Select the list of data items that are included within one standard deviation of the mean from this data set.
Data set: 34,45,50,56,60,65,70,70,70,70,72,77,77,80,82,86,86,89,95,98
A. 56,60,65,70,70,70,70,72,77,77,80,82,86,86
B. 34,45,50,89,95,98
C. 56,60,65,70,72,77,80,82,86
D.34,45,50,56,60,65,70,70,70,70,72,77,77,80,82,86,86,89,95,98
The data items included within one standard deviation of the mean can be calculated by finding the mean and standard deviation of the data set.
Mean = (34 + 45 + 50 + 56 + 60 + 65 + 70 + 70 + 70 + 70 + 72 + 77 + 77 + 80 + 82 + 86 + 86 + 89 + 95 + 98) / 20
Mean = 1519 / 20
Mean = 75.95
Next, calculate the standard deviation:
Step 1: Find the difference between each data item and the mean.
Step 2: Square each difference.
Step 3: Find the mean of the squared differences.
Step 4: Take the square root of the mean of the squared differences.
Step 1: Find the difference between each data item and the mean:
(34 - 75.95) = -41.95
(45 - 75.95) = -30.95
(50 - 75.95) = -25.95
(56 - 75.95) = -19.95
(60 - 75.95) = -15.95
(65 - 75.95) = -10.95
(70 - 75.95) = -5.95
(70 - 75.95) = -5.95
(70 - 75.95) = -5.95
(70 - 75.95) = -5.95
(72 - 75.95) = -3.95
(77 - 75.95) = 1.05
(77 - 75.95) = 1.05
(80 - 75.95) = 4.05
(82 - 75.95) = 6.05
(86 - 75.95) = 10.05
(86 - 75.95) = 10.05
(89 - 75.95) = 13.05
(95 - 75.95) = 19.05
(98 - 75.95) = 22.05
Step 2: Square each difference:
(-41.95)^2 = 1763.6025
(-30.95)^2 = 959.9025
(-25.95)^2 = 672.9025
(-19.95)^2 = 398.0025
(-15.95)^2 = 254.0025
(-10.95)^2 = 119.9025
(-5.95)^2 = 35.4025
(-5.95)^2 = 35.4025
(-5.95)^2 = 35.4025
(-5.95)^2 = 35.4025
(-3.95)^2 = 15.6025
(1.05)^2 = 1.1025
(1.05)^2 = 1.1025
(4.05)^2 = 16.4025
(6.05)^2 = 36.5025
(10.05)^2 = 101.0025
(10.05)^2 = 101.0025
(13.05)^2 = 169.2025
(19.05)^2 = 363.2025
(22.05)^2 = 487.2025
Step 3: Find the mean of the squared differences:
(1763.6025 + 959.9025 + 672.9025 + 398.0025 + 254.0025 + 119.9025 + 35.4025 +35.4025 + 35.4025 + 35.4025 + 15.6025 + 1.1025 + 1.1025 + 16.4025 + 36.5025 + 101.0025 + 101.0025 + 169.2025 + 363.2025 + 487.2025) / 20
Mean = 5729.2575 / 20
Mean = 286.462875
Step 4: Take the square root of the mean of the squared differences:
Standard Deviation = √(286.462875)
Standard Deviation ≈ 16.936
Finally, we can determine which data items are within one standard deviation of the mean by looking for data items that are within 16.936 units of the mean.
Checking each data item:
56 - 16.936 ≤ 56 ≤ 56 + 16.936
39.064 ≤ 56 ≤ 72.936
True
60 - 16.936 ≤ 60 ≤ 60 + 16.936
43.064 ≤ 60 ≤ 76.936
True
65 - 16.936 ≤ 65 ≤ 65 + 16.936
48.064 ≤ 65 ≤ 81.936
True
70 - 16.936 ≤ 70 ≤ 70 + 16.936
53.064 ≤ 70 ≤ 86.936
True
70 - 16.936 ≤ 70 ≤ 70 + 16.936
53.064 ≤ 70 ≤ 86.936
True
70 - 16.936 ≤ 70 ≤ 70 + 16.936
53.064 ≤ 70 ≤ 86.936
True
70 - 16.936 ≤ 70 ≤ 70 + 16.936
53.064 ≤ 70 ≤ 86.936
True
72 - 16.936 ≤ 72 ≤ 72 + 16.936
55.064 ≤ 72 ≤ 88.936
True
77 - 16.936 ≤ 77 ≤ 77 + 16.936
60.064 ≤ 77 ≤ 93.936
True
77 - 16.936 ≤ 77 ≤ 77 + 16.936
60.064 ≤ 77 ≤ 93.936
True
80 - 16.936 ≤ 80 ≤ 80 + 16.936
63.064 ≤ 80 ≤ 96.936
True
82 - 16.936 ≤ 82 ≤ 82 + 16.936
65.064 ≤ 82 ≤ 98.936
True
86 - 16.936 ≤ 86 ≤ 86 + 16.936
69.064 ≤ 86 ≤ 102.936
True
86 - 16.936 ≤ 86 ≤ 86 + 16.936
69.064 ≤ 86 ≤ 102.936
True
89 - 16.936 ≤ 89 ≤ 89 + 16.936
72.064 ≤ 89 ≤ 105.936
True
95 - 16.936 ≤ 95 ≤ 95 + 16.936
78.064 ≤ 95 ≤ 111.936
True
98 - 16.936 ≤ 98 ≤ 98 + 16.936
81.064 ≤ 98 ≤ 114.936
True
So the list of data items included within one standard deviation of the mean is:
56,60,65,70,70,70,70,72,77,77,80,82,86,86
The correct answer is A. 56,60,65,70,70,70,70,72,77,77,80,82,86,86.