Statistics: Bias and Sample Bias

In statistics we come across two types of errors, random errors and systematic errors. Random error is the error due to sampling variability or at times measurement precision. It occurs essentially in all quantitative studies and can be minimized to an extent but not avoided. We might wonder, What is Bias? Bias or Systematic error is a reproducible inaccuracy that produces a consistently false pattern of differences between observed (estimated) and true values. We get Definition Bias or definition of Bias as the systematic or average difference between the true value of the parameter and the experimental estimate value.

Bias Statistics or Statistical Bias is an error we cannot correct by repeating an experiment many times and averaging together the results, it is a directional error in an estimator.In a population which includes 50% males and 50% females, if we know that an important variable is the gender. Then the sample needs to include the same proportion. But if the sample includes 30% males then we can say that the results are likely biased as there are not enough responses from the male gender. This is how we know Sample Bias or in other words if the given sample is biased.

In statistics, Bias is used to refer the directional error in an estimator. Though we repeat the same experiment many times there would be some randomness existing.  By repeating an experiment we might see a slight variation in the estimation values. The bias is a systematic or average difference between the experimental estimate and the true value.

Examples of Bias
For example, if the national census could be completed online using the internet, then the sampling is said to be biased as only those people who could afford a computer and internet connection would be included and the people who cannot afford a computer or have the necessary skills to operate one will be excluded.

Jason was assigned by his editor to determine what most Americans think about the new law that will place a special tax on all electronic goods purchased and was told to mention in the survey form that he revenue returns collected from the tax would be made use of to enforce new online decency laws. Jason decided to use the email poll for convenience. In this poll 90% of those surveyed opposed the tax. Jason was quite surprised when 65% of all Americans voted for the taxes. So, online and call-in polls are particularly biased as the respondents are self-selected.

For example, a movie is released and a survey is done on the first day first show to know how people liked it. In the survey some say the movie was awesome and they all liked it and some say the movie was boring and they did not like it at all. Here ‘they’ might refer to only their group.

Inverse Function

What is an Inverse Function?

Inverse Function

In real life, did you ever undo the activity you or some other person has performed? If yes, then you have applied inverse function in your life. The inverse function is a function which will undo another function.  For example, let us consider that you have to call your friend using your mobile phone. When you wanted to dial to your friend, you will open the phone book, select his name and click the call button. What happens internally? The phone number of the corresponding name is retrieved and a call is established to that number. In the inverse way, what happens when your friend calls you? You get his name listed in the caller id. This is because; your friend’s phone number reaches your phone. The number gets converted into name and gets displayed. Thus, while calling your friend, the name gets converted into number. While receiving the call, the inverse of function happens and the number gets converted into the name.

However, the inverse of a function must result in the function itself i.e. assume that John is in your contact list and his number is 9904567345. If you call John from your phone book, then the call goes to 9904567345. In that case, when you receive a call back from 9904567345, it should be displayed as John in your phone and not some other name.

Here is another example for you: If a function represents the statement: “John is the father of Dave” then the inverse function will be: “Dave is the son of John”.

How to find Inverse Function?
Now that we know what the inverse of a function is, let us concentrate on how to find inverse function. Finding the inverse of a function can be demonstrated using the algebraic equation x = 4y + 3.

For finding inverse functions of x=4y+3, follow the steps given below:

Step 1: Move the + 3 on right side to the left side. Inverse of addition is subtraction. Thus when the +3 goes to the left side, it becomes -3. Thus the function will be evolved into x-3 = 4y.

Step 2: Now the number 4 has to be moved from right side to left side. Number 4 which is multiplied with y has to move to left side. Inverse of multiplication is division. Thus the 4 on the right hand side will perform division when moving to the left side. The function will now be transformed into (x-3)/4 = y.

Thus the inverse function of x=4y+3 is y=(x-3)/4.

Learn Exponential functions

Definition of Exponential functions: Exponential functions  are those function in which independent variable appear as an exponent .
exponential functions examples : y = 3x , f(x) = ex , y = 32x+5  are an example of exponential function.
Exponential function
Y = ax, where a is a positive real number and a ?0  is the most popular exponential function . It is an increasing function (as the value of x increasing the value of y increases ) when a>1, It is a decreasing function (as the value of x decreases the value of y decreases ) when a<1.
Derivative of exponential function :
We will use the following property to derivate  exponential function such as
  (ex) = ex

Least common multiple and Least common denominator

Least common multiple or LCM of two numbers is the smallest number that is multiple of both.
How to find the least common multiple?
Methods to find the LCM:
1. Prime factorization method – Following steps are followed to find LCM by prime factorization method:
• Find the factors of numbers whose LCM has to be obtained.
• Find the common factors of all the numbers.
• Find the factors that are not common.
• LCM is the product of common factors and the factors that are not common
LCM = (Product of common factors) X (Product of uncommon factors)
2. Common division method – Following steps are followed to find LCM by common division method: -
• Arrange numbers together separated by commas.
• Start with smallest prime number and keep on dividing till none of the numbers can further be divided.
• Multiply all the factors together to find the LCM.
The common division method is useful for finding the LCM of more than two numbers.
Least common denominator
Least common denominator is calculated for adding and subtracting the fractions. We cannot add or subtract any fraction before finding the least common denominator.
 How to find the least common denominator? (Finding the least common denominator)
Least common denominator finder
Suppose if we have two fractions such as 7/4 and 5/6.
To add these fractions, we need to find the least common denominator and o find the least common denominator of these two fractions, we find the least common multiple of their denominators i.e. LCM of 4 and 6.
4 = 2 X 2
6 = 2 X 3
2 is the only common factor and 2, 3 are the factors that are not common.
Product of common factors = 2
Product of factors that are not common = 2 X 3 = 6
LCM = (Product of common factors) X (Product of factors that are not common)
= (2) X (2 X 3) = 2 X 6 = 12
Hence the LCM of 4 and 6 is 12.
The least common denominator of 7/4 and 5/6 is 12
7/4 + 5/6 = 7*3/4*3 + 5*2/6*2 = 21/12 + 10/12 = 31/12

Properties of Real Numbers

What is a Real Number?
You can define real number as any valid number, be it whole number or rational number or irrational number. For example: 1, 1.234, 1/8, π, √7 are real numbers.

Properties of Real Numbers
There are certain properties that can be applied to all the real numbers. The different properties of real numbers are:

Cumulative Property
• Cumulative Property of Addition: This property of real numbers states that if there are two numbers, they can be added in any order. For example, 10 + 5 = 5 + 10

• Cumulative Property of Multiplication: This property states that the numbers can be multiplied in any order. For example, 10 X 5 = 5 X 10, both return the same value.

Associative Property
• Associative Property of Addition: If more numbers have to be added together, then you can associate any of them together in any way. For example, 10 + (5 + 2) = (10 + 5) + 2.

• Associative Property of Multiplication: If more numbers have to be multiplied together, then they can be associated in any way. For example, 10 X (5 X 2) = (10 X 5) X2.

Identity Property
• Identity Property of Addition: Any number added to zero will result in the number itself.  For example, 10 + 0 = 10.

• Identity Property of Multiplication: Any number added to one will result in the number itself. For example, 10 X 1 = 10.

Inverse Property
• Inverse Property of Addition: A positive number when added to its inverse results in zero. For example, 10 + (-10) = 0.

• Inverse Property of Multiplication: A number when multiplied by (1/same number) will result in 1. For example, 10 X (1 / 10) = 1.

Zero Property
Any number multiplied with zero results in zero. For example, 10 X 0 = 0

Density Property
 As per density property, it is always feasible to find a number existing between two real numbers. For example, between 10.1 and 10.2 you have a lot of numbers like 10.11, 10.12, and 10.13 and so on.

Distributive Property
Distributive property is applied when an expression includes addition and also multiplication. If a number is multiplied with a result of addition, then the multiplication has to be distributed over all the numbers participating in addition. For example, 2 X (5 + 10) = (2 X 5) + (2 X 10)

If you understand these properties clearly, then you can easily solve the algebra problems that include even complex expressions.

Derivatives of cotangent function

Derivative of cot:
Based on the above discussion we see that cot function exists (or is defined) only if sin(x) at that point is not equal to zero. Therefore for the derivative of cot function to exist, the following condition has to be met:
 Interval (a,b) should not contain n(pi) for any n Є Z, (Z = set of all integers). That is because, for x = n(pi), sin (x) = sin (n(pi)) = 0. And we already established that sin function cannot be equal to zero for cot function to exist at that point.

Derivative of cot x:
If (a,b) does not contain 

Assuming that sin and cos are differentiable and x ≠ n(pi), then sin x ≠ 0.


Proof for derivative of cotx:

The above derivation was using the quotient rule. We can also obtain the above form using the limit definition of derivative as follows:
Derivative of cot x (using limit definition):

Derivative of cot(-x):

(using chain rule, therefore multiplying by (d/dx) (-x) = -1)

= csc^2(x)

Improper Fractions

What is an improper fraction?
An improper fraction is a fraction in which the numerator, (the top number) is greater than the denominator (the bottom number).  Like in a fraction a/b, it is improper when a>b and b not zero.

For example, 5/3 is an improper fraction (5>3)
Improper fractions can be written as a mixed number, mixed number is a fraction in which we have a whole part and fraction part. 3 ½ is a mixed number, with 3 as the whole part and ½ , the fraction part
Let us convert a mixed numbers to improper fractions.

To convert a mixed number to improper fraction, we first find the product of the whole part and the denominator and then add the numerator to this product. This gives us the numerator of the improper fraction, the denominator remains the same

Convert  the mixed number, 5 ¾ to an improper fraction
(5=whole part, 3=numerator, 4=denominator)

Improper Fraction example

we find the product of the whole part and the denominator
Product = (whole part x denominator) = (5 x 4) = 20
then, we add the numerator to the product
      = 20 +3 = 23 (numerator)
While converting, the denominator remains the same
So, the improper fraction will be 23/4

Converting improper fraction to mixed number
Example: Convert the improper fraction, 8/3 to a mixed number
To convert, we need to divide the improper fraction using the long division method. The quotient will be the whole part, the remainder the numerator and the denominator remains the same
When we divide 8 with 3, we get
Quotient = 2 and remainder = 2
So, the required improper fraction would be 2 ⅔      
(2 is the whole part, ⅔ is the fraction part)

Improper fraction to mixed number
To understand the converting of an improper fraction to a mixed number, let us take a look at the example given below:
Convert the improper fraction 24/5 to mixed number
Divide 24 with 5 using the long division method
24 ÷ 5 = 4 (quotient) Remainder = 4
The mixed number is written as 4 4/5