The **standard deviation** is a measure of the amount of variation or dispersion in a data set. A low standard deviation means the data points are close to the mean, while a high standard deviation means the data points are spread out over a wide range of values.
Follow these steps to calculate the standard deviation:
Step 1: Find the mean (average) of the data set. Step 2: Subtract the mean from each data point and square the result. Step 3: Find the average of these squared differences. Step 4: Take the square root of this average to get the standard deviation.
Example 1: Find the standard deviation of the numbers 4, 8, 6, 5, 3.
Step 1: The mean is (4 + 8 + 6 + 5 + 3) / 5 = 26 / 5 = 5.2. Step 2: Subtract the mean from each number: (4 - 5.2), (8 - 5.2), (6 - 5.2), (5 - 5.2), (3 - 5.2). Step 3: Square the differences: (-1.2)^2, (2.8)^2, (0.8)^2, (-0.2)^2, (-2.2)^2. Step 4: Find the average of these squares: (1.44 + 7.84 + 0.64 + 0.04 + 4.84) / 5 = 14.8 / 5 = 2.96. Step 5: Take the square root of 2.96: √2.96 ≈ 1.72. Standard Deviation ≈ 1.72
Example 2: Find the standard deviation of the numbers 12, 15, 18, 14, 16.
Step 1: The mean is (12 + 15 + 18 + 14 + 16) / 5 = 75 / 5 = 15. Step 2: Subtract the mean from each number: (12 - 15), (15 - 15), (18 - 15), (14 - 15), (16 - 15). Step 3: Square the differences: (-3)^2, (0)^2, (3)^2, (-1)^2, (1)^2. Step 4: Find the average of these squares: (9 + 0 + 9 + 1 + 1) / 5 = 20 / 5 = 4. Step 5: Take the square root of 4: √4 = 2. Standard Deviation = 2
Find the standard deviation of the following numbers:
10, 12, 9, 14, 13
Hint: Follow the steps outlined above for calculating the standard deviation.