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Question: The AM of n observations is\[\overline x \] . If the sum of \[n - 4\;\]observations is K, then the m...

The AM of n observations isx‾\overline x . If the sum of n−4  n - 4\;observations is K, then the mean of the remaining 4 observations is
A. x‾−K4\dfrac{{\overline x - K}}{4}
B. nx‾−Kn−4\dfrac{{n\overline x - K}}{{n - 4}}
C. nx‾−K4\dfrac{{n\overline x - K}}{4}
D. nx‾−(n−4)K4\dfrac{{n\overline x - (n - 4)K}}{4}

Explanation

Solution

We are given arithmetic mean of some observations and also the sum of some of the numbers. To find the solution we will first consider the numbers as x1,x2,x3,...................xn{x_1},{x_2},{x_3},...................{x_n} and put them into values of K and x‾\overline x . Then we would exclude the last 4 terms to get their sum and also the average.

Complete step by step answer:

We will start by saying,
Let n observationsbe, x1,x2,x3,...................xn{x_1},{x_2},{x_3},...................{x_n}
Now, we have their arithmetic mean as, x‾\overline x
So,
x1+x2+x3+...................+xnn=x‾\dfrac{{{x_1} + {x_2} + {x_3} + ................... + {x_n}}}{n} = \overline x ........eq(1)
And also given, the sum of n−4  n - 4\; observations is K.
Then we also get,
x1+x2+x3+...................+xn−4=K{x_1} + {x_2} + {x_3} + ................... + {x_{n - 4}} = K.........eq(2)
Equation 1 can be written as,
x1+x2+x3+...................+xn−4+xn−1+xn−2+xn−3+xnn=x‾\dfrac{{{x_1} + {x_2} + {x_3} + ................... + {x_{n - 4}} + {x_{n - 1}} + {x_{n - 2}} + {x_{n - 3}} + {x_n}}}{n} = \overline x
On substituting the value of equation (2) we get,
⇒K+xn−1+xn−2+xn−3+xnn=x‾\Rightarrow\dfrac{{K + {x_{n - 1}} + {x_{n - 2}} + {x_{n - 3}} + {x_n}}}{n} = \overline x
On Simplifying we get,
⇒K+xn−1+xn−2+xn−3+xn=nx‾\Rightarrow K + {x_{n - 1}} + {x_{n - 2}} + {x_{n - 3}} + {x_n} = n\overline x
On subtracting K from both sides we get,
⇒xn−1+xn−2+xn−3+xn=nx‾−K\Rightarrow {x_{n - 1}} + {x_{n - 2}} + {x_{n - 3}} + {x_n} = n\overline x - K
Hence, the mean of last four observation is,
xn−1+xn−2+xn−3+xn4=nx‾−K4\dfrac{{{x_{n - 1}} + {x_{n - 2}} + {x_{n - 3}} + {x_n}}}{4} = \dfrac{{n\overline x - K}}{4}
Hence, option (c) is correct.

Note: The arithmetic mean gives us the idea of simply the mean or the average (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results of an experiment or an observational study, or frequently a set of results from a survey.