Maths Q9 – JEE-MAIN 2025

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

Clue 1 of 3: Understand the impact of correction

When an observation is corrected, both the sum of observations and the sum of squares of observations need to be adjusted.

🧠 Full Solution Path

Step 1: Calculate Corrected Sum and Mean✦ Active

First, use the given incorrect mean to find the incorrect sum of observations. Then, adjust this sum by subtracting the incorrect observation and adding the correct one to find the correct sum. Finally, calculate the correct mean $μ$ .

\sum x_{i n c} = n \times {\bar{x}}_{i n c} = 100 \times 40 = 4000

\sum x_{c o r r} = \sum x_{i n c} - 50 + 40 = 4000 - 10 = 3990

μ = \frac{\sum x_{c o r r}}{n} = \frac{3990}{100} = 39.9

Step 2: Calculate Corrected Sum of Squares and Standard Deviation○ Expand

Next, use the incorrect standard deviation and mean to find the incorrect sum of squares. Adjust this sum of squares by subtracting the square of the incorrect observation and adding the square of the correct one. Use the corrected sum of squares and the correct mean to find the correct standard deviation $σ$ .

σ_{i n c}^{2} = \frac{\sum x_{i n c}^{2}}{n} - ({\bar{x}}_{i n c})^{2}

{5.1}^{2} = \frac{\sum x_{i n c}^{2}}{100} - 40^{2}

26.01 = \frac{\sum x_{i n c}^{2}}{100} - 1600

\sum x_{i n c}^{2} = 100 (1600 + 26.01) = 162601

\sum x_{c o r r}^{2} = \sum x_{i n c}^{2} - 50^{2} + 40^{2} = 162601 - 2500 + 1600 = 161701

σ^{2} = \frac{\sum x_{c o r r}^{2}}{n} - μ^{2} = \frac{161701}{100} - (39.9)^{2} = 1617.01 - 1592.01 = 25

σ = \sqrt{25} = 5

Step 3: Calculate the Final Expression○ Expand

Finally, substitute the calculated values of $μ$ and $σ$ into the expression $10 (μ + σ)$ .

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

✦ STEM Console utilizes AI models to generate step-by-step explanations and math clues. AI can make mistakes.

Maths Question 9 – JEE-MAIN 2025

Clue 1 of 3: Understand the impact of correction

🧠 Full Solution Path