3.3 4.87 -1.57 – ≈ 4.5 4.87 -0.37 – ≈ 4.7 4.87 -0.17 – ≈ Notice that some of the differences are negative and some are positive. In fact, if we were to add the distances now, they would add up to approximately zero. (Remember the mean is the balancing point.) The negative numbers are a problem. To average the distances we need to get rid of the negatives. There are two ways to deal with negative numbers in mathematics, absolute value or squaring the numbers. Absolute value can have issues with calculus applications, so early statisticians preferred to square all the numbers and then eventually take a square root. Squares of the distances (1.33)^2 1.7689 ≈ (0.23)^2 0.0529 ≈ (2.53)^2 6.4009 ≈ (-1.97)^2 3.8809 ≈ (-1.57)^2 2.4649 ≈ (-0.37)^2 0.1369 ≈ (-0.17)^2 0.0289 ≈ Now we will add up all the squared distances and calculate the of Squares” . This is a “Sum very important technique in statistics and occurs in many different applications. Sum of Squares 1.7689 + 0.0529 + 6.4009 + 3.8809 + 2.4649 + 0.1369 + 0.0289 14.6814 ≈ ≈ We now want to take an average of the sum of squares. When dealing with spread, we will divide by one less than the sample size. This is called of freedom” in statistics. “degrees Therefore, we will divide by n-1 instead of the frequency n. There are seven numbers in the data set, so we will divide by 7 1 or 6. Then we will take the square root of the answer. –
3.3 – 4.87 ≈ -1.57 4.5 – 4.87 ≈ -0.37 4.7 – 4.87 ≈ -0.17 Notice that some of the differences are negative and some are positive. In fact, if we were to add the distances now, they would add up to approximately zero. (Remember the mean is the balancing point.) The negative numbers are a problem. To average the distances we need to get rid of the negatives. There are two ways to deal with negative numbers in mathematics, absolute value or squaring the numbers. Absolute value can have issues with calculus applications, so early statisticians preferred to square all the numbers and then eventually take a square root. Squares of the distances (1.33)^2 ≈ 1.7689 (0.23)^2 ≈ 0.0529 (2.53)^2 ≈ 6.4009 (-1.97)^2 ≈ 3.8809 (-1.57)^2 ≈ 2.4649 (-0.37)^2 ≈ 0.1369 (-0.17)^2 ≈ 0.0289 Now we will add up all the squared distances and calculate the “Sum of Squares” . This is a very important technique in statistics and occurs in many different applications. Sum of Squares ≈ 1.7689 + 0.0529 + 6.4009 + 3.8809 + 2.4649 + 0.1369 + 0.0289 ≈ 14.6814 We now want to take an average of the sum of squares. When dealing with spread, we will divide by one less than the sample size. This is called “degrees of freedom” in statistics. Therefore, we will divide by n-1 instead of the frequency n. There are seven numbers in the data set, so we will divide by 7 – 1 or 6. Then we will take the square root of the answer.
Ch3_Intro_to_data_analysis_1st_ed_Oct_2017.docx