To get the standard deviation of a data set, follow these steps:
- Calculate the mean (average) : Add all the data points together and divide by the number of data points.
- Find each data point's deviation from the mean : Subtract the mean from each data point.
- Square each deviation : Multiply each deviation by itself to get positive values.
- Sum the squared deviations : Add all the squared deviations together.
- Calculate the variance :
- For a sample , divide the sum of squared deviations by n−1n-1n−1, where nnn is the sample size.
- For a population , divide by nnn.
- Take the square root of the variance : This gives the standard deviation.
The formula for the sample standard deviation sss is:
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 number of data points
. Alternatively, a computational formula can be used for easier calculation:
s=∑Xi2−(∑Xi)2nn−1s=\sqrt{\frac{\sum X_i^2-\frac{(\sum X_i)^2}{n}}{n-1}}s=n−1∑Xi2−n(∑Xi)2
Both formulas yield the same result
. In summary, standard deviation measures how spread out the data points are around the mean, with a higher standard deviation indicating greater variability
