Standard Deviation Calculator
! Let's say we have a dataset that represents the scores of students on a math test. The dataset contains the following scores:
82, 75, 90, 68, 88
To calculate the standard deviation, follow these steps:
Step 1: Find the mean (average) of the dataset. To find the mean, add up all the scores and divide the sum by the total number of scores: (82 + 75 + 90 + 68 + 88) / 5 = 403 / 5 = 80.6
Step 2: Subtract the mean from each score and square the result. Subtract the mean (80.6) from each score, and then square the difference: (82 - 80.6)^2 = 1.6^2 = 2.56 (75 - 80.6)^2 = (-5.6)^2 = 31.36 (90 - 80.6)^2 = 9.4^2 = 88.36 (68 - 80.6)^2 = (-12.6)^2 = 158.76 (88 - 80.6)^2 = 7.4^2 = 54.76
Step 3: Find the mean of the squared differences. Add up all the squared differences and divide the sum by the total number of scores: (2.56 + 31.36 + 88.36 + 158.76 + 54.76) / 5 = 335.8 / 5 = 67.16
Step 4: Take the square root of the mean of squared differences. Take the square root of the result from Step 3 to get the standard deviation: √67.16 ≈ 8.19
So, the standard deviation of the scores is approximately 8.19. The standard deviation measures the amount of variation or dispersion in the dataset. In this example, a higher standard deviation indicates that the scores are more spread out from the mean, while a lower standard deviation suggests that the scores are closer together.
Let's say we have a set of numbers representing the scores of students in a class on a certain test. Here are the scores:
85, 90, 92, 88, 95
To find the standard deviation of these scores, we follow these steps:
Calculate the mean (average) of the scores. Mean = (85 + 90 + 92 + 88 + 95) / 5 = 90
Calculate the difference between each score and the mean. Differences = (85 - 90, 90 - 90, 92 - 90, 88 - 90, 95 - 90) = (-5, 0, 2, -2, 5)
Square each difference. Squares = (-5)^2, 0^2, 2^2, (-2)^2, 5^2 = (25, 0, 4, 4, 25)
Calculate the mean of the squared differences. Mean of Squares = (25 + 0 + 4 + 4 + 25) / 5 = 10.8
Take the square root of the mean of the squared differences. Standard Deviation = sqrt(10.8) ≈ 3.29
Therefore, the standard deviation of the scores is approximately 3.29. The standard deviation represents the spread or variability of the scores around the mean. In this example, it indicates how much the individual scores differ from the average score of 90.
