To find the standard deviation of a data set, follow these steps:
- Calculate the mean (average) of the data by adding all the values together and dividing by the number of values.
- Find each value's deviation from the mean by subtracting the mean from each data point.
- Square each deviation to make them positive.
- Sum all the squared deviations.
- Calculate the variance by dividing the sum of squared deviations by:
- n−1n-1n−1 if you have a sample (where nnn is the number of data points),
- or by NNN if you have the entire population.
- Take the square root of the variance to get the standard deviation.
This process can be summarized by the formula for sample standard deviation:
s=∑(xi−xˉ)2n−1s=\sqrt{\frac{\sum (x_i-\bar{x})^2}{n-1}}s=n−1∑(xi−xˉ)2
where xix_ixi are the data points, xˉ\bar{x}xˉ is the mean, and nnn is the sample size
. For example, if you have data points 46, 69, 32, 60, 52, and 41:
- Mean = (46 + 69 + 32 + 60 + 52 + 41) / 6 = 50
- Deviations: -4, 19, -18, 10, 2, -9
- Squared deviations: 16, 361, 324, 100, 4, 81
- Sum of squares = 886
- Variance = 886 / (6 - 1) = 177.2
- Standard deviation = 177.2≈13.31\sqrt{177.2}\approx 13.31177.2≈13.31
You can also use online calculators to compute standard deviation quickly by inputting your data
. This method applies whether you are calculating for a sample or a population, with the main difference being the divisor in the variance step.
