The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is

Q: Wait, why do we need to calculate the sum? Can't we just adjust the mean directly by adding and subtracting the values?

That is a classic trap! The mean is not an additive property. You cannot simply add or subtract values from the mean itself because the mean is a ratio. You must always revert to the total sum, perform your operations there, and then re-calculate the ratio with the updated count.

Q: Does the order of operations matter when updating the count?

It absolutely does. Always track the count as a separate variable. Start with your initial count, subtract the number of items removed, and add the number of items introduced. In this case, $16 - 1 + 3 = 18$. If you lose track of the count, the entire calculation collapses.

Q: Why did the mean decrease? Is that expected?

Yes, it is perfectly logical. You removed a value of $16$ and replaced it with three values ($3, 4, 5$) whose average is only $4$. Since you replaced a high-value observation with much smaller ones, the overall 'center of mass' of your data must shift downwards. Trust the math!

Question

The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data, is:

Options:

(A) 15.8

(B) 14.0

(C) 16.8

(D) 16.0

Sigma Percentile · Accepted Answer

### Analyzing the Initial State

Imagine you are standing in a room with sixteen people, and each person is holding a certain number of apples. The average number of apples per person is sixteen. In the world of statistics, this is our initial state: $n_1 = 16$ and $\bar{x}_1 = 16$.

It means that if we were to collect all the apples into one giant pile, the total number of apples would be the product of the number of people and the average number of apples each person holds. Using the formula $\bar{x} = \frac{\sum x}{n}$, we can find this total sum:

$$\sum x = \bar{x} 	imes n$$

So, our initial sum is $S_1 = 16 	imes 16 = 256$. Think of this $256$ as the total energy or the total weight of our dataset.

### The Dynamic Shift

Now, let's introduce change. One person, who happens to be holding sixteen apples, decides to leave the room. We must subtract their contribution from our total sum:

$$S_2 = 256 - 16 = 240$$

But we aren't done yet! Three new people enter the room, carrying $3$, $4$, and $5$ apples respectively. We add these to our pile:

$$S_3 = 240 + (3 + 4 + 5) = 240 + 12 = 252$$

We have successfully updated the total weight of our system.

### The Final Equilibrium

Now, we must account for the change in the number of people. We started with $16$, one left, and three arrived. The new count is:

$$n_{new} = 16 - 1 + 3 = 18$$

We now have a new total of $252$ apples distributed among $18$ people. To find the new mean, we simply divide the final sum by the new count:

$$\bar{x}_{new} = \frac{252}{18}$$

Performing this division, we find that the **new mean is $14.0$**.

Notice the elegance of this process. We didn't need to know the individual values of the original sixteen observations; we only needed the aggregate sum. By treating the mean as a balance point, we can navigate any modification to a dataset with absolute precision.

Animated Solution for Mathematics - Statistics: The mean of the data set comprising of 16 observations is 16. If one of the observation valued 16 is deleted and three new observations valued 3, 4 and 5 are added to the data, then the mean of the resultant data, is:

Select Answer:

Visualized Solution

Initial State: $n = 16, \overset{x}{ˉ} = 16$

Logic: Sum of Observations

Calculating Initial Sum

Modification 1: Deleting an Observation

Modification 2: Adding Three Observations

Calculating New Count

New Mean Setup

Final Result: Mean $= 14.0$

The Sigma Insight: Measures of Central Tendency (Mean, Median, Mode)

Analyzing the Initial State

The Dynamic Shift

The Final Equilibrium

Similar Questions

The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by 2, then the median of the new set

If the sum of the deviations of 50 observations from 30 is 50, then the mean of these observation is :

The mean of 10 numbers $7 \times 8, 10 \times 10, 13 \times 12, 16 \times 14, \dots$ is .

If for some $x \in R$ , the frequency distribution of the marks obtained by 20 students in a test is : Marks 2, 3, 5, 7; Frequency $(x + 1)^{2}, 2 x - 5, x^{2} - 3 x, x$ . then the mean of the marks is :

The mean of a set of 30 observations is 75. If each observation is multiplied by a non-zero number $λ$ and then each of them is decreased by 25, their mean remains the same. Then $λ$ is equal to

Consider the data on $x$ taking the values $0, 2, 4, 8, \dots, 2^{n}$ with frequencies $^{n} C_{0},^{n} C_{1},^{n} C_{2}, \dots,^{n} C_{n}$ respectively. If the mean of this data is $\frac{728}{2 ^{n}}$ then $n$ is equal to

The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If the mean age of the teachers in this school now is 39 years, then the age (in years) of the newly appointed teacher is ______.

Consider the following frequency distribution: $Class : Frequency : 0 - 6 a 6 - 12 b 12 - 18 12 18 - 24 9 24 - 30 5$ If mean = $\frac{309}{22}$ and median = 14, then the value $(a - b)^{2}$ is equal to _____.

Let $M$ denote the median of the following frequency distribution. $Class Frequency 0 - 4 3 4 - 8 9 8 - 12 10 12 - 16 8 16 - 20 6$ Then $20 M$ is equal to :

A variable takes value $x$ with frequency $^{n + x - 1} C_{x}$ , $x = 0, 1, 2, \dots, n$ . The mode of the variable is .........