Standard Deviation 

 Standard deviation is a widely used measurement of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the mean. A low standard deviation indicates that the data points tend to be very close to the mean, whereas high standard deviation indicates that the data are spread out over a large range of values.

Calculating Standard Deviation 

 
s_N = \sqrt{\frac{1}{N} \sum_{i=1}^N (x_i - \overline{x})^2}.

Definition provided by Wikipedia  

Example 

Consider an experiment that measures how far a ball rolls in 2 seconds. The results from this experiment are expressed in centimeters:

2,4,4,4,5,5,7,9 

these eight data points have a mean of 5 centimeters:

(2+4+4+4+5+5+7+9)/8 = 5

To calculate the standard deviation, first compute the difference of each data point from the mean, and square the result of each:

 Next compute the average of these values, and take the square root:

  2 centimeters is the standard deviation, this means that about 70% of all data that will be collected will fall 2 centimeters from the mean, 5 centimeters.  95% of all data will fall with in two standard deviations, 4 centimeters from the mean. 

Example provided by Wikipedia 

Error Bars 

What are error bars 

    Error bars are used on graphs to indicate the error, or uncertainty in a reported measurement. They give a general idea of how accurate a measurement is, or conversely, how far from the reported value the true (error free) value might be. Error bars often indicate one standard deviation of uncertainty, but may also indicate the standard error. These quantities are not the same and so the measure selected should be stated explicitly in the graph or supporting text.

   When graphing, the error bars should be equal to one standard deviation of uncertainty, and are critical to determine the range of the gradient.  Using the smallest and largest measured values, connecting the lowest possible value (the smallest measured value minus the uncertainty) with the highest value (the largest measured value plus the uncertainty) with a straight line you can measure the highest gradient value.  Using smallest measured value and add the uncertainty, and the largest measured value and subtracting the uncertainty, connecting those two point with a straight line will give you the lowest gradient value. 

Definition Provided by Wikipedia 

   Why do we put error bars on a graph