Question

\(\sum_{i=1}^{18}\left(x_{i}-8\right)=9\) and \(\sum_{i=1}^{18}\left(x_{i}-8\right)^{2}=45\), then standard deviation of \(x_{1}, x_{2}, \ldots, x_{18}\) is (1) \(\frac{4}{9}\) (2) \(\frac{9}{4}\) (3) \(\frac{3}{2}\) (4) None of these

Short answer

The standard deviation is \(\frac{\sqrt{10}}{2}\), so the correct answer is (4).

Step-by-step solution

Understand the Given Equations

We are given two summations: \ \(\begin{aligned} \ \text{1) } & \ \ \ \ \ \ \ \ \sum_{i=1}^{18}\big(x_{i}-8\big)=9 \ \text{2) } \ \sum_{i=1}^{18}\big(x_{i}-8\big)^{2}=45 \ \ \ \ \ \ \ \ \end{aligned}\).

Find the Mean

The first equation represents the deviation from 8: \(x_{i} - 8\). Since the mean of \(x_{i} - 8\) equals \(\frac{\sum_{i=1}^{18}(x_{i} - 8)}{18}\), we get \(\frac{9}{18} = 0.5\). The mean of \(x_{i}\), therefore, is \(8 + 0.5 = 8.5\).

Calculate the Variance

Variance is given by \(\sigma^{2} = \frac{\sum_{i=1}^{n}(x_{i} - \bar{x})^2}{n}\). Rewriting it using our deviation from 8, we have: \(\frac{\sum_{i=1}^{18}(x_{i} - 8)^2}{18}\). Using the second equation: \(\frac{45}{18} = 2.5\).

Find the Standard Deviation

Standard deviation, \(\sigma\), is the square root of the variance: \( \sqrt{2.5} = \frac{\sqrt{10}}{2} \approx 1.58 \). Thus, it is \(\frac{\sqrt{10}}{2}\). This corresponds to none of the given options. Therefore, the correct answer is option (4).

Key concepts

Mean Calculation

Understanding the mean is crucial for solving many statistical problems, including calculating standard deviation. The mean, often referred to as the average, is the sum of all data points divided by the number of data points. In this exercise, we were given the sum of the deviations from a number (8).
The equation provided \(\begin{aligned} \sum_{i=1}^{18}\big(x_{i}-8\big)=9 \end{aligned}\) simplifies to \( (x_{1} - 8) + (x_{2} - 8) + ... + (x_{18} - 8) = 9\). This sum reflects how much all data points deviate from the number 8 overall. To find the mean deviation, we divide that summation by the number of terms:
\( \frac{9}{18} = 0.5\).
Hence, the mean of the deviation is 0.5. To find the actual mean of the data set, we add this deviation to 8, obtaining \( 8 + 0.5 = 8.5\).

Variance

Variance measures how spread out the numbers in a data set are. It is the average of the squared differences from the Mean. Using the details provided in the problem, we know the sum of the squared deviations: \(\begin{aligned} \sum_{i=1}^{18}\big(x_{i}-8\big)^{2}=45 \end{aligned}\).
To find the variance, we divide this summation by the number of data points, which is: \( \frac{45}{18} = 2.5\).
Thus, the variance (\( \sigma^{2} \)) is 2.5. Variance is essential for understanding the dispersion within a data set and is a critical step in finding the standard deviation.

Summation

Summation, represented by the symbol \( \sum \), is the process of adding up a series of numbers. It’s a fundamental concept that is heavily utilized in statistics.
In this exercise, we see two types of summations: one for the deviations from the number 8 and another for the squared deviations. These are represented as:
\( \sum_{i=1}^{18}\big(x_{i}-8\big)=9 \) and \( \sum_{i=1}^{18}\big(x_{i}-8\big)^{2}=45 \).
Summation helps in breaking down complex calculations into simpler parts. It helps compile an overview of significant data points, streamlining the mean and variance calculations. To apply summation appropriately, always ensure you understand what the terms within the summation represent and how they relate to the problem at hand.