To find the standard deviation we make a standard deviation chart. We shall try learning how to make a standard deviation chart using an example.
Example: The runs scored by a cricket player in 6 innings are 60, 45, 25, 40, 70 and 30. Find the standard deviation.
Solution: First we find the mean of the data given to us. We see that the number of observations is 6. Therefore our n = 6.
The sum of observations = ∑x_i = x_1 + x_2 + x_3 + x_4 + x_5 + x_6
∑x_i = 60 + 45 + 25 + 40 + 70 + 30 = 270
Therefore the mean = x ̅ = ∑x_i/n = 270/6 = 45
Next we make the chart for the standard deviation calculation. The first column of the chart would have the observations x_i. The second column would be x_i - x ̅ for each of the observations. The third column would be (x_i - x ̅)^2 for each of the observations. The table would look as follows for this problem:
x_i x_i - x ̅ (x_i - x ̅)^2
60 60-45 = 15 (15)^2 = 225
45 45-45 = 0 (0)^2 = 0
25 25-45 = -20 (-20)^2 = 400
40 40-45 = -5 (-5)^2 = 25
70 70-45 = 25 (25)^2 = 625
30 30-45 = -15 (-15)^2 = 225
Next we sum up the third column so we have:
∑(x_i - x ̅)^2 = 225 + 0 + 400 + 25 + 625 + 225 = 1500
Lastly we use the formula to find the standard deviation which is like this:
Standard deviation = σ = sqrt[(∑(x_i - x ̅)^2)/n] = sqrt[1500/6] = sqrt [250] = 15.81
The above steps remain the same for any number of observations.
Standard deviation graph: A standard deviation graph would look like the picture below. This is a typical graph for a normal distribution. A normal distribution is symmetric about the mean. In a normal distribution, about 68% of the observations would lie between one standard deviation from the mean. That means 68% of the observations would lie between the range x ̅ – σ to x ̅ + σ. 95% of the data would lie within 2 standard deviations of the mean, that is in the range between x ̅ – 2σ to x ̅ + 2σ. 99.7% of the data would lie with 3 standard deviations of the mean.
My forthcoming post is on Frequency Table will give you more understanding about Statistics
