All questions of Measures of Central Tendency for Commerce Exam

Given data set: 3, 8, 6, 7, 1, 6, 10, 6, 7, 2k, 5, 9, 7, and 13

Finding the mode:
The mode is the value that appears most frequently in a data set. In this case, the mode is given as 7.

To find the value of k:
We need to determine the value of k in the data set. To do this, we can use the information given and apply it to the concept of mode.

Analysis:
- The mode is 7, which means that 7 appears most frequently in the data set.
- Looking at the given data set, we see that 7 appears three times: 7, 7, and 7.
- We also see that there are two 6s in the data set.
- All other numbers appear only once.

Deducing the value of k:
- Since the mode is 7 and 7 appears three times in the data set, we can conclude that the value of k must be 7 as well to maintain the mode.
- If we substitute 7 for k in the data set, we get: 3, 8, 6, 7, 1, 6, 10, 6, 7, 7, 5, 9, 7, and 13.
- In this revised data set, the mode is still 7, as it appears four times, which is more than any other number.

Therefore, the value of k in the data set is 7.

Explanation of the correct answer:
- Option 'D' is the correct answer, which states that the value of k is 1.
- This answer is incorrect because we have determined that the value of k is 7, not 1.
- It seems there may be a mistake or confusion in the given options, as option 'D' does not match the correct answer based on the analysis and calculations above.

In conclusion, the correct value of k in the data set 3, 8, 6, 7, 1, 6, 10, 6, 7, 2k, 5, 9, 7, and 13 is 7, not 1 as stated in the given options.

The Correct Option is (C)
decile divide any series into 10 parts.

For calculating median it is required to set the data either in ascending order or descending order.

Given:
Let x be the mean of squares of the first n natural numbers.
Let y be the square of the mean of the first n natural numbers.
It is given that x/y = 55/42.

To Find:
The value of n.

Explanation:
Let's start by finding the values of x and y.

Finding the value of x:
The squares of the first n natural numbers are 1^2, 2^2, 3^2, ..., n^2.
The sum of these squares can be expressed as:
x = 1^2 + 2^2 + 3^2 + ... + n^2

Using the formula for the sum of squares, we can rewrite this as:
x = n(n + 1)(2n + 1)/6

Finding the value of y:
The mean of the first n natural numbers is the sum of the numbers divided by n.
The sum of the first n natural numbers can be expressed as:
sum = 1 + 2 + 3 + ... + n

Using the formula for the sum of an arithmetic series, we can rewrite this as:
sum = n(n + 1)/2

The mean is sum/n, so we can write the mean as:
mean = n(n + 1)/2n = (n + 1)/2

The square of the mean is:
y = (mean)^2 = [(n + 1)/2]^2 = (n + 1)^2/4

Calculating x/y:
Now, we can calculate x/y using the values we found for x and y:
x/y = (n(n + 1)(2n + 1)/6) / ((n + 1)^2/4)
Simplifying this expression:
x/y = (4n(n + 1)(2n + 1)) / (6(n + 1)^2)
x/y = (4n(2n + 1)) / (6(n + 1))
x/y = (2n(2n + 1)) / (3(n + 1))

Given that x/y = 55/42, we can set up the equation:
(2n(2n + 1)) / (3(n + 1)) = 55/42

Solving the equation:
Cross multiplying:
42 * 2n(2n + 1) = 55 * 3(n + 1)
84n(2n + 1) = 165(n + 1)
168n^2 + 84n = 165n + 165
168n^2 - 81n - 165 = 0

Factoring the quadratic equation:
(8n - 11)(21n + 15) = 0

Setting each factor to zero and solving for n:
8n - 11 = 0 or 21n + 15 = 0
8n = 11 or 21n = -15
n = 11/8 or n = -15/21

Since n represents the number of natural numbers, it cannot be negative. Therefore, n = 11/8 is not a valid solution.

Conclusion:
The valid solution for n