Mean

We are given the initial mean and standard deviation for 100 candidates:
- Mean
- Standard deviation
However, two scores were misread:
- The score 25 was misread as 52.
- The score 35 was misread as 53.
Step 1: Correcting the Mean The initial total sum of the scores is given by: Now, we need to correct this sum. The error in the sum comes from the two misread scores:
- The misread score 52 should have been 25, so the difference is .
- The misread score 53 should have been 35, so the difference is .
Thus, the corrected sum of the scores is: The corrected mean is: Step 2: Correcting the Standard Deviation The formula for standard deviation is: where is the number of observations and are the individual data points. We know that the squared deviations for 25 and 35 with respect to the original mean (40) are:
- For score 25:
- For score 35:
Similarly, the squared deviations for the corrected scores are:
- For score 52:
- For score 53:
Now, we calculate the total sum of squared deviations for the corrected data. Initially, the total squared deviations would have been the same as for the original scores, with the adjustments made for the two incorrect data points: The corrected standard deviation can then be calculated by plugging the new total squared deviations into the standard deviation formula, and the result comes out to be approximately 14.97. Thus, the corrected standard deviation is . Final Answer: The corrected mean and standard deviation are:

Let the two numbers be and . The arithmetic mean and geometric means between and are defined as follows:
Step 1: Arithmetic Mean (AM) Formula
The arithmetic mean (AM) of two numbers and is given by: Step 2: Geometric Means (GM) Formula
The geometric means inserted between and are and , and the formula for the geometric mean between two numbers is: Step 3: Use of Arithmetic Mean and Geometric Mean Relationship Since and are geometric means, we know the relationship between the AM and GM: Step 4: Compute
We are asked to find .
Using the identity for the sum of cubes: Since , this simplifies to: Now, express in terms of and . Using the relation between AM and GM, we get: Thus, the correct answer is:

Let the two numbers be and . The arithmetic mean and geometric means between and are defined as follows:
Step 1: Arithmetic Mean (AM) Formula
The arithmetic mean (AM) of two numbers and is given by: Step 2: Geometric Means (GM) Formula
The geometric means inserted between and are and , and the formula for the geometric mean between two numbers is: Step 3: Use of Identity for Sum of Cubes
We are asked to find the value of .
Since , we can use the identity for the sum of cubes: Since , this simplifies to: Step 4: Express in Terms of and
From the AM-GM inequality, we know that . So, we compute : Step 5: Final Expression
We now have the expression for , which is: Thus, the correct answer is: