Question

For a data there are 3n observations in which first \(n\) observations are \(a-d\), second n observation are a and last n observations are \(a+d\) and there variance is \((4 / 3)\) then \(|\mathrm{d}|=\) (a) 1 (b) \(\sqrt{2}\) (c) \(\sqrt{(2 / 3)}\) (d) \(\sqrt{(3 / 2)}\)

Short answer

\( |d| = \sqrt{\frac{4}{3}} \) Solution: (d) \(\sqrt{(3 / 2)}\).

Step-by-step solution

In order to work with variance, we need to know its formula. The formula for variance is defined as: \[ \text{Variance} = \frac{\sum_{i=1}^n (x_i - \mu)^2}{n} \] where n is the number of observations, \(x_i\) is each observation, and \(\mu\) is the mean of the set. #Step 2: Calculate The Mean#

We need the mean of the dataset to use in the variance formula. The dataset contains 3n observations: n as \(a-d\), n as \(a\), and n as \(a+d\). So, we can write the mean as follows: \[ \mu = \frac{\sum_{i=1}^n (a-d) + \sum_{i=1}^n a + \sum_{i=1}^n (a+d)}{3n} \] Calculating the sums and simplifying the expression, we get: \[ \mu = \frac{3an}{3n} = a \] #Step 3: Apply The Variance Formula#

Now that we have the mean value, we can apply the variance formula on the dataset to get an expression for the variance: \[ \text{Variance} = \frac{\sum_{i=1}^n [(a-d)-a]^2 + \sum_{i=1}^n [a-a]^2 + \sum_{i=1}^n [(a+d)-a]^2}{3n} \] Substituting the given value of variance, \((4/3)\), and simplifying the expression, we get: \[ \frac{4}{3} = \frac{\sum_{i=1}^n d^2 + 0 + \sum_{i=1}^n d^2}{3n} \] #Step 4: Solve For d#

Next, we solve for d using the expression from Step 3: \[ \frac{4}{3} = \frac{2nd^2}{3n} \] Canceling out terms and simplifying, we get: \[ d^2 = \frac{4}{3} \] Now, we take the square root on both sides to find the absolute value of d: \[ |d| = \sqrt{\frac{4}{3}} \] Comparing the result with the given options, we find that our answer matches the option (d). Solution: (d) \(\sqrt{(3 / 2)}\).

Key concepts

Variance

Variance is a crucial statistical measure that tells us how much the data is spread out around the mean. It shows the level of variability or diversity in a data set. The formula for variance is:\[\text{Variance} = \frac{\sum_{i=1}^n (x_i - \mu)^2}{n}\]Here, \(n\) is the number of observations, \(x_i\) are the data points, and \(\mu\) is the mean of the data set.

Let's break it down:

• First, subtract the mean from each observation to find the deviation.
• Square these deviations to ensure they are positive and amplify larger deviations.
• Find the average of these squared deviations.

In this exercise, the variance is given as \(\frac{4}{3}\). This value is important because it provides insight into how the data points deviate from the mean. By setting this given variance equal to the calculated variance, we solve for \(|d|\).
Understanding variance helps you grasp how much the observations differ from the typical value and is fundamental in comparing data sets.

Mean

The mean is one of the most basic but essential concepts in statistics. It calculates the average of a set of numbers, providing a central value that represents the data.
It's calculated by summing all the data points and dividing by the number of observations:\[\mu = \frac{\sum_{i=1}^n x_i}{n}\]In our exercise, we had 3n observations, divided into three groups: \(n\) observations for each of \(a-d\), \(a\), and \(a+d\).

Steps to find the mean:

• Compute the total sum of each group's observations.
• Add these sums together to find the complete sum of all observations.
• Divide by the total number of observations \(3n\) to get the mean.

The exercise shows the mean \(\mu\) as \(a\) since \[\mu = \frac{\sum_{i=1}^n (a-d) + \sum_{i=1}^n a + \sum_{i=1}^n (a+d)}{3n} = a\]Understanding the mean helps us establish a reference point for variance calculations, offering a clear picture of where data points lie relative to this central value.

Observations

Observations are the individual data points collected in an experiment or study. They form the backbone of statistical analysis and provide insight into the overall trends and patterns in the data.

In our exercise, we looked at a scenario where there are 3n observations. These were broken into:

• \(n\) observations of \(a-d\)
• \(n\) observations of \(a\)
• \(n\) observations of \(a+d\)

Having observations divided into these groups allowed us to compute both the mean and variance effectively.
Understanding observations involves recognizing:

• The role each observation plays in influencing the mean and variance.
• The significance of each observation in representing the data set.

Each observation contributes to the overall pattern and helps calculate descriptive statistics like mean and variance, which are crucial for data interpretation. In statistical exercises like this one, clear understanding and correct use of observations are pivotal for accurate calculations.