77, 45, 72, 56, 98, 60, 70, 89, 50, 80, 95, 70, 34, 77, 65, 70, 70, 82, 86, 86
Part 1: Find the standard deviation of the scores. Please round the final answer to two decimal places. Be sure to show all work.
To find the standard deviation of the scores, we need to follow these steps:
1. Find the mean of the scores.
2. Calculate the difference between each score and the mean.
3. Square each difference.
4. Find the mean of the squared differences.
5. Take the square root of the mean squared differences to get the standard deviation.
Let's find the standard deviation step by step:
Step 1: Find the mean of the scores.
To find the mean, add up all the scores and divide by the number of scores.
Mean = (77 + 45 + 72 + 56 + 98 + 60 + 70 + 89 + 50 + 80 + 95 + 70 + 34 + 77 + 65 + 70 + 70 + 82 + 86 + 86) / 20
Mean = 1440 / 20
Mean = 72
Step 2: Calculate the difference between each score and the mean.
For each score, subtract the mean from the score.
Difference for each score:
77 - 72 = 5
45 - 72 = -27
72 - 72 = 0
56 - 72 = -16
98 - 72 = 26
60 - 72 = -12
70 - 72 = -2
89 - 72 = 17
50 - 72 = -22
80 - 72 = 8
95 - 72 = 23
70 - 72 = -2
34 - 72 = -38
77 - 72 = 5
65 - 72 = -7
70 - 72 = -2
70 - 72 = -2
82 - 72 = 10
86 - 72 = 14
86 - 72 = 14
Step 3: Square each difference.
Square each difference calculated in the previous step.
Squared differences:
5^2 = 25
(-27)^2 = 729
0^2 = 0
(-16)^2 = 256
26^2 = 676
(-12)^2 = 144
(-2)^2 = 4
17^2 = 289
(-22)^2 = 484
8^2 = 64
23^2 = 529
(-2)^2 = 4
(-38)^2 = 1444
5^2 = 25
(-7)^2 = 49
(-2)^2 = 4
(-2)^2 = 4
10^2 = 100
14^2 = 196
14^2 = 196
Step 4: Find the mean of the squared differences.
Add up all the squared differences and divide by the total number of scores.
Mean squared differences = (25 + 729 + 0 + 256 + 676 + 144 + 4 + 289 + 484 + 64 + 529 + 4 + 1444 + 25 + 49 + 4 + 4 + 100 + 196 + 196) / 20
Mean squared differences = 6732 / 20
Mean squared differences = 336.6
Step 5: Take the square root of the mean squared differences to get the standard deviation.
Standard deviation = √(336.6)
Standard deviation ≈ 18.34 (rounded to two decimal places)
Therefore, the standard deviation of the scores is approximately 18.34.