Question

Mean of n observations is \(\mathrm{m}\) and sum of \(\mathrm{n}-3\) observations is b then mean of remaining 3 observations is (a) \(n m+b\) (b) \([(\mathrm{nm}-\mathrm{b}) / 3]\) (c) \([(n m+b) / 3]\) (d) \(n m-b\)

Short answer

The mean of the remaining three observations is \(\boxed{[(nm-b)/3]}\).

Step-by-step solution

Find the sum of n observations

First, we need to find the sum of all n observations. To do this, we can use the formula for the mean: mean = (sum of observations) / number of observations We are given that the mean of n observations is m, so we can write this equation as: \( m = \frac{sum\,of\,n\,observations}{n} \) Now, we can solve for the sum of n observations: \( sum\,of\,n\,observations = n\times m \)

Find the sum of the remaining three observations

To find the sum of the remaining three observations, we subtract the sum of the n-3 observations (which is given as b) from the sum of all n observations. So, we have: \( sum\,of\,remaining\,3\,observations = (n\times m) - b \)

Find the mean of the remaining three observations

Now that we have the sum of the remaining three observations, we can find their mean by dividing the sum by the number of remaining observations (which is 3). So, we get: \( mean\,of\,remaining\,3\,observations = \frac{sum\,of\,remaining\,3\,observations}{3} \) Substituting the expression for the sum of the remaining three observations from Step 2, we get: \( mean\,of\,remaining\,3\,observations = \frac{(n\times m) - b}{3} \) So, the mean of the remaining three observations is: \( \boxed{[(nm-b)/3]} \) Hence, the correct answer is (b).

Key concepts

Sum of Observations

In statistics, the **sum of observations** is a crucial step when calculating the mean. The mean is essentially an average, and to find it, you first need the total sum of all the values you're studying. This is done by adding up each individual observation, or data point, in your dataset.

For example, if we have a dataset consisting of several observations and we know their mean and number, the sum of these observations can be calculated using the formula:

• Mean (\( m \)) = (Sum of Observations) / Number of Observations (\( n \)).

Rearranging the formula to solve for the sum of observations gives us:

• Sum of Observations = \( n \times m \).

This method relies heavily on knowing either the mean or the number of observations and allows you to transfer between these two pieces of information with ease. Calculating the sum is fundamental for further statistical analysis, such as comparing subsets of data or calculating new means when the dataset changes.

n Observations

The concept of **n observations** in statistics refers to the total number of data points that you have in a given dataset. This is represented by the letter \( n \) and plays a significant role in calculating the mean or average since it determines how the sum of observations is divided to find the mean.

For example, if a dataset's mean is known to be \( m \), and you know the total number of observations (\( n \)), you can easily find all sorts of valuable statistics like the sum. In many exercises, knowing \( n \) helps in identifying whether a data segment or partition needs further analysis or is sufficient as is.
Being aware of the number of observations is crucial as it can influence the statistical significance of your results. A larger \( n \) generally provides more reliable averages, which can lead to better decision-making or understanding of the dataset's underlying traits.

Remaining Observations

In statistical exercises, often you'll need to examine a subset of your dataset, leading you to consider **remaining observations**. These are data points left after other parts of the data have been taken away or isolated for particular analysis.

For instance, say you are given a sum of \( n-3 \) observations. To find the sum of the \( 3 \) remaining observations, you subtract this part from the total sum calculated from all \( n \) observations:

• Sum of Remaining 3 Observations = \((n \times m) - b \),

where \( b \) is the sum of the \( n-3 \) observations.
Understanding the remaining observations is vital for secondary calculations. You often use it to find new means or test hypotheses about that specific part of your data. Clarity around remaining observations helps ensure the accuracy of smaller, detailed statistical interpretations.