JEE Main 2008

LEVELJEE Main

Animated Solution for Mathematics - Statistics: The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80. Then which one of the following gives possible values of a and b?

Q: I got $a=3, b=4$ and $a=4, b=3$. Which one is the answer?

Both are mathematically correct! The set of numbers $\{3, 4, 8, 5, 10\}$ and $\{4, 3, 8, 5, 10\}$ have the exact same mean and variance. In the context of this problem, the options provided will only contain one of these pairs. If both were present, the question would be ambiguous, but usually, you just pick the pair that matches your options.

Select Answer:

Visualized Solution

Understanding the Data

Given numbers: $a, b, 8, 5, 10$
Total observations $n = 5$
Mean $\overset{x}{ˉ} = 6$
Variance $σ^{2} = 6.80$

The Mean Formula

The formula for the mean is: $\overset{x}{ˉ} = \frac{\sum x _{i}}{n}$
This represents the sum of all observations divided by the total number of observations.

Substituting Values into Mean

Substitute the known values:
$6 = \frac{a + b + 8 + 5 + 10}{5}$

Simplifying the Mean Equation

Multiply both sides by $5$ :
$30 = a + b + 23$
Isolate $a + b$ :
$a + b = 7$
Express $b$ in terms of $a$ :
$b = 7 - a$

The Variance Formula

The formula for variance is:
$σ^{2} = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n}$
It measures the average squared deviation from the mean.

Substituting Values into Variance

Substitute $σ^{2} = 6.80$ and $\overset{x}{ˉ} = 6$ :
$6.80 = \frac{( a - 6 ) ^{2} + ( b - 6 ) ^{2} + ( 8 - 6 ) ^{2} + ( 5 - 6 ) ^{2} + ( 10 - 6 ) ^{2}}{5}$

Simplifying the Numerical Terms

Multiply by $5$ : $34 = (a - 6)^{2} + (b - 6)^{2} + (2)^{2} + (- 1)^{2} + (4)^{2}$
Calculate squares: $34 = (a - 6)^{2} + (b - 6)^{2} + 4 + 1 + 16$
Sum the constants: $34 = (a - 6)^{2} + (b - 6)^{2} + 21$

Isolating the Unknowns

Subtract $21$ from both sides:
$34 - 21 = (a - 6)^{2} + (b - 6)^{2}$
$13 = (a - 6)^{2} + (b - 6)^{2}$

Substituting $b = 7 - a$

Recall from the mean equation: $b = 7 - a$
Substitute this into the variance equation:
$(a - 6)^{2} + ((7 - a) - 6)^{2} = 13$
Simplify the second term:
$(a - 6)^{2} + (1 - a)^{2} = 13$

Expanding the Squared Terms

Expand $(a - 6)^{2}$ : $a^{2} - 12 a + 36$
Expand $(1 - a)^{2}$ : $1 - 2 a + a^{2}$
Combine them:
$(a^{2} - 12 a + 36) + (1 - 2 a + a^{2}) = 13$

Forming the Quadratic Equation

Combine like terms:
$2 a^{2} - 14 a + 37 = 13$
Subtract $13$ from both sides:
$2 a^{2} - 14 a + 24 = 0$
Divide the entire equation by $2$ :
$a^{2} - 7 a + 12 = 0$

Solving for $a$

Factorize the quadratic equation:
$a^{2} - 3 a - 4 a + 12 = 0$
$a (a - 3) - 4 (a - 3) = 0$
$(a - 3) (a - 4) = 0$
Therefore, $a = 3$ or $a = 4$

Finding $b$ and Final Answer

If $a = 3$ , then $b = 7 - 3 = 4$
If $a = 4$ , then $b = 7 - 4 = 3$
The possible values for the pair $(a, b)$ are $(3, 4)$ or $(4, 3)$ .
Looking at the options, $a = 3, b = 4$ is the correct choice.

00:00 / 00:00

The Sigma Insight: Variance and Standard Deviation

Solution Diagram

Analyzing the Setup

Imagine you are a detective, but instead of chasing criminals, you are chasing the hidden DNA of a dataset. We are given five numbers:

a, b, 8, 5, 10

We know the mean is

6

and the variance is

6.80

. These two numbers are not just random statistics; they are the constraints that define the very existence of

a

and

b

The Linear Constraint

The mean is the balance point of our data, representing the center of gravity. The formula is defined as:

\overset{x}{ˉ} = \frac{\sum x _{i}}{n}

With

n = 5

and

\overset{x}{ˉ} = 6

, we have:

6 = \frac{a + b + 8 + 5 + 10}{5}

Multiplying by

5

gives us

30 = a + b + 23

. This simplifies beautifully to

a + b = 7

This is our first clue. It tells us that

a

and

b

are locked in a dance where their sum must always be

7

. We can express

b

b = 7 - a

The Quadratic Constraint

Now, we turn to the variance, the measure of how spread out our data is. The formula is:

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

Plugging in our values:

6.80 = \frac{( a - 6 ) ^{2} + ( b - 6 ) ^{2} + ( 8 - 6 ) ^{2} + ( 5 - 6 ) ^{2} + ( 10 - 6 ) ^{2}}{5}

Multiply by

5

to get

34 = (a - 6)^{2} + (b - 6)^{2} + (2)^{2} + (- 1)^{2} + (4)^{2}

. Calculating the squares, we find

4 + 1 + 16 = 21

So,

34 = (a - 6)^{2} + (b - 6)^{2} + 21

. Subtracting

21

from both sides, we get:

(a - 6)^{2} + (b - 6)^{2} = 13

This is the geometric reality of our data: the sum of the squared deviations of

a

and

b

from the mean must be

13

The Algebraic Synthesis

Now, we combine our two worlds. We have

a + b = 7

and

(a - 6)^{2} + (b - 6)^{2} = 13

Substitute

b = 7 - a

into the variance equation:

(a - 6)^{2} + ((7 - a) - 6)^{2} = 13

This simplifies to

(a - 6)^{2} + (1 - a)^{2} = 13

. Expanding these binomials, we get:

(a^{2} - 12 a + 36) + (1 - 2 a + a^{2}) = 13

Combining like terms, we arrive at

2 a^{2} - 14 a + 37 = 13

. Subtracting

13

gives

2 a^{2} - 14 a + 24 = 0

Dividing by

2

, we find the elegant quadratic:

a^{2} - 7 a + 12 = 0

Factoring this is a joy:

(a - 3) (a - 4) = 0

. Thus,

a

can be

3

4

a = 3

, then

b = 4

. If

a = 4

, then

b = 3

The mystery is solved. The values are $3$ and $4$ .

Common Traps & FAQs

Why can't we use the formula

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

You absolutely can! It is a valid alternative. However, be careful with the arithmetic. Squaring large numbers like

1 0^{2}

8^{2}

can sometimes lead to calculation errors compared to the deviation method. Both paths lead to the same destination, but the deviation method used here is often cleaner for these types of problems.

I got

a = 3, b = 4

and

a = 4, b = 3

. Which one is the answer?

Both are mathematically correct! The set of numbers

{3, 4, 8, 5, 10}

and

{4, 3, 8, 5, 10}

have the exact same mean and variance. In the context of this problem, the options provided will only contain one of these pairs. If both were present, the question would be ambiguous, but usually, you just pick the pair that matches your options.

Is there a shortcut to avoid the quadratic equation?

Not really. Because variance involves squared terms, you are inherently dealing with a quadratic relationship. Embrace the quadratic; it's the signature of variance! Once you see the structure, solving it becomes second nature.

Animated Solution for Mathematics - Statistics: The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80. Then which one of the following gives possible values of a and b?

Select Answer:

Visualized Solution

Understanding the Data

The Mean Formula

Substituting Values into Mean

Simplifying the Mean Equation

The Variance Formula

Substituting Values into Variance

Simplifying the Numerical Terms

Isolating the Unknowns

Substituting $b = 7 - a$

Expanding the Squared Terms

Forming the Quadratic Equation

Solving for $a$

Finding $b$ and Final Answer

The Sigma Insight: Variance and Standard Deviation

Analyzing the Setup

The Linear Constraint

The Quadratic Constraint

The Algebraic Synthesis

Similar Questions

If the mean and the variance of 6, 4, a, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then $a + b + ab$ is equal to :

The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1, 3 and 8, then a ratio of other two observations is :

If the mean and variance of six observations 7, 10, 11, 15, a, b are 10 and $\frac{20}{3}$ , respectively, then the value of $∣ a - b ∣$ is equal to :

The mean and variance of 7 observations are 8 and 16 respectively. If two observations are 6 and 8, then the variance of the remaining 5 observations is :

The mean and variance of 7 observations are 8 and 16 respectively. If one observation 14 is omitted and $a$ and $b$ are respectively mean and variance of remaining 6 observation, then $a + 3 b - 5$ is equal to ______.

If the mean and variance of eight numbers $3, 7, 9, 12, 13, 20, x$ and $y$ be 10 and 25 respectively then $x y$ is equal to ________.

If the mean of the data : 7, 8, 9, 7, 8, 7, $λ$ , 8 is 8, then the variance of this data is :-

Let the Mean and Variance of five observations $x_{1} = 1, x_{2} = 3, x_{3} = a, x_{4} = 7$ and $x_{5} = b, a > b$ , be 5 and 10 respectively. Then the Variance of the observations $n + x_{n}, n = 1, 2, \dots, 5$ is

Let the six numbers $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}$ be in A.P. and $a_{1} + a_{3} = 10$ . If the mean of these six numbers is $\frac{19}{2}$ and their variance is $σ^{2}$ , then $8 σ^{2}$ is equal to

The mean and variance of a data of 10 observations are 10 and 2, respectively. If an observations $α$ in this data is replaced by $β$ , then the mean and variance become 10.1 and 1.99, respectively. Then $α + β$ equals

Animated Solution for Mathematics - Statistics: The mean of the numbers a, b, 8, 5, 10 is 6 and the variance is 6.80. Then which one of the following gives possible values of a and b?

Select Answer:

Visualized Solution

Understanding the Data

The Mean Formula

Substituting Values into Mean

Simplifying the Mean Equation

The Variance Formula

Substituting Values into Variance

Simplifying the Numerical Terms

Isolating the Unknowns

Substituting .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b=7−a

Expanding the Squared Terms

Forming the Quadratic Equation

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

Finding .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } b and Final Answer

The Sigma Insight: Variance and Standard Deviation

Analyzing the Setup

The Linear Constraint

The Quadratic Constraint

The Algebraic Synthesis

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the mean and the variance of 6, 4, a, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then a+b+ab is equal to :

The mean of five observations is 5 and their variance is 9.20. If three of the given five observations are 1, 3 and 8, then a ratio of other two observations is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the mean and variance of six observations 7, 10, 11, 15, a, b are 10 and 320​, respectively, then the value of ∣a−b∣ is equal to :

The mean and variance of 7 observations are 8 and 16 respectively. If two observations are 6 and 8, then the variance of the remaining 5 observations is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the mean and variance of eight numbers 3,7,9,12,13,20,x and y be 10 and 25 respectively then xy is equal to ________.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the mean of the data : 7, 8, 9, 7, 8, 7, λ, 8 is 8, then the variance of this data is :-

Substituting $b = 7 - a$

Solving for $a$

Finding $b$ and Final Answer

If the mean and the variance of 6, 4, a, 8, b, 12, 10, 13 are 9 and 9.25 respectively, then $a + b + ab$ is equal to :

If the mean and variance of six observations 7, 10, 11, 15, a, b are 10 and $\frac{20}{3}$ , respectively, then the value of $∣ a - b ∣$ is equal to :

If the mean and variance of eight numbers $3, 7, 9, 12, 13, 20, x$ and $y$ be 10 and 25 respectively then $x y$ is equal to ________.

If the mean of the data : 7, 8, 9, 7, 8, 7, $λ$ , 8 is 8, then the variance of this data is :-