The mean and standard deviation of 15 observations are found to be 8 and 3 respectively. On rechecki

Q: Wait, why do we need to calculate the sum of squares? Can't we just adjust the variance directly?

That is a brilliant instinct, but variance is not a linear operator. Because variance depends on the square of the deviations from the mean, it is sensitive to the distribution of every single point. You cannot simply 'add' the difference to the variance; you must reconstruct the raw data's 'sum of squares' to account for the non-linear impact of the correction.

Q: Why did we subtract the square of the wrong number and add the square of the right one?

Think of $\sum x^2$ as a total 'energy' reservoir of the dataset. When you realize a value was misread, you must remove the 'energy' contribution of the incorrect value (which is $5^2$) and inject the 'energy' contribution of the correct value (which is $20^2$). It is a simple accounting procedure for the dataset's total squared magnitude.

Q: Is the mean always going to change when we correct an observation?

Yes, unless the incorrect value and the correct value happen to be identical! Since the mean is the average of all observations, changing even one value shifts the center of gravity of your data. Always update your mean first, as it is a required component in the variance formula.

Q: What if the number of observations $n$ had changed? Would the formula still work?

If $n$ changed, the problem would become much more complex because the denominator in your variance formula would shift. In this specific problem, we are replacing one value with another, so $n$ remains constant at $15$. Always check if your $n$ is stable before proceeding!

Q: I keep getting a negative variance. What am I doing wrong?

A negative variance is physically impossible! If you get a negative result, it usually means you swapped the order of the terms in the variance formula: $\frac{\sum x^2}{n} - (\bar{x})^2$. Remember, the mean of the squares must always be greater than or equal to the square of the mean.

Question

The mean and standard deviation of $15$ observations are found to be $8$ and $3$ respectively. On rechecking it was found that, in the observations, $20$ was misread as $5$. Then, the correct variance is equal to _______.

Sigma Percentile · Accepted Answer

### The Detective Work of Statistics: Uncovering the Truth

Welcome, future engineer! Today, we are not just solving a math problem; we are acting as data detectives. In the world of JEE Advanced, statistics is often dismissed as 'easy,' but it is precisely here that the most subtle traps are laid.

We have a dataset of $15$ observations, but there is a ghost in the machine—a misread value. We were told $5$ was the observation, but the truth is $20$. Our mission is to restore the integrity of our variance.

### Phase 1: The Ledger of Sums

Before we can find the variance, we must understand the 'DNA' of our dataset. We know the number of observations $n = 15$, the old mean $\bar{x}_{old} = 8$, and the old standard deviation $\sigma_{old} = 3$.

Think of the mean as the balance point of your data. The formula $\bar{x} = \frac{\sum x}{n}$ tells us that the total sum of our observations is simply $\sum x = n 	imes \bar{x}$.

Initially, our ledger showed: 
$$\sum x_{old} = 15 	imes 8 = 120$$

But this sum is tainted by the presence of the number $5$. To fix it, we perform a surgical correction: remove the imposter and insert the truth.

$$\sum x_{new} = 120 - 5 + 20 = 135$$

With this corrected sum, our new mean becomes $\bar{x}_{new} = \frac{135}{15} = 9$. We have successfully shifted our center of gravity!

### Phase 2: The Power of Squares

Now, we face the real challenge: the variance. The variance formula, $\sigma^2 = \frac{\sum x^2}{n} - (\bar{x})^2$, is our most powerful tool. It tells us how spread out our data is.

To find the new variance, we need the new sum of squares, $\sum x_{new}^2$. We start by extracting the old sum of squares from the old variance. Using $\sigma_{old}^2 = 9$, we write:

$$9 = \frac{\sum x_{old}^2}{15} - 8^2$$

$$9 = \frac{\sum x_{old}^2}{15} - 64$$

$$\frac{\sum x_{old}^2}{15} = 73 \implies \sum x_{old}^2 = 1095$$

This $1095$ is the 'total squared energy' of our original, flawed dataset. Just as we corrected the sum, we must correct this sum of squares by subtracting the square of the wrong value and adding the square of the correct one:

$$\sum x_{new}^2 = 1095 - 5^2 + 20^2$$

$$\sum x_{new}^2 = 1095 - 25 + 400 = 1470$$

### Phase 3: The Final Reveal

We are now at the finish line. We have the corrected sum of squares ($1470$) and the corrected mean ($9$). All that remains is to plug these into our variance formula one last time:

$$\sigma_{new}^2 = \frac{1470}{15} - 9^2$$

$$\sigma_{new}^2 = 98 - 81$$

$$\sigma_{new}^2 = 17$$

Look at that elegance! The math cleans itself up perfectly. By carefully tracking the impact of that single misread value through the mean and the sum of squares, we have arrived at the correct variance of **$17$**.

Remember, in JEE, it is not just about the final number; it is about the discipline of the process. You have mastered the art of data correction—keep that focus, and no problem will ever be too complex for you.

Animated Solution for Mathematics - Statistics: The mean and standard deviation of $15$ observations are found to be $8$ and $3$ respectively. On rechecking it was found that, in the observations, $20$ was misread as $5$ . Then, the correct variance is equal to _______.

Enter Numerical Value:

Visualized Solution

Initial Parameters

Identifying the Error

Formula for Mean

Old Sum of Observations

Correcting the Sum

Calculating Correct Mean

Formula for Variance

Using Old Variance

Old Sum of Squares

Correcting the Sum of Squares

Calculating Correct Sum of Squares

Final Variance Setup

Final Answer

The Sigma Insight: Variance and Standard Deviation

The Detective Work of Statistics

Phase 1

Phase 2

Phase 3

Similar Questions

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