Final answer:
The standard deviation of the data set is calculated by first finding the mean, subtracting the mean from each score, squaring the differences, averaging these squared differences to get the variance, and then taking the square root of the variance. The standard deviation in this case is approximately 12.8.
Step-by-step explanation:
To find the standard deviation of the given data set, we first need to calculate the mean (average) of the numbers. Adding all the performance scores together, 83 + 87 + 90 + 92 + 93 + 100 + 104 + 111 + 115 + 121, we get a total of 996. Dividing this sum by the number of scores, which is 10, gives us a mean of 99.6.
Next, we subtract the mean from each of the individual scores and square the result for each:
(83 - 99.6)² = 277.44
(87 - 99.6)² = 159.04
(90 - 99.6)² = 92.16
(92 - 99.6)² = 57.76
(93 - 99.6)² = 43.56
(100 - 99.6)² = 0.16
(104 - 99.6)² = 19.36
(111 - 99.6)² = 129.96
(115 - 99.6)² = 237.16
(121 - 99.6)² = 457.44
We then add up all those squared differences:
277.44 + 159.04 + 92.16 + 57.76 + 43.56 + 0.16 + 19.36 + 129.96 + 237.16 + 457.44 = 1474.48
Now we divide this sum by the number of values minus one, which is 10 - 1 = 9. This gives us the variance:
1474.48 / 9 = 163.83
The standard deviation is the square root of the variance. Therefore:
Square root of 163.83 ≈ 12.8
The standard deviation of the data set is approximately 12.8.