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 :)

First part is correct.

not familiar with B) part of your question.

To calculate the mean of the scaled marks, you need to apply the scaling equation y = 1.2x - 10 to each individual mark, and then calculate the mean of the scaled values.

First, let's calculate the mean of the original marks. We are given that the sum of the marks (Sum of x) is 3026, and there are 56 students (N).
Mean (x̄) = Sum of x / N
Mean (x̄) = 3026 / 56
Mean (x̄) = 54.04 (rounded to 2 decimal places)

Now, let's calculate the standard deviation of the original marks. We are given the sum of squared deviations (sum of (x - mean)^2) is 15378.
Standard Deviation (s) = √(sum of (x - mean)^2 / (N - 1))
Standard Deviation (s) = √(15378 / 55)
Standard Deviation (s) = 16.61 (rounded to 2 decimal places)

Now, let's move on to scaling the marks. We are given the scaling equation y = 1.2x - 10.

To find the mean of the scaled marks:
1. Substitute each original mark into the scaling equation and calculate the corresponding scaled mark.
Example: Suppose the original mark is 60.
Scaled mark = 1.2 * 60 - 10 = 62
2. Repeat this process for all 56 original marks.
3. Calculate the mean of the 56 scaled marks using the same formula as before.

Next, let's calculate the standard deviation of the scaled marks:
1. Substitute each original mark into the scaling equation and calculate the corresponding scaled mark.
2. Calculate the sum of squared deviations of the scaled marks from the mean of the scaled marks.
3. Divide the sum of squared deviations by (N - 1) and take the square root to calculate the standard deviation.

I hope this helps! Let me know if you have any further questions.

A) To calculate the mean and standard deviation of a sample, you need to follow these steps:

1. Mean (Average):
The mean of a sample is obtained by summing up all the values in the sample and then dividing it by the number of values (N).

Mean = sum of x / N

In this case, you have the sum of x, which is 3026, and the sample size N is 56. Therefore,

Mean = 3026 / 56 = 54.0357 (rounded to 2 decimal places)

So, your mean value 54.04 (2 dp) is correct.

2. Standard Deviation:
The standard deviation measures the spread or variability of the data points around the mean. To calculate it, you need to follow these steps:

i. Calculate the sum of squares of differences between each observation and the mean.

Sum of squares (SS) = sum of (x - mean)^2

In this case, you have the value for the sum of squares which is 15378.

SS = 15378

ii. Calculate the variance (s^2) by dividing the sum of squares by the sample size (N).

Variance = SS / N

Variance = 15378 / 56 = 274.9643 (rounded to 4 decimal places)

iii. Finally, take the square root of the variance to get the standard deviation (s).

Standard Deviation (s) = √(variance)

Standard Deviation (s) = √(274.9643) = 16.5805 (rounded to 4 decimal places)

So, your standard deviation value 16.72 (2 dp) is incorrect. The correct value is 16.58 (4 dp).

B) To find the mean and standard deviation of the scaled marks using the formula y = 1.2x - 10, follow these steps:

1. Calculate the new mean:
Multiply the original mean by 1.2 and then subtract 10 from it.

New Mean = (1.2 * Original Mean) - 10

New Mean = (1.2 * 54.0357) - 10

New Mean = 64.8428 - 10

New Mean = 54.8428

So, the mean of the scaled marks is 54.8428.

2. Calculate the new standard deviation:
Multiply the original standard deviation by the scaling factor (1.2) since the standard deviation is affected by linear transformations.

New Standard Deviation = Original Standard Deviation * Scaling Factor

New Standard Deviation = 16.5805 * 1.2

New Standard Deviation = 19.8966

So, the standard deviation of the scaled marks is 19.8966.

Therefore, the mean and standard deviation of the scaled marks are 54.8428 and 19.8966, respectively.