Statistics
The marks scored by a sample of 56 students are summarised by:
N=56
Sum of x =3026
sum of(x-the mean)^2 =15378
A) calculate the mean and the standard deviation
Mean I got - 54.04 (2.dp)
Standard deviation I got - 16.72 (2.dp)
Is this correct?
B) the formula y=1.2x-10 is used to scale the marks. Find the mean and the standard deviation of the scaled marks.
I am not sure how to do part B of this question. Please help.
Thank you :)
A) To calculate the mean and standard deviation, we can use the formulas:
Mean (µ) = Sum of x / N
Standard deviation (σ) = √(sum of (x - the mean)^2 / N)
Plugging in the given values:
N = 56
Sum of x = 3026
Sum of (x - the mean)^2 = 15378
Mean (µ) = 3026 / 56 = 54.04 (correct)
Standard deviation (σ) = √(15378 / 56) = √(274.963) ≈ 16.57 (correct to 2 decimal places, not 16.72 as you mentioned)
So, the mean is 54.04 and the standard deviation is 16.57.
B) To find the mean and standard deviation of the scaled marks, we need to apply the given formula y = 1.2x - 10 to each individual mark.
Mean of scaled marks (µ') = 1.2 * mean of original marks - 10
Substituting the mean of the original marks (54.04) into the formula:
Mean of scaled marks (µ') = 1.2 * 54.04 - 10 ≈ 54.84
To find the standard deviation of the scaled marks (σ'), we need to consider that scaling the marks does not change the spread or variability of the data, only the center. So, the standard deviation of the scaled marks remains the same:
Standard deviation of scaled marks (σ') ≈ 16.57
Therefore, the mean of scaled marks is approximately 54.84, and the standard deviation remains approximately 16.57.
A) To calculate the mean and standard deviation, follow these steps:
1. Mean:
The formula for the mean is:
mean (μ) = sum of x / N
Given:
Sum of x = 3026
N = 56
Apply the formula:
mean (μ) = 3026 / 56 = 54.036 (rounded to 2 decimal places)
Therefore, the mean is approximately 54.04.
2. Standard Deviation:
The formula for the standard deviation is:
standard deviation (σ) = sqrt(sum of (x - mean)^2 / (N - 1))
Given:
Sum of (x - mean)^2 = 15378
N = 56
Apply the formula:
standard deviation (σ) = sqrt(15378 / (56 - 1)) = sqrt(277.9636) = 16.664 (rounded to 3 decimal places)
Therefore, the standard deviation is approximately 16.664.
Your calculations are almost correct; however, there's a slight rounding error on the standard deviation. It should be 16.664, not 16.72.
B) To find the mean and standard deviation of the scaled marks using the formula y = 1.2x - 10, follow these steps:
1. Mean:
To find the new mean (μ'), substitute the formula for y into the original mean formula:
μ' = 1.2μ - 10
Given:
μ = 54.04
Apply the formula:
μ' = 1.2 * 54.04 - 10 = 54.048 (rounded to 3 decimal places)
Therefore, the mean of the scaled marks is approximately 54.048.
2. Standard Deviation:
To find the new standard deviation (σ'), substitute the formula for y into the original standard deviation formula:
σ' = 1.2 * σ
Given:
σ = 16.664
Apply the formula:
σ' = 1.2 * 16.664 = 19.997 (rounded to 3 decimal places)
Therefore, the standard deviation of the scaled marks is approximately 19.997.
Thus, the mean and standard deviation of the scaled marks are approximately 54.048 and 19.997, respectively.