13Dsd

= 13D Standard Deviation =


 * The ** standard deviation ** is a measure of the spread of the data
 * It is a more sophisticated measure than Interquartile Range
 * It is widely used in business and industry
 * It is linked with the mean (in the same way IQR is linked with the median)
 * The symbol for standard deviation is s .. {(the greek letter __sigma__ (lower case)}

Recall that the **__mean__** of a set of data is given by:

math \\ . \qquad \bar{x} = \dfrac{ \sum{x} }{n} \\. \\ . \qquad \qquad \text{where } n = \text{number of values} \\. \\ . \qquad \qquad \text{and } \sum{x} = \text{sum of all values} math

The formula for __**standard deviation**__ is:

math . \qquad \sigma = \sqrt{ \dfrac{ \sum \big( \bar{x} - x \big)^2}{n} } math


 * Standard Deviation by hand **

To calculate the standard deviation by hand:
 * 1) calculate the mean
 * 2) make a list of the differences between each value and the mean
 * 3) square each entry in the list from step 2
 * 4) sum all the results from step 3
 * 5) divide by n (the number of values)
 * 6) take the square root of the result


 * Example 1 **

Find the standard deviation of the following values:




 * Solution:**

... ... ** 1. ** .. ** Calculate the mean **

math . \qquad \bar{x} = \dfrac{ \sum{x} }{n} = \dfrac{181}{10} = 18.1 math

... ... ** 2. ** .. ** Make a list of the differences between each value and the mean **

... ... First entry = 18.1 – 12 = 6.1



... ... ** 3. ** .. ** Square each entry from the list in step 2 ** ... ... First entry = 6.1 2 = 37.21



... ... ** 4. ** .. ** Sum all the results from step 3 **

math . \qquad \sum \big( \bar{x} - x \big)^2 = 148.90 math

... ... ** 5. ** .. ** Divide by n ** (the number of values)

math . \qquad \dfrac{148.90}{10} = 14.89 math

... ... ** 6. ** .. ** Take the square root **

math . \qquad \sigma = \sqrt{14.89} = 3.86 math

So, for these values
 * mean = 18.1
 * standard deviation = 3.86

Sample Standard Deviation


 * Provided a randomly selected sample is sufficiently large (minimum square root of the population)
 * It has been found that
 * the mean of the sample is a good estimate for the mean of the population
 * the standard deviation of the sample will be slightly larger than the s.d. of the population.
 * For that reason we use a slightly different formula when using a sample to estimate the s.d. of a population.

math . \qquad \sigma(s) = \sqrt{ \dfrac{ \sum \big( \bar{x} - x \big)^2}{n-1} } math


 * The only difference is that at step 5 we divide by (n – 1) instead of dividing by n.


 * Finding Standard Deviation using Technology **


 * Because of the amount of arithmetic involved in finding a standard deviation we usually use some form of technology to calculate it.
 * You can use
 * Any spreadsheet software
 * Any CAS calculator
 * Most scientific calculators


 * It is important to ensure the correct function is chosen
 * If you want the standard deviation of the data, look for s n
 * If the data is from a sample and you want an estimate of the population, look for s n–1.


 * The actual steps to enter the data and find the statistics vary quite a lot between calculators. You will need to consult the operating manual for your particular calculator.

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