5.2. Calculating the Standard Deviation http://www.ck12.org
standard deviations from the mean. There is a value for the standard deviation that tells you how big your steps must
be to move from one tile to the other. This value can be calculated for a given set of data and it is added three times
to the mean for moving to the right and subtracted three times from the mean for moving to the left. If the mean of
the tiles was 65 and the standard deviation was 4, then you could put numbers on all the tiles.
For normal distribution, 68% of the data would be located between 61 and 69. This is within one standard deviation
of the mean. Within two standard deviations of the mean, 95% of the data would be located between 57 and 73.
Finally, within three standard deviations of the mean, 99.7% of the data would be located between 53 and 77. Now
let’s see what this entire explanation means on a normal distribution curve.
Now it is time to actually calculate the standard deviation of a set of numbers. To make the process more organized,
it is best to use a table to record your work. The table will consist of three columns. The first column will contain
the data and will be labeledx. The second column will contain the differences between the data value of the mean
of the data. This column will be labelled(x−x ̄). The final column will contain the square of each of the values in
the second column.(x−x ̄)^2.
To find the standard deviation you subtract the mean from each data score to determine how much the data varies
from the mean. This will result in positive values when the data point is greater than the mean and in negative values
when the data point is less than the mean.
If we continue now, what would happen is that when we sum the variations (Data –Mean(x−x ̄)column both
negative and positive variations would give a total of zero. The sum of zero implies that there is no variation in
the data and the mean. In other words, if we were conducting a survey of the number of hours that students watch
television in one day, and we relied upon the sum of the variations to give us some pertinent information, the only
thing that we would learn is that all students watch television for the exact same number of hours each day. We know
that this is not true because we did not receive the same answer from every student. In order to ensure that these
variations will not lose their significance when added, the variation values are squared prior to adding them together.
What we need for this normal distribution is a measure of spread that is proportional to the scatter of the data,
independent of the number of values in the data set and independent of the mean. The spread will be small when the
data values are close but large when the data values are scattered. Increasing the number of values in a data set will
increase the values of both the variance and the standard deviation even if the spread of the values is not increasing.
These values should be independent of the mean because we are not interested in this measure of central tendency
but rather with the spread of the data. For a normal distribution, both the variance and the standard deviation fit the
above profile and both values can be calculated for the set of data.
To calculate the variance(σ^2 )for a set of normally distributed data: