How to Calculate Standard Deviation How to Calculate Standard Deviation The Standard deviation is the concept of statistics and probability theory that describes the quantity of changes that exists in the average (mean, or expected value). Generally two types of deviation we study first one is low standard deviation that denotes the data points are very close to the point and second one is high standard deviation that denotes the data points are very far to the mean or circulate out over a broad area of values. The standard deviation is the square root for the random variable, statistical populations, data set or probability distribution. Most important property of the standard deviation is that, unlike variance, standard deviation is specify in the same units as the data. Standard deviation is very useful in finance, science, geometric representation and in many more. Although there are various types for measuring the Variations or dispersions property of deviation such as range, mean deviation, standard deviation and quartile deviation. Standard deviation is denoted by the sigma sine Ďƒ. In statistics the standard deviation is the measure of dispersion. Calculating Standard Deviation is not a tough task, only we have to follow the formula and some steps for calculating the standard deviation. Know More About Z score Table Math.Tutorvista.com
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Step 1 : - First of all find the arithmetic mean Step 2 : - Then determine the deviation of every element from the mean. Step 3 : - Square the obtained deviations and add them. Step 4 : - Now we get ∑(x – x')2 Step 5 : - After that divide the obtained sum by the total number of elements or items. Step 6 : - And at last take the square root of obtained result, This is the standard deviation. Lets take some examples that will show the whole calculating process of standard deviation. Method 1 Example 1 : - Find the standard deviation of 5, 6, 7, 8, 9 Solution : Step 1 : - First of all we have to find the arithmetic mean x M (x – M) (x – M)2 57-24
67-11
7700
8711
9724 Step 2 : - Find the sum of (x – M)2 that is 4 + 1 +0 + 1 + 4 = 10 Learn More z score chart Math.Tutorvista.com
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Step 3 : - We have total number of items 5 so n = 5 Step 4 : - Now find the standard deviation from the standard deviation formula that is S = √( ∑(x – M)2 / n - 1) S = √10 / 4 = 2.5 S = √2.5 S = 1.58113 Method 2 Example 2 : - Find the standard deviation 5, 6, 7, 8, 9 Solution : - First of all we have to square each of the scores. x x2
5 25
6 36
7 49
8 64
9 81
By using the formula S = √(∑ (x2) - (∑(x). ∑(x / n)) / n – 1)
= √(255 – (35.(35 / 5)) / 5 - 1)
= √(255 - (1225 / 5) / 4
= √(255 - 245 / 4)
= √(10/4) = √2.5
S = 1.58113
so the standard deviation for the above example is 1.58113
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