Standard deviation for the following sets
and compare the differences.
first 7 #'s 57 61 57 57 58 57 61
second 7#'s 61 52 69 64 46 54 47
To find the standard deviation for a set of numbers, you can follow these steps:
1. Calculate the mean (average) of the set.
2. Subtract the mean from each number in the set.
3. Square each of the differences obtained from step 2.
4. Find the mean of the squared differences obtained from step 3.
5. Take the square root of the mean from step 4.
Now, let's calculate the standard deviation for the first set of numbers:
1. Mean: (57 + 61 + 57 + 57 + 58 + 57 + 61) / 7 = 58
2. Differences from the mean: (57 - 58), (61 - 58), (57 - 58), (57 - 58), (58 - 58), (57 - 58), (61 - 58) = -1, 3, -1, -1, 0, -1, 3
3. Squared differences: (-1)^2, (3)^2, (-1)^2, (-1)^2, (0)^2, (-1)^2, (3)^2 = 1, 9, 1, 1, 0, 1, 9
4. Mean of squared differences: (1 + 9 + 1 + 1 + 0 + 1 + 9) / 7 = 2.29 (rounded to two decimal places)
5. Square root of the mean: √2.29 = 1.51 (rounded to two decimal places)
The standard deviation for the first set of numbers is approximately 1.51.
Now, let's calculate the standard deviation for the second set of numbers using the same steps:
1. Mean: (61 + 52 + 69 + 64 + 46 + 54 + 47) / 7 = 56.71 (rounded to two decimal places)
2. Differences from the mean: (61 - 56.71), (52 - 56.71), (69 - 56.71), (64 - 56.71), (46 - 56.71), (54 - 56.71), (47 - 56.71) = 4.29, -4.71, 12.29, 7.29, -10.71, -2.71, -9.71
3. Squared differences: (4.29)^2, (-4.71)^2, (12.29)^2, (7.29)^2, (-10.71)^2, (-2.71)^2, (-9.71)^2 = 18.36, 22.12, 150.92, 53.05, 114.84, 7.34, 94.63
4. Mean of squared differences: (18.36 + 22.12 + 150.92 + 53.05 + 114.84 + 7.34 + 94.63) / 7 = 56.43 (rounded to two decimal places)
5. Square root of the mean: √56.43 = 7.51 (rounded to two decimal places)
The standard deviation for the second set of numbers is approximately 7.51.
Comparing the differences in the standard deviations, we can see that the first set of numbers has a smaller standard deviation (1.51) compared to the second set of numbers (7.51). This indicates that the numbers in the first set are more closely packed around the mean, while the numbers in the second set are more spread out.