$, which corresponds to option (B).