Question

The mean of the data set consisting of 16 observations is 16. If one of the observations valued 16 is deleted and three new observations valued 3,4,5 are added to the data, then the mean of the resultant data, is
(a) 16.8
(b) 16.0
(c) 15.8
(d) 14.0

Answer 1

Hint: To solve this question, we will first assume some variable for the sum of the data set and we know that there are 16 observations. We are also given the mean of these observations. We can find the sum of values of observations from the available data. Now, subtract 16 from the total sum of the data set. The total number of observations will also reduce by 1. Then we will add three observations valued 3, 4 and 5 into the data set. The number observations will now increase by 3. Thus, we can find the mean of the new data set.

Complete step-by-step answer:
We know that the mean of a data set is defined as the ratio of the sum of the values of all the values of observations in the data set and the number of data sets.
We are given that the number of observations in our data set is 16 and the mean is also 16.
Now, let the sum of all the observations of the data set be x.
$\begin{array}{rl}& \Rightarrow {\displaystyle \frac{x}{16}}=16\\ & \Rightarrow x=256\end{array}$
Thus, the sum of the values of the observations in the original data set is x = 256.
Now, we will delete an observation valued 16. Thus, we have to subtract 16 from the total sum of values and the number of observations will also decrease by 1.
Thus, the total sum of values after observation valued 16 is deleted is (256 – 16) = 240 and the number of observations is 15.
$\begin{array}{rl}& \Rightarrow mean={\displaystyle \frac{240}{15}}\\ & \Rightarrow mean=16\end{array}$
Now, it is given that 3 observations valued 3, 4 and 5 are added to the data set.
The total sum of values of observations after 3 observations are added (240 + 3 + 4 + 5) = 252 and number of observations is (15 + 3) = 3.
$\begin{array}{rl}& \Rightarrow mean={\displaystyle \frac{252}{18}}\\ & \Rightarrow mean=14\end{array}$
Therefore, the mean of the resultant observations is 14.

So, the correct answer is “Option d”.

Note: In statistics, only mean if affected if the values of the observations change. Median is defined as the central value of the data set and remains the same as long as the number of observations is the same and the central value is not disturbed. Whereas, mode is a value which occurs maximum number of times and hence doesn’t change if any value other than the value which occurs maximum number of times changes.