How to calculate deviation, variance and standard deviation

Dispersion and variation

Dispersion and variation are words that are used to describe the spread of values in a given set of data.

Before you proceed, I recommend you check this post out, it's a prerequisite.

Consider the set of examination marks of ten students in the  image below



The marks are the same. They do not vary. No dispersion.

Check the image below



You can see that the marks are not the same. They vary. Their mean is 60. Some of the marks are greater than 60 while others are less. This set of marks has a greater variation or dispersion than those in table 1

Also, consider this table which shows a third set of marks of ten students.



Although the mean is also 60. Don't know how to calculate mean? click here. However, it is clear that the marks are more varied than those in table 2. Thus, the marks here have a greater dispersion than the other marks.

Let's talk about range.

Range

The range of a set of numbers is the difference between the largest and the smallest numbers.

Example 1

Find the range of the following set of marks:

          79, 60, 52, 34, 58, 60.

Solution

To get the range, arrange the set in descending order.

          79, 60, 60, 58, 52, 34

Now, subtract the last number from the first number.

           = 79 - 34

Range   = 45

NB: Range is the subtraction of the lowest value from the highest value.

What's the range of this set of marks?

          48, 56, 57, 59, 69, 60, 88

Deviation from the mean.

If the mean of a distribution is subtracted from any value in the distribution, the result is called the deviation of the value from the mean. For example, the mean of the scores in the table below is 60. The deviation of the scores from the mean are:

        +5, +2, +2, +1, +1, 0, 0, -1, -2, -8

This means that the difference of each score from the mean (60).

Another thing is that the sum of these deviations is 0 (zero). For any distribution, the sum of the deviation is always zero.

Now that we understand deviation, let's talk about variance.

Variance

Variance or mean squared deviation is a single value that gives a convenient measure of dispersion or spread of a distribution.

Variance is the mean of the squares of the deviations from the mean of the distribution. Therefore, to find the variance for a set of n numbers:

Step 1: Calculate the mean, m. click here to know-how

Step 2: Find the deviation, d, of each number, x, from the mean (d = x - m)

Step 3: Square the deviation and find the sum £d²/n




Where £ (pronounced sigma) is short for the sum of.

Example

Calculate the variance of these marks

Solution

The mean is 60.

Deviations from the mean are:

       +5, +2, +2, +1, +1, 0, 0, -1, -2, -8

Variance = [(+5)² + (+2)² + (+2)² + (+1)² + (+1)² + (0)² + (0)² + (-1)² + (-2)² + (-8)²]/10

= (25+4+4+1+0+0+1+4+64)/10

= 104/10

= 10.4

Now that we understand this, let's move to standard deviation.

Standard Deviation

Standard deviation is the most helpful and most reliable measure of dispersion. It is derived from variance:

Standard deviation = ✓variance.



Basically, S.D is the square root of variance.

Example

Calculate the standard deviation of the set of numbers
                    2,5,6,7,3,8,9,8

Solution

Mean = 6

Let's find variance through deviation using our formula



Where x represents the number in the given set, m the mean and d the deviation from the mean.



The sum of d² = 44

Variance = 44/8
            =  5.5

Standard deviation is the square root of variance.

Therefore,
S.D = ✓(5.5
      = 2.35

P.A

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