The variance of the numbers 8, 21, 34, 47, \dots, 320, is:
Official Solution
Correct Option: (1)
To find the variance of the sequence of numbers 8, 21, 34, 47, ..., 320, we start by identifying it as an arithmetic sequence. The first term , the common difference , and the last term .
Step 1: Determine the Number of Terms (n)
The nth term of an arithmetic sequence is given by:
Setting :
Step 2: Calculate the Mean ( )
The mean is:
The sum of an arithmetic sequence is calculated by:
Thus:
The mean is:
Step 3: Calculate the Variance ( )
Variance is defined as:
For an arithmetic sequence, the variance formula simplifies, and we can calculate using:
Substituting in the values:
The calculated variance is 8788.
02
PYQ 2026
medium
mathematicsID: jee-main
The mean deviation about the mean for the data 56
is equal to:
1
2
3
4
Official Solution
Correct Option: (3)
Step 1: Understanding the Question:
We are given a frequency distribution table and asked to calculate the mean deviation about the mean. Step 2: Key Formula or Approach:
The process involves three main steps:
1. Calculate the mean ( ) of the distribution: .
2. Calculate the absolute deviations from the mean: .
3. Calculate the mean deviation (M.D.) about the mean: M.D. = . Step 3: Detailed Explanation:
Let's organize the calculations in a table. Step 3.1: Calculate the Mean ( )
First, find the total number of observations, .
.
Next, find .
.
Now, calculate the mean:
. Step 3.2: Calculate the Mean Deviation
Now we build a table to calculate , with .
From the table, . Finally, calculate the mean deviation:
M.D. = .
Simplify the fraction:
M.D. = . Step 4: Final Answer:
The mean deviation about the mean is .
