Find the change score that is 2.2 standard deviations below the mean. (Round your answer to one decimal place.)

Question

One hundred teachers attended a seminar on mathematical problem-solving. The attitudes of a representative sample of 12 of teachers were measured before and after the seminar. A positive number for change in mentality indicates that a teacher's attitude toward math became more positive. The twelve change scores are as follows.

3; 8; −1; 2; 0; 4; −3; 1; −1; 5; 4; −2

Answer 1

Standard Deviation, or S.D. = sqrt(summation of absolute value | x - mu |² / N)

where x: a value in your dataset

N: # of values in your dataset (12 in this sample)

mu is the mean of values in the data set.

Your question asks you to find the change score, i.e., the score indicating the change in mentality of teacher's attitude toward math from before to after undergoing the seminar on mathematical problem-solving, that is 2.2 standard deviations below the mean. Thus, the 2.2 standard deviations below the mean may correspond to a S.D. of -2.2. You already know N is 12 from the word problem question stem. The mean can be found by summing all the data points dividing by 12.

Notably, the summation term in the numerator under the square root requires finding the absolute value of x - mu (squared) 12 times since there are 12 different datapoints, which is a lot. This will look like | 3 - mu |² + | 8 - mu |² + | -1 + mu |² + ... so on and so forth.

mu is the mean of the dataset, which can be found via:

(3+8+(-1)+2+0+4+(-3)+1+(-1)+5+4+(-2)) / 12 = 20/12 ~ 10/6 ~ 1 4/6 or 1 2/3 ~ 1.6667

You would need a calculator to solve for the standard deviation of the data set, then use that number to find what value corresponds to 2.2 standard deviations below the mean (where the mean is ~1.6667). Let's say the standard deviation comes out to be 0.5, then two standard deviations below the mean would be 2(0.5) = 1 below 1.6667 which is 0.6667. Hope this helps!