How to apply Mean and Standard Deviation?

What is Mean?

Mean of any given distribution is a measure of central tendency of that distribution. Mean is also known as Arithmetic Mean or Average Value or Expected Value.

For a data-set with discrete real values mean can simply be computed as sum of all the values in data-set divided by the number of values. Below is the formula

Arithmetic Mean Formula

Example: Mean of 1,1,2,3,4,4 is given by the expression = (1+1+2+3+4+4)/6

    Mean = μ  = = 2.5

What is Standard Deviation?

Standard Deviation of any given distribution is a measure of dispersion of that data-set. It tells you how spread out the distribution is with-respect-to its mean.

For a data-set with discrete real values standard-deviation can be computed by below formula.

Standard Deviation Formula

Example: Mean of 1,1,2,3,4,4 is given by the expression = (1+1+2+3+4+4)/6

Mean = μ  = = 2.5

Using the Mean Value computed above we will compute the Standard-Deviation

σ = s = 1.258306

We have so far defined mean and standard deviations. But the ideas still seem abstract. How do we apply them to a real life problem?

Let us look at an example. Zen is an under-graduate student who wants to evaluate and choose between two post-graduate colleges (College A and College B) she got accepted into. The student thinks of  her three most important criterion for choosing grad-school.

  1. Faculty
  2. Location
  3. Expected salary after graduating

Zen does not find any significant difference between colleges A & B along her first two criterion. Now the only deciding factor will be the “Expected Salary”. Zen collects 1000 salary data points from each college to evaluate which college will help her earn a higher salary.  The issue that Zen faces here is that she has to evaluate 1000 data points from each college to come to a decision. To make sense of such huge number of data points seemed challenging. But then, Zen remembered that she learnt about statistical tools that can help her make a better decision. Below is how she went about evaluating:

Applying the Measure or Central Tendency (or) Applying Mean:

Zen asks herself, “If I were to come up with a numeric value for each college, that is representative of the salary earned in that college what will it be?”. She realizes that she can compute mean salary for each college. The value of the mean salary will indicate the central tendency of the data-set. Which means if you were to visualize distribution of the salary, then mean salary will be in between the lowest and highest salaries with its value gravitating towards the salary bucket which is most frequent.

The Mean Salary value for both the colleges A & B turns out to be 95,000 and 97,000 respectively. If Zen were to only use the mean value as deciding factor then she can simply join college B. But then the she wonders, “Mean Salary only represents the central tendency of those 1000 data points from each college. What about how spread out the salary numbers are?”. To get an understanding of how spread out the salary numbers are, she finds the highest and lowest salaries for both the colleges. Below is how the data looks like

Lowest Salary Highest Salary
College A 80,000 110,000
College B 50,000 150,000

As can be seen, College A has lower highest salary and lowest salary for both the colleges is same. Does it mean Zen should simply take College B. Not Just Yet! There is an important measure she needs to think about. That is Standard Deviation.

Applying the Measure or Dispersion (or) Applying Standard Deviation:

Using the formulas learnt above Zen computes Standard Deviations for both college A and B. Below is how the Standard Deviation looks like.

Standard Deviation of Salary
College A 3000
College B 7000

From the above numbers it is evident that College B has higher Standard Deviation which means that salaries in College B are more spread out as compared to College A. This means, taking college B can mean much higher or much lower salary compared to College A. Zen thinks that College B presents a high risk and high reward opportunity while College A is more likely to fetch a salary closer to its mean which is 95000. Zen being a risk averse person, chooses to take up College A.

Such simple tools like Mean and Standard Deviation can enable you to think along dimensions which were hitherto non-existent. Thank you!

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