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

Q: Wait, I thought the sum of deviations from the mean is always zero. Why is the sum here 50?

That is a brilliant observation! You are absolutely correct: the sum of deviations from the *actual* mean is always zero. However, the problem states the deviations are measured from 30, not the mean. This tells us that 30 is not the mean, but an 'assumed mean'. The non-zero sum (50) is exactly what allows us to calculate how far the true mean is from our assumed anchor point of 30.

Q: Can I just use the formula $\bar{x} = A + \frac{\sum d_i}{n}$ directly?

Yes, absolutely! This is the 'Assumed Mean Method'. It is a powerful shortcut in statistics. By using $\bar{x} = A + \frac{\sum d_i}{n}$, you are essentially saying: 'The true mean is my starting guess (30) plus the average error (deviation) I made in that guess.' It is mathematically equivalent to the longer algebraic method and saves time.

Q: What if the number of observations was not 50? Would the formula still work?

Yes, the logic is universal. The formula $\bar{x} = A + \frac{\sum d_i}{n}$ holds for any number of observations $n$. The key is that the sum of deviations $\sum d_i$ must be divided by the *total count* of observations to find the average deviation. As long as you have $n$, the sum of deviations, and the assumed mean, you can always find the true mean.

Question

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

Options:

(A) 50

(B) 51

(C) 30

(D) 31

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on a vast, flat plain with fifty distinct markers. We are given their collective relationship to a fixed anchor point at the number $30$.

This is the essence of statistics: finding the hidden center of gravity in a sea of data. The mean represents the balance point where the sum of all deviations is zero.

### The Algebraic Translation

We have $n = 50$ observations, denoted as $x_1, x_2, \dots, x_{50}$. We are given that the sum of their deviations from $30$ is $50$.

Mathematically, this is expressed as:

$$\sum_{i=1}^{50} (x_i - 30) = 50$$

Using the linearity of the summation operator, we distribute the sigma sign:

$$\sum_{i=1}^{50} x_i - \sum_{i=1}^{50} 30 = 50$$

The second term, $\sum_{i=1}^{50} 30$, represents a constant added to itself $50$ times, which equals $50 	imes 30 = 1500$. Our equation simplifies to:

$$\sum_{i=1}^{50} x_i - 1500 = 50$$

By adding $1500$ to both sides, we determine the total sum of all observations:

$$\sum_{i=1}^{50} x_i = 1550$$

### The Final Step: Calculating the Mean

Now that we have the total sum, finding the mean $\bar{x}$ is straightforward. The mean is defined as the total sum divided by the number of observations:

$$\bar{x} = \frac{\sum x_i}{n} = \frac{1550}{50}$$

Performing this division, the zeros cancel out, leaving us with $155 / 5$, which equals **31**.

### The Pro Shortcut: The Assumed Mean Method

There is a faster, more elegant way to view this problem. Think of the mean as the 'Assumed Mean' ($A$) plus the average of the deviations.

If the sum of deviations is $50$ across $50$ observations, the average deviation is:

$$	ext{Average Deviation} = \frac{50}{50} = 1$$

Therefore, the true mean is simply:

$$\bar{x} = 30 + 1 = 31$$

This 'Assumed Mean' method is a favorite among top-tier examiners because it tests your conceptual understanding of how the mean shifts. Whether you use the long-form algebra or this intuitive shortcut, you arrive at the same result: the mean is **31**.

Animated Solution for Mathematics - Statistics: If the sum of the deviations of 50 observations from 30 is 50, then the mean of these observation is :

Select Answer:

Visualized Solution

Visualizing the Data

Defining Deviation

Setting up the Equation

Distributing the Summation

Evaluating the Constant Sum

Finding the Sum of Observations

Calculating the Mean

The Assumed Mean Shortcut

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

Analyzing the Setup

The Algebraic Translation

The Final Step

The Pro Shortcut

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