Statistics

Data is a representation of real life. It’s an abstraction, and it’s impossible to encapsulate everything in a spreadsheet, which leads to uncertainty in the numbers. How well does a sample represent a full population? How likely is it that a dataset represents the truth? How much do you trust the numbers? Statistics is a game where you figure out these uncertainties and make estimated judgements based on your calculations.

Statistics is all about “compressing” a lot of information into a few numbers so our brain can process it more easily.

For example: More than 500 million people are living in the European Union’s 28 countries. If we want to compare salaries between countries, we won’t compare each of the salaries one by one. That’s nonsense. We will calculate and compare average salaries instead. That’s a helpful abstraction.

Now, let’s move one step ahead and say that we want to find a single value, which will in some way represent all the values in our distribution. A value which could be useful for us, and do exactly what we want is called an average. Here is a list of the types of averages used in statistics also called or measures of central tendency. We start with the common type with their explanation.

Arithmetic mean – This is the most commonly used average. Usually, it is referred as the average or simply mean. The arithmetic mean is defined as the value obtained by dividing the sum of values by their number or quantity.Following is the mathematical formula for the arithmetic mean: Let x1, x2, … , xn be a list. Then,

               -     mean  = (x1+x2+x3+...+xn)/n

One of the most known examples of the use of this mean (at least among us students and of course, the teachers) is the process of calculating the average grade for a course. Let’s say that we have 5 courses: statistics, cryptography, distributed, c programming and software engineering. For these 5 courses, one student has 5 grades (on the scale of 18 to 30): 27, 28, 29, 30, 26. The average grade for this student would be calculated as: (27+28+29+30+26)/5=28.

Weighted Arithmetic Mean – When the values are not of equal importance, we assign them certain numerical values to express their relative importance. These numerical values are called weights. The weighted artimetic mean is found by adding the product of each weight and each value and then dividing by the sum of all weight.If X1, X2, ……, Xk have weights W1, W2, ……., W3, then the weighted arithmetic mean or the weighted mean, can be calculated by the following formula:

                      -weighted artimetic mean  =ΣWiXi/ΣW

Let’s use the same example as before, just with a little modification. Let’s add the weights to the subjects respectively: 5,4,3,1,2. Thus, the weighted mean can be calculated as: (275+284+293+301+26*2)/(5+4+3+1+2)= 27.73

Geometric mean – Given n values that are positive, the geometric mean is the nth root of their product.

                       - Geometric mean = nth root of(x1 * x2 * ...*xn)

Example:- Return, or growth, is one of the important parameters used to determine the profitability of an investment, either in the present or the future. When the return or growth amount is compounded, the investor needs to use the geometric mean to calculate the final value of the investment.

Case example: an investor is offered two different investment options. The first option is a $20,000 initial deposit with a 3% interest rate for each year over 25 years. The second option is a $20,000 initial deposit, and after 25 years the investor will get $40,000. Which investment should the investor choose? The investor will use the future value or the present value formula, which is derived from the geometric mean.

Harmonic mean – The harmonic mean is found by dividing the numbers of values by the sum of the reciprocals of all values. Mathematically, the formula for harmonic mean will be as follows:

                     -harmonic mean  = n/(ΣW/X)

For example, You are a stock analyst in an investment bank. Your manager asked you to determine the P/E ratio of the index that tracks the stocks of Company A and Company B. Company A reports a market capitalization of $1 billion and earnings of $20 million while Company B reports a market capitalization of $20 billion and earnings of $5 billion. The index consists of 40% of Company A and 60% Company B. Firstly, we need to find the P/E ratios of each company. Remember that the P/E ratio is essentially the market capitalization divided by the earnings.

                - P/E (Company A) = ($1 billion) / ($20 million) = 50
                - P/E (Company B) = ($20 billion) / ($5 billion) = 4 We must use the weighted harmonic mean to calculate the P/E ratio of the index. Using the formula for the weighted harmonic mean, the P/E ratio of the index can be found in the following way:

                  - P/E (Index) = (0.4+0.6) / (0.4/50 + 0.6/4) = 6.33

Quadratic mean – The quadratic mean (also called the root mean square) is another type of average. It represents the square root of the mean of the squares of the numbers in the set. It can be calculated with the following formula:

            - quadratic mean = sqrt((Σ)/n)

A real life example of the use of the quadratic mean can be found in calculating the size of a signal. The term “size of a signal” is used to represent “strength of the signal”. It is crucial to know the “size” of a signal used in a certain application.