Problem 20 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(\sum_{i=1}^{n}\left(x_{i}-a\right)=n\) and \(\sum_{i=1}^{n}\left(x_{i}-a\right)^{2}=n a,(n, a>1)\), then the standard deviation of \(n\) observations \(x_{1}, x_{2}, \ldots, x_{n}\) is: [Sep. 06, 2020 (I)] (a) \(a-1\) (b) \(n \sqrt{a-1}\) (c) \(\sqrt{n(a-1)}\) (d) \(\sqrt{a-1}\)

Step-by-Step Solution

Answer

The standard deviation is \( \sqrt{a-1} \) which is option (d).

Step 1: Analyze Given Information

We are given two equations: 1. \( \sum_{i=1}^{n}(x_i - a) = n \)2. \( \sum_{i=1}^{n} (x_i - a)^2 = n \cdot a \) These will help us find the standard deviation of the observations \( x_1, x_2, \ldots, x_n \).

Step 2: Relate First Equation to Mean

The first equation \( \sum_{i=1}^{n}(x_i - a) = n \) can be rewritten to find the mean \( \bar{x} \):\( \sum_{i=1}^{n} x_i = na + n \)This implies that the mean \( \bar{x} = a + 1 \).

Step 3: Use Second Equation for Variance

According to the second equation, \( \sum_{i=1}^{n}(x_i - a)^2 = n \cdot a \).To find the variance, we calculate:\[ \text{Variance} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]

Step 4: Simplify Expression for Variance

Now let's expand \( (x_i - \bar{x})^2 \):\( (x_i - (a+1))^2 = (x_i - a)^2 - 2(x_i - a) + 1 \)Thus,\[ \sum_{i=1}^{n} (x_i - (a+1))^2 = \sum_{i=1}^{n} (x_i - a)^2 - 2\sum_{i=1}^{n} (x_i - a) + n \]

Step 5: Full Substitution

Replace the terms using given equations:\[ \sum_{i=1}^{n}(x_i - (a+1))^2 = n a - 2(n) + n = n(a-1) \]Thus, the variance is:\[ \text{Variance} = \frac{n(a-1)}{n} = a-1 \]

Step 6: Determine Standard Deviation

The standard deviation is the square root of the variance:\( \text{Standard Deviation} = \sqrt{a-1} \)

Step 7: Conclusion

Based on the provided options, the standard deviation of the observations is answer (d): \( \sqrt{a-1} \).

Key Concepts

Variance

Variance is a central concept in statistics that measures how much observations in a data set differ from the arithmetic mean. It's basically about understanding the spread or dispersion of a given set of numbers.
The formula for variance is:

• \( \text{Variance} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 \)

This formula calculates the average of the squared differences between each data point \( x_i \) and the mean \( \bar{x} \).
Variance is important because:

• It provides insights into the consistency of data.
• A higher variance implies greater dispersion, whereas a lower variance suggests that data points are closer to the mean.

In the context of our exercise, once we had the variance value of \( a-1 \), finding the standard deviation became straightforward.

Arithmetic Mean

The arithmetic mean, often just called the "mean," is one of the most common measures of central tendency. It gives us a single value representing the central point of a data set.
Simply put, it's the sum of all values divided by the number of values. Mathematically, it's:

• \( \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \)

The mean provides a useful summary about the overall dataset.

• It's straightforward to calculate and easy to understand.
• However, it can be affected by extreme values or outliers, which might skew the results.

In our step-by-step solution, knowing the arithmetic mean \( \bar{x} = a + 1 \) was crucial for calculating the variance, as it allowed us to understand how each data point compared to the average.

Statistical Formulas

Statistical formulas are tools that allow us to quantify and analyze patterns in data. They're essential for making informed conclusions and inferences based on numerical information.
In our exercise, we primarily dealt with formulas linked to variance, mean, and standard deviation:

• **Variance Formula:** Helps measure the spread of the data.
• **Mean Formula:** Provides the average value of the data.
• **Standard Deviation Formula:** \( \text{Standard Deviation} = \sqrt{\text{Variance}} \), gives a sense of the average distance of each data point from the mean.

These formulas are interconnected:

• Understanding the mean is necessary to compute the variance.
• Variance is foundational for calculating the standard deviation, which adds interpretative value by bringing it back to the units of the original data.

Mastery of these statistical formulas equips us with the tools to analyze data effectively, as seen with our example where they led us to pin down the correct standard deviation as \( \sqrt{a-1} \).