How to compare standard deviation

How to compare standard deviation of two sets

Consider two sets A and B as follows

  A = {2, 4, 6, 8, 10}

B = {102, 104, 106, 108, 110}

  Which of the two sets has a greater Standard Deviation?

On careful observation, the two sets have a same S.D.

The reason:

 Standard deviation, as its name connotes, does not depend on the magnitude of the numbers.  

Whether the terms are small or big will not affect S.D. 

What matters is the deviations of each term from its average, that’s all!

Now, the terms in the respective sets are at same distances or differences or statistically speaking, Deviations from their respective averages.

Therefore, the standard deviation of the two sets are same.

First rule for how to to compare  standard deviation.

Speaking properly, S.D., as its name implies, will depend on deviations of terms from their Average.

Still, to compare standard deviation of two sets fast, we can check deviations of “terms from each other” instead of “terms from average”

In the two sets

A = {2, 4, 6, 8, 10} and B = {102, 104, 106, 108, 110}

Just check if the terms are equally spaced from each other in the two sets.

We see that the terms are indeed equally spaced from each other in the two sets,

having as a consequence, therefore a same S.D., in spite of different magnitudes of the terms.

Since, S.D. of set A is 2√2, so, S.D. of set B is also 2√2

Example 2

Which of the two sets below has a greater S.D.?

A = {2, 4, 6, 8, 10}
B = {3, 6, 9, 12, 15}

Solution:

Without calculating the standard deviation of the two sets, just check, in which set are terms close to or separated from each other.

The terms in set A are closer to each other than what they are in set B.

So, we see elements in A are dense but scattered in set B.

We prefer to say deviated to scattered to sound statistical in language.

Standard deviation is greater in the set in which elements are more deviated.

Therefore, set B has greater S.D. as its elements are more deviated from each other.

The above example leads us to the following rule

  First rule of how to compare standard deviation of two sets
  Standard deviation is high if terms are deviated from each other and low if they are close to each other.  

Consider the following three sets A, B, C:

  A = {1, 2, 3, 4, 5}
  B = {2, 4, 6, 8, 10}
  C = {3, 6, 9, 12, 15} 

Check deviations of the terms from each other in each set.

  We see the following in the three sets 

Deviation of terms in set A <

 Deviation of terms in set B <

Deviation of terms in set C.

Therefore, we have the order below:

S.D. of set A < S.D. of set B < S.D of set C

Second Rule of how to Compare standard deviation of two sets
If every term is either increased or decreased by a same number, then Standard Deviation does not change.

If A = {2, 4, 6, 8, 10} and

B = {102, 104, 106, 108, 100}

Which set has a greater S.D.?

Actually, both have a same S.D.

Every element of set A is increased by a same number, 100 to yield elements of set B.

Since the original space between the terms in set A is retained in set B even after increasing every term by a same number, therefore Standard deviation does not change.

Therefore, S.D. of set B is same as that of set A.

Note:

If every element of set B is decreased by the same number 100, then elements of set A are regained resulting in no change in the S.D.

Third rule of how to compare Standard deviation of two sets

Consider the two sets

A = {2, 4, 6, 8, 10}
B = {4, 8, 12, 16, 20}

From the first rule on S.D., set B will have a greater S.D.

But what will the exact value of S.D. be?

The elements of set B can be formed by multiplying each element of set A by 2.

This will change by widening the original space between the elements and therefore will increase S.D.

It must be noted that S.D. will increase proportionately, i.e. it will also multiply by 2.

A similar change should be envisioned when every term is divided by a same number.