JEE Main 2021 (March)

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Animated Solution for Mathematics - Statistics: Let in a series of $2 n$ observations, half of them are equal to $a$ and remaining half are equal to $- a$ . Also by adding a constant $b$ in each of these observations, the mean and standard deviation of new set become 5 and 20, respectively. Then the value of $a^{2} + b^{2}$ is equal to:

Q: Is there a faster way to calculate the variance than the standard formula?

Yes, using the formula $\sigma^2 = \frac{\sum x_i^2}{N} - (\bar{x})^2$. Since we determined $\bar{x}=0$, the formula simplifies beautifully to $\sigma^2 = \frac{\sum x_i^2}{N}$. This saves you from tedious subtractions and keeps your algebra clean.

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Visualized Solution

Initial Observations

Total observations: $2 n$
First half: $n$ observations equal to $a$
Second half: $n$ observations equal to $- a$

Formula for Mean $\overset{x}{ˉ}$

Mean $\overset{x}{ˉ} = \frac{Sum of all observations}{Total number of observations}$
$\overset{x}{ˉ} = \frac{\sum _{i = 1}^{2 n} x _{i}}{2 n}$

Substituting Values for Mean

Sum of first half $= n \times a$
Sum of second half $= n \times (- a)$
$\overset{x}{ˉ} = \frac{n ( a ) + n ( - a )}{2 n}$

Calculating Original Mean

$\overset{x}{ˉ} = \frac{na - na}{2 n}$
$\overset{x}{ˉ} = \frac{0}{2 n} = 0$

Formula for Variance $σ_{x}^{2}$

Variance $σ_{x}^{2} = \frac{\sum x _{i}^{2}}{N} - (\overset{x}{ˉ})^{2}$
Here, $N = 2 n$ and $\overset{x}{ˉ} = 0$

Substituting Values for Variance

Sum of squares $= n (a)^{2} + n (- a)^{2}$
$σ_{x}^{2} = \frac{n ( a ) ^{2} + n ( - a ) ^{2}}{2 n} - (0)^{2}$

Calculating Original Variance

$σ_{x}^{2} = \frac{n a ^{2} + n a ^{2}}{2 n}$
$σ_{x}^{2} = \frac{2 n a ^{2}}{2 n} = a^{2}$

Shifting the Observations

A constant $b$ is added to each observation.
New observations: $y_{i} = x_{i} + b$
New points: $- a + b$ and $a + b$

Property of Mean Under Translation

Property: If $y_{i} = x_{i} + b$ , then new mean $\overset{y}{ˉ} = \overset{x}{ˉ} + b$
The mean shifts by the exact same constant $b$ .

Calculating the Constant $b$

Given new mean $\overset{y}{ˉ} = 5$
We know $\overset{x}{ˉ} = 0$
$0 + b = 5 ⟹ b = 5$

Property of Standard Deviation Under Translation

Property: Standard deviation (and variance) is invariant under translation.
Shifting data does not change its spread.
New standard deviation $σ_{y} = σ_{x}$

Calculating $a^{2}$

Given new standard deviation $σ_{y} = 20$
Therefore, original standard deviation $σ_{x} = 20$
Original variance $σ_{x}^{2} = (20)^{2} = 400$
Since $σ_{x}^{2} = a^{2}$ , we get $a^{2} = 400$

Final Value of $a^{2} + b^{2}$

We found $a^{2} = 400$ and $b = 5$
Calculate $a^{2} + b^{2} = 400 + (5)^{2}$
$a^{2} + b^{2} = 400 + 25 = 425$
Final Answer: $425$

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The Sigma Insight: Variance and Standard Deviation

Solution Diagram

Analyzing the Initial Distribution

We are given

2 n

observations, split perfectly down the middle:

n

points resting at

a

, and

n

points resting at

- a

. Because these points are symmetrically placed around the origin, the mean

\overset{x}{ˉ}

must be zero.

Mathematically, we calculate the mean

\overset{x}{ˉ}

as the sum of all observations divided by the total count

2 n

\overset{x}{ˉ} = \frac{n ( a ) + n ( - a )}{2 n} = 0

Calculating the Variance

The variance

σ_{x}^{2}

measures the spread of the data and is defined as the mean of the squared deviations from the mean. Since our mean is zero, the variance is simply the average of the squared values:

σ_{x}^{2} = \frac{n ( a ^{2} ) + n ( - a ) ^{2}}{2 n} = \frac{2 n a ^{2}}{2 n} = a^{2}

This result,

σ_{x}^{2} = a^{2}

, represents the intrinsic spread of our original data set.

Applying the Transformation

We add a constant

b

to every single observation to create a new set of data. This transformation acts as a rigid shift of the distribution along the number line.

When a distribution is shifted by a constant, the internal spread remains invariant. Therefore, the standard deviation of the new data is identical to the standard deviation of the original data.

The new mean, however, shifts by exactly

b

. Given the new mean is

5

, we have:

\overset{y}{ˉ} = \overset{x}{ˉ} + b \Rightarrow 0 + b = 5 \Rightarrow b = 5

Final Calculation

We are given that the new standard deviation is

20

. Since the standard deviation is invariant under translation, the original standard deviation must also be

20

Given

σ_{x} = 20

, the variance is:

σ_{x}^{2} = (20)^{2} = 400

Since we previously established that

σ_{x}^{2} = a^{2}

, it follows that

a^{2} = 400

. We now compute the target value:

a^{2} + b^{2} = 400 + (5)^{2} = 400 + 25 = 425

The final result is 425.

Common Traps & FAQs

Wait, why doesn't the standard deviation change when we add

b

Think of standard deviation as the 'width' of the distribution. If you slide a rigid ruler across a table, does its length change? No. Adding

b

is just a translation of the entire set. The distance between any two points

x_{i}

and

x_{j}

remains

(x_{i} + b) - (x_{j} + b) = x_{i} - x_{j}

. Since the relative distances are preserved, the spread is invariant.

Is there a faster way to calculate the variance than the standard formula?

Yes, using the formula

σ^{2} = \frac{\sum x _{i}^{2}}{N} - (\overset{x}{ˉ})^{2}

. Since we determined

\overset{x}{ˉ} = 0

, the formula simplifies beautifully to

σ^{2} = \frac{\sum x _{i}^{2}}{N}

. This saves you from tedious subtractions and keeps your algebra clean.

What if the number of observations were not equal for both sets?

That is a great question! If the counts were different, say

n_{1}

points at

a

and

n_{2}

points at

- a

, the mean would no longer be zero. You would have to calculate the weighted average. Always check the symmetry of your data before diving into the summation formulas!

Select Answer:

Visualized Solution

Initial Observations

Formula for Mean $\overset{x}{ˉ}$

Substituting Values for Mean

Calculating Original Mean

Formula for Variance $σ_{x}^{2}$

Substituting Values for Variance

Calculating Original Variance

Shifting the Observations

Property of Mean Under Translation

Calculating the Constant $b$

Property of Standard Deviation Under Translation

Calculating $a^{2}$

Final Value of $a^{2} + b^{2}$

The Sigma Insight: Variance and Standard Deviation

Analyzing the Initial Distribution

Calculating the Variance

Applying the Transformation

Final Calculation

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Select Answer:

Visualized Solution

Initial Observations

Formula for Mean .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } xˉ

Substituting Values for Mean

Calculating Original Mean

Formula for Variance .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } σx2​

Substituting Values for Variance

Calculating Original Variance

Shifting the Observations

Property of Mean Under Translation

Calculating the Constant .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b

Property of Standard Deviation Under Translation

Calculating .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a2

Final Value of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a2+b2

The Sigma Insight: Variance and Standard Deviation

Analyzing the Initial Distribution

Calculating the Variance

Applying the Transformation

Final Calculation

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Formula for Mean $\overset{x}{ˉ}$

Formula for Variance $σ_{x}^{2}$

Calculating the Constant $b$

Calculating $a^{2}$

Final Value of $a^{2} + b^{2}$

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