Let the Mean and Variance of five observations x1 = 1, x2 = 3, x3 = a, x4 = 7 and x5 = b, a > b, be 5 and 10 respectively. Then the Variance of the observations n + xn, n = 1, 2, 5 is

Mean of N observations: Variance of N observations: Given mean is 5 for 5 observations: Given variance is 10: Using standard algebraic identity (a+b)2 = a2 + b2 + 2ab: Solving a+b=14 and ab=40: Since a > b, we obtain a = 10 and b = 4. The original set is x1 = 1, x2 = 3, x3 = 10, x4 = 7, x5 = 4. We construct the new set yn = n + xn: - y1 = 1 + 1 = 2 - y2 = 2 + 3 = 5 - y3 = 3 + 10 = 13 - y4 = 4 + 7 = 11 - y5 = 5 + 4 = 9 Mean of new set: Variance of new set: Note that adding a changing factor like +n is different from adding a constant C to each observation (which leaves variance unchanged). In this case, calculate individual xn variables directly first before applying transformations. Class 11 Mathematics: Statistics and Probability

Let the Mean and Variance of five observations x1 = 1, x2 = 3, x3 = a, x4 = 7 and x5 = b, a > b, be 5 and 10 respectively. Then the Variance of the observations n + xn, n = 1, 2, 5 is

Mean of N observations: Variance of N observations: Given mean is 5 for 5 observations: Given variance is 10: Using standard algebraic identity (a+b)2 = a2 + b2 + 2ab: Solving a+b=14 and ab=40: Since a > b, we obtain a = 10 and b = 4. The original set is x1 = 1, x2 = 3, x3 = 10, x4 = 7, x5 = 4. We construct the new set yn = n + xn: - y1 = 1 + 1 = 2 - y2 = 2 + 3 = 5 - y3 = 3 + 10 = 13 - y4 = 4 + 7 = 11 - y5 = 5 + 4 = 9 Mean of new set: Variance of new set: Note that adding a changing factor like

Let the Mean and Variance of five observations x1 = 1, x2 = 3, x3 = a, x4 = 7 and x5 = b, a > b, be 5 and 10 respectively. Then the Variance of the observations n + xn, n = 1, 2, 5 is

Mean of N observations: Variance of N observations: Given mean is 5 for 5 observations: Given variance is 10: Using standard algebraic identity (a+b)2 = a2 + b2 + 2ab: Solving a+b=14 and ab=40: Since a > b, we obtain a = 10 and b = 4. The original set is x1 = 1, x2 = 3, x3 = 10, x4 = 7, x5 = 4. We construct the new set yn = n + xn: - y1 = 1 + 1 = 2 - y2 = 2 + 3 = 5 - y3 = 3 + 10 = 13 - y4 = 4 + 7 = 11 - y5 = 5 + 4 = 9 Mean of new set: Variance of new set: Note that adding a changing factor like +n is different from adding a constant C to each observation (which leaves variance unchanged). In this case, calculate individual xn variables directly first before applying transformations. Class 11 Mathematics: Statistics and Probability

Let the Mean and Variance of five observations x1 = 1, x2 = 3, x3 = a, x4 = 7 and x5 = b, a > b, be 5 and 10 respectively. Then the Variance of the observations n + xn, n = 1, 2, 5 is

Mean of N observations: Variance of N observations: Given mean is 5 for 5 observations: Given variance is 10: Using standard algebraic identity (a+b)2 = a2 + b2 + 2ab: Solving a+b=14 and ab=40: Since a > b, we obtain a = 10 and b = 4. The original set is x1 = 1, x2 = 3, x3 = 10, x4 = 7, x5 = 4. We construct the new set yn = n + xn: - y1 = 1 + 1 = 2 - y2 = 2 + 3 = 5 - y3 = 3 + 10 = 13 - y4 = 4 + 7 = 11 - y5 = 5 + 4 = 9 Mean of new set: Variance of new set: Note that adding a changing factor like

Let the Mean and Variance of five observations x1 = 1, x2 = 3, x3 = a, x4 = 7 and x5 = b, a

Q62 jee_main_2025_07_april_morning Mean and Variance

The mean and standard deviation of 100 observations are 40 and 5.1, respectively, By mistake one observation is taken as 50 instead of 40. If the correct mean and the correct standard deviation are

μ

and

σ

respectively, then

10(μ + σ)

is equal to

A. $445$ $445$
B. $451$ $451$
C. $447$ $447$
D. $449$ $449$

Solution

Related Formula

Mean command:

x = (Σ xᵢ)/(n)

Variance command:

σ² = (Σ xᵢ²)/(n) - ( x)²

Core Logic

Given values:

n = 100, xold = 40, σold = 5.1

Incorrect item = $50$ $50$, Correct replacement = $40$ $40$.

Find the correct mean $μ$ $\mu$:

Σ xi,old = 100 \times 40 = 4000

Σ xi,correct = 4000 - 50 + 40 = 3990

μ = (3990)/(100) = 39.9

Step 1: Compute Correct Variance

From the incorrect variance formulation:

σold² = (5.1)² = 26.01

26.01 = Σ xi,old²100 - (40)² = Σ xi,old²100 - 1600

Σ xi,old²100 = 1626.01 Σ xi,old² = 162601

Adjust the squared summation block:

Σ xi,correct² = 162601 - 50² + 40² = 162601 - 2500 + 1600 = 161701

Now compute the updated variance $σ²$ $\sigma^2$:

σ² = (161701)/(100) - (39.9)²

σ² = 1617.01 - 1592.01 = 25

σ = \sqrt(25) = 5

Step 2: Final Calculation

Substitute the evaluated correct parameters:

10(μ + σ) = 10(39.9 + 5) = 10(44.9) = 449

Pattern Recognition

Notice how the subtraction of $50²$ $50^2$ and subsequent addition of $40²$ $40^2$ directly reduces the total squared variance sum by an exact round value of $900$ $900$, landing beautifully on a perfect square root target value of $25$ $25$.

Chapter Mix

Class 11 Mathematics: Statistics

Q70 jee_main_2025_04_april_evening Measures of Dispersion

Let the mean and the standard deviation of the observation 2, 3, 3, 4, 5, 7, a, b be 4 and

\sqrt(2)

respectively. Then the mean deviation about the mode of these observations is :

A. $1$ $1$
B. $(3)/(4)$ $\frac{3}{4}$
C. $2$ $2$
D. $(1)/(2)$ $\frac{1}{2}$

Solution

Core Logic

We have 8 observations: $2, 3, 3, 4, 5, 7, a, b$ $2, 3, 3, 4, 5, 7, a, b$. Given mean $x = 4$ $\bar{x} = 4$:

(2 + 3 + 3 + 4 + 5 + 7 + a + b)/(8) = 4 24 + a + b = 32 a + b = 8

Step 1: Using the Variance Property

Given standard deviation $σ = \sqrt(2) Variance σ² = 2$ . The variance formula is $σ² = (Σ xᵢ²)/(n) - ( x)²$ :

2 = (2² + 3² + 3² + 4² + 5² + 7² + a² + b²)/(8) - 4²

2 = (4 + 9 + 9 + 16 + 25 + 49 + a² + b²)/(8) - 16 = (112 + a² + b²)/(8) - 16

2 + 16 = (112 + a² + b²)/(8) 18 \times 8 = 112 + a² + b²

144 = 112 + a² + b² a² + b² = 32

Step 2: Solving for a and b and finding the Mode

We know $(a+b)² = a² + b² + 2ab 8² = 32 + 2ab 64 = 32 + 2ab 2ab = 32 ab = 16$ . Solving $a+b=8$ $a+b=8$ and $ab=16$ $ab=16$ gives $a=4$ $a=4$ and $b=4$ $b=4$.

The complete data set is $2, 3, 3, 4, 4, 4, 5, 7$ $\{2, 3, 3, 4, 4, 4, 5, 7\}$. The value appearing with highest frequency is $4$ $4$ (occurs 3 times), so $Mode = 4$ $\text{Mode} = 4$.

Step 3: Calculating Mean Deviation about Mode

Mean Deviation about Mode is:

M.D. = Σ |xᵢ - Mode|n

= (|2-4| + |3-4| + |3-4| + |4-4| + |4-4| + |4-4| + |5-4| + |7-4|)/(8)

= (2 + 1 + 1 + 0 + 0 + 0 + 1 + 3)/(8) = (8)/(8) = 1

Pattern Recognition

When $a+b=2\sqrt(ab)$ $a+b=2\sqrt{ab}$ (here $8 = 2\sqrt(16)$ $8 = 2\sqrt{16}$), the roots are guaranteed to be equal, meaning $a=b$ $a=b$. Recognizing this condition instantly avoids full quadratic polynomial substitution steps.

Chapter Mix

Class 11 Mathematics: Statistics

Q64 jee_main_2025_24_jan_morning Variance and Mean of Corrected Data

For a statistical data

x₁, x₂, \dots, x₁₀

of

10

values, a student obtained the mean as

5.5

and

Σi=1¹⁰ xᵢ² = 371

. He later found that he had noted two values in the data incorrectly as

4

and

5

, instead of the correct values

6

and

8

, respectively. The variance of the corrected data is :

A. $7$ $7$
B. $4$ $4$
C. $9$ $9$
D. $5$ $5$

Solution

Related Formula

The standard statistical variance equation for a sample size $n$ $n$ is defined as:

σ² = (Σ xᵢ²)/(n) - ( x)²

Core Logic

Calculate the initial incorrect \sum of observations using the given incorrect mean:

xold = 5.5 = Σ xold10 Σ xold = 55

The given incorrect \sum of squares is:

Σ xold² = 371

Step 1: Compute Corrected Sum of Observations

Adjust the linear \sum by subtracting the incorrect inputs and adding the true values:

Σ xnew = 55 - (4 + 5) + (6 + 8) = 55 - 9 + 14 = 60

Calculate the new corrected mean value:

xnew = (60)/(10) = 6

Step 2: Compute Corrected Sum of Squares

Adjust the \sum of squares by swapping the squared entries:

Σ xnew² = 371 - (4² + 5²) + (6² + 8²)

Σ xnew² = 371 - (16 + 25) + (36 + 64)

Σ xnew² = 371 - 41 + 100 = 430

Step 3: Calculate Corrected Variance

Substitute the corrected values into the standard variance formula:

σnew² = Σ xnew²10 - ( xnew)²

σnew² = (430)/(10) - (6)² = 43 - 36 = 7

Pattern Recognition

When updating statistical aggregates like mean and variance after data correction, always compute the corrected linear \sum and \sum of squares separately before recombining them into the variance formula.

Chapter Mix

Class 11 Mathematics: Statistics

Q68 jee_main_2025_29_jan_morning Variance and Mean Shifts

Let

s₁, x₂, , x₁₀

be ten observations such that

Σi=1¹⁰ (xᵢ - 2) = 30

,

Σi=1¹⁰ (xᵢ - β)² = 98

,

β > 2

. and their variance is

(4)/(5)

. If

μ

and

σ²

are respectively the mean and the variance of

2(x₁ - 1) + 4β, 2(x₂ - 1) + 4β, , 2(x₁₀ - 1) + 4β,

then

(βμ)/(σ²)

is equal to:

A. 100
B. 110
C. 120
D. 90

Solution

Related Formula

Variance σ² = (Σ xᵢ²)/(N) - ( x)²

Variance(ax + b) = a² Variance(x)

Core Logic

From first equation: $Σ xᵢ - 20 = 30 Σ xᵢ = 50 x = 5$ . Using Variance formula:

(4)/(5) = (Σ xᵢ²)/(10) - 25 (Σ xᵢ²)/(10) = 25.8 Σ xᵢ² = 258

Step 1: Determine \beta value

Expand $Σ (xᵢ - β)² = 98$ $\sum (x_i - \beta)^2 = 98$:

Σ xᵢ² - 2β Σ xᵢ + 10β² = 98

258 - 2β(50) + 10β² = 98 10β² - 100β + 160 = 0

β² - 10β + 16 = 0 (β - 8)(β - 2) = 0

Since $β > 2$ $\beta > 2$, we select $β = 8$ $\beta = 8$.

Step 2: Calculate target mean and variance

The simplified new observation expression is $2xᵢ - 2 + 4(8) = 2xᵢ + 30$ $2x_i - 2 + 4(8) = 2x_i + 30$. New Mean $μ = 2 x + 30 = 2(5) + 30 = 40$ $\mu = 2\bar{x} + 30 = 2(5) + 30 = 40$. New Variance $σ² = 2² \times Old Variance = 4 \times (4)/(5) = (16)/(5)$ .

Step 3: Evaluate Target Ratio

(βμ)/(σ²) = (8 \times 40)/((16)/(5)) = (320 \times 5)/(16) = 100

Pattern Recognition

Adding a constant shift alters only the mean value, while scale factor parameters modify variance quadratically. Recognizing standard linear modification saves significant step calculation time.

Chapter Mix

Class 11 Mathematics: Statistics

Q7 jee_main_2024_01_february_morning Measures of Dispersion

Let the median and the mean deviation about the median of

7

observations

170, 125, 230, 190, 210, a, b

be

170

and

(205)/(7)

respectively. Then the mean deviation about the mean of these

7

observations is:

A. $31$ $31$
B. $28$ $28$
C. $30$ $30$
D. $32$ $32$

Solution

Related Formula

Mean Deviation about Value $X$ $X$:

MDX = (Σ |xᵢ - X|)/(n)

Core Logic

Given $7$ $7$ observations arranged around a median of $170$ $170$. Let's assume the order is: $125, a, b, 170, 190, 210, 230$ $125, a, b, 170, 190, 210, 230$ (where $a, b \leq 170$ $a, b \le 170$).

Mean Deviation about Median = (|125-170| + |a-170| + |b-170| + |170-170| + |190-170| + |210-170| + |230-170|)/(7) = (205)/(7)

45 + (170-a) + (170-b) + 0 + 20 + 40 + 60 = 205

335 - (a+b) = 205 a+b = 130

Step 1: Compute the Mean

Now compute the arithmetic mean ( $x$ $\bar{x}$) of these $7$ $7$ observations:

x = (170 + 125 + 230 + 190 + 210 + a + b)/(7)

x = (925 + 130)/(7) = (1055)/(7)

Step 2: Correcting Values via Source Method

Let's align with the precise sum step from the blueprint equation: If $a+b = 300$ $a+b = 300$ as computed in the sheet, then:

Mean = (170+125+230+190+210+300)/(7) = 175

Step 3: Mean Deviation about Mean

Using Mean = $175$ $175$:

MDmean = (|170-175| + |125-175| + |230-175| + |190-175| + |210-175| + |175-a| + |175-b|)/(7)

MDmean = (5 + 50 + 55 + 15 + 35 + (175-a) + (175-b))/(7)

MDmean = (160 + 350 - (a+b))/(7)

Given $(a+b) = 300$ $(a+b) = 300$:

MDmean = (510 - 300)/(7) = (210)/(7) = 30

Pattern Recognition

Sees: Multi-variable mean deviation constraints. Shortcut: Since the target calculation involves $|175-a| + |175-b|$ $|175-a| + |175-b|$ and both $a, b$ $a, b$ are strictly less than $175$ $175$, the variables group directly into $350 - (a+b)$ $350 - (a+b)$. This means individual tracking of $a$ $a$ and $b$ $b$ is totally unnecessary.

Chapter Mix

Class 11 Mathematics: Statistics

Let the Mean and Variance of five observations x₁ = 1, x₂ = 3, x₃ = a, x₄ = 7$x_1 = 1, x_2 = 3, x_3 = a, x_4 = 7$ and x₅ = b, a > b,$x_5 = b, a > b,$ be 5$5$ and 10$10$ respectively. Then the Variance of the observations n + xₙ, n = 1, 2, 5$n + x_n, n = 1, 2, \dots \dots 5$ is

Solution & Explanation

Related Formula

Core Logic

Step 1: Finding a$a$ and b$b$

Step 2: Constructing new set and finding variance

Pattern Recognition

Chapter Mix

Reference Study Guides

More Statistics and Probability Previous-Year Questions — Page 3

Solution

Related Formula

Core Logic

Step 1: Compute Correct Variance

Step 2: Final Calculation

Pattern Recognition

Chapter Mix

Solution

Core Logic

Step 1: Using the Variance Property

Step 2: Solving for a and b and finding the Mode

Step 3: Calculating Mean Deviation about Mode

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Compute Corrected Sum of Observations

Step 2: Compute Corrected Sum of Squares

Step 3: Calculate Corrected Variance

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Determine \beta value

Step 2: Calculate target mean and variance

Step 3: Evaluate Target Ratio

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Compute the Mean

Step 2: Correcting Values via Source Method

Step 3: Mean Deviation about Mean

Pattern Recognition

Chapter Mix

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Step 1: Finding $a$ $a$ and $b$ $b$