Median-Measure of Central Tendency

Median in Statistics a branch of Mathematics is one of the measures of central tendency. We can define Median as the middle value of the given list when arranged in the numerical order.  For instance, let us consider the data, 10, 10, 9, 9, 8, 10, 7. Let us first arrange the given data in the numerical order that gives us 7,8,9,9,10,10,10. There are 7 data values, so the middle value would be the fourth value. Here the fourth value is 9 and hence the median.

To find the Median of a given set of data, first we need to arrange the given data list in the numerical order, then count the number of values in the given list. This is important as to know whether the count is even or odd. If the count is odd then we do not need to worry as the middle value in the arranged list would be the median, but if the count is even then we will have two middle values.  So, how do we go about, the middle of the two middle values in simple words the average of the two middle values gives us the median.  For instance, consider the data, 8, 7, 4, 3, 9, 10, 9, 8. First we arrange the given data in the increasing order or numerical order, 3, 4, 7, 8, 8, 9, 9, 10.

Now we count the data values, there are 8 data values which is an even number and hence we have two middle values. The two middle values are the fourth and fifth values which are 8, 8. To find median when the count of the data values is even is to find the average of the two middle values which would be (8+8)/2 = 16/2 = 8. So, the median is 8.

We can calculate median of the given data list using the Median Formula which is one more than the total number of data values divided by 2. Median Formula is given by (n+1)/2 , the value got is the position of the data value that would be the median when the list is arranged in the numerical order. For instance if there are 7 data values in the list, then (7+1)/2 = 8/2 = 4, which means the fourth value will be the median. If there are 8 values that is even number of values in the list, then (8+1)/2 = 9/2 = 4.5, a bit confusing, as we do not have 4.5th value. Here we take the average of the fourth and fifth values to calculate median as 4.5 lies between 4 and 5.

A short note on how to do standard deviation

To find the standard deviation we make a standard deviation chart. We shall try learning how to make a standard deviation chart using an example.

Example: The runs scored by a cricket player in 6 innings are 60, 45, 25, 40, 70 and 30. Find the standard deviation.
Solution: First we find the mean of the data given to us. We see that the number of observations is 6. Therefore our n = 6.

The sum of observations = ∑x_i = x_1 + x_2 + x_3 + x_4 + x_5 + x_6
∑x_i = 60 + 45 + 25 + 40 + 70 + 30 = 270
Therefore the mean = x ̅ = ∑x_i/n = 270/6 = 45

Next we make the chart for the standard deviation calculation. The first column of the chart would have the observations x_i. The second column would be x_i - x ̅ for each of the observations. The third column would be (x_i - x ̅)^2 for each of the observations. The table would look as follows for this problem:
x_i x_i - x ̅ (x_i - x ̅)^2
60 60-45 = 15 (15)^2 = 225
45 45-45 = 0 (0)^2 = 0
25 25-45 = -20 (-20)^2 = 400
40 40-45 = -5 (-5)^2 = 25
70 70-45 = 25 (25)^2 = 625
30 30-45 = -15 (-15)^2 = 225
Next we sum up the third column so we have:
∑(x_i - x ̅)^2 = 225 + 0 + 400 + 25 + 625 + 225 = 1500
Lastly we use the formula to find the standard deviation which is like this:
Standard deviation = σ = sqrt[(∑(x_i - x ̅)^2)/n] = sqrt[1500/6] = sqrt [250] = 15.81
The above steps remain the same for any number of observations.

Standard deviation graph: A standard deviation graph would look like the picture below. This is a typical graph for a normal distribution. A normal distribution is symmetric about the mean. In a normal distribution, about 68% of the observations would lie between one standard deviation from the mean. That means 68% of the observations would lie between the range x ̅ – σ to x ̅ + σ. 95% of the data would lie within 2 standard deviations of the mean, that is in the range between x ̅ – 2σ to x ̅ + 2σ. 99.7% of the data would lie with 3 standard deviations of the mean.

My forthcoming post is on Frequency Table will give you more understanding about Statistics

P a B Conditional Probability

Conditional Probability Definition:-
If B is any arbitrary event in a sample space and P ( B ) > 0. The probability of A given B is defined by
P ( A / B ) = P ( A n B ) / P ( B ).
This is called conditional probability of A given B and is written as P ( A / B ).

Proof of Conditional Probability
Let A and B be any two events in a random experiment associated with the sample space "S" then
if n ( S ) = number of elements in S
n ( B ) = number of elements in B
n ( A n B ) = number of elements in A n B.
P ( B ) = n ( B ) / n ( S )
P ( A n B ) = n ( A n B ) / n ( S )

Now for the conditional event A after B the favorable outcomes must be one of the sample points "B"
In other words, the sample space for the event A after B is B.
P ( A / B ) = n ( A ) / n ( B )
P ( A / B ) =n ( A n B ) / n ( B )                [ A n B = A ]
P ( A / B ) =  [ P ( A n B ) / n ( S ) ] / [ P ( B ) / n ( S ) ]
P ( A / B ) = P ( A n B ) / P ( B )
 Hence Proved.

Example of Conditional Probability
If the probability that a communication system will have fidelity is 81 and the probability that it will have fidelity and selectivity is 18. What is the probability that a system will high fidelity will also have high selectivity?
Solution:-
Let the event B will be the communication system to have high fidelity
P ( B ) = 81
Let the event to have high fidelity and selectivity is A n B
P ( A n B ) = 18
By the definition of Conditional probability
P (A / B) = P (A n B) / P (B)
Probability to have selectivity if it will have fidelity
P (A / B) = 18 / 81
P (A / B) = 2 / 9  = 0.22.....

I have recently faced lot of problem while learning statistics, But thank to online resources of math which helped me to learn myself easily on net.

Statistics Answers Explained

In Statistics, a branch of mathematics there are numerous Statistics Problems for which we can calculate Statistics Answers. The following are some of the Statistics math Answers which use stepwise explanation for the given problems.

Consider the scores obtained by 10 students in mathematics test is given as, 55 36 95 73 60 42 25 78 75 62. Calculate and give the statistics Answer for the mean, median, mode and range of the given scores. First we need to arrange the data in the ascending order (preferably). That gives us, 25 36 42 55 60 62 73 75 78 95. The mean of the scores is given by adding the scores and dividing it by the number of scores, which would be, [25+36+42+55+60+62+73+75+78+95]/10 = 601/10=60.1, the median is the middle value in the ordered data set. Here, the data has even number of scores and hence there are two middle values, the median in such case would be the mean of the two middle values, [60+62]/2= 61. The mode is the number that occurs the most number of times, here there is no such score and hence there is no mode for the given data set. Range is the difference of the highest value and the lowest value of the ordered data set, range = 95-25= 70

Find the mean deviation about the mean for the following data: 12,3, 18, 17, 4, 9, 17, 19, 20, 15, 8, 17, 2, 3, 16, 11, 3, 1, 0,  5
Let us first find the mean of the (x bar) of the given data
(xbar) = 1/20*summation(i=1to20)[x(i)]= 200/20 = 10
The respective absolute values of the deviations from the mean , modulus[x(i) – xbar]are
2, 7, 8,7, 6,1,7,9,10,5,2,7,8,7,6,1,7,9,10,5
Summation[(i=1to20)[modulus(x(i) – xbar)] = 124
So, the mean deviation about the mean would be, M.D.(x bar) = 124/20 =6.2

Given the frequency table of the monthly salaries of 20 employees of a company:
Monthly salary(in dollars)   frequency of number of employees[f(i)]
     3500 5
     4000 8
     4200 5
     4300 2
Find the statistics answer of the mean of the salaries of the 20 people and also calculate the standard deviation
Let x(i) be the i th salary and f(i) be the corresponding frequency
Mean of the grouped data can be calculated as mu= summation [x(i)f(i)]/summation[f(i)] which gives us
= [3500x5 + 4000x8 +4200x5 +4300x2]/5+8+5+2
= [17500+32000+21000+8600]/20
= $3955
Now, let us find the standard deviation of the 20 people which can be calculated by the formula
Sqrt of {[summation [x(i)-mu]^2 times f(i)]/summation[f(i)]}.
= sqrt {[5*(3500-3955)^2+ 8*(4000-3955)^2+5*(4200-3955)^2+2*(4300-3955)^2]/20} on simplification, we get 282, rounded to the nearest unit.

Solving indefinite integrals

The following theorems and standard forms are used for finding indefinite integrals:
Theorem I: If f and g are differentiable on (a,b) and f’(x) = g’(x) for all x belonging to (a,b), then f(x) = g(x) + c, where c is a constant called the arbitrary constant.

Basics of Limit

What is limit?
A number suppose (x) is said to be limit of function at a particular point if for any arbitrary chosen for any positive integer, however small but not zero there exist a corresponding number greater then zero for all value of that particular point. In simple way we can understand limit that the absolute value at that particular point without any regard to sign.

Limits are both right handed limit and left handed limit. For better knowing if any function is at x
=a then, (x) approaches (a) from the right that is(x) is larger than a then the limit of function is known as right hand limit.
Similarly if(x) approaches from the left that is (x) is lower than a then the limit of function is known as a left hand limit.

Laws of limits
As we know about limit. We have basic knowledge of limits. Laws of limits are very simple means we easily understand the laws.

Addition law-if limit is for any function such as addition of two or more function then by using this law limits goes in right hand side individually for each function.
Subtraction law-this is similar as addition law but only difference is that here we have to choose subtract function.
Constant law- if (x) is a constant and limit for any function is existing with constant then constant should me out from the function.
Multiplication law-if limit exit for multiplication of two functions then by using this law limits goes to each function separately.

Division law-this law is similar to multiplication law but here the difference is that the second function should not equal to zero.
Power law-if power is only in function then by using this law power should be in function with limit.

Limit laws calculus
Calculus may be derivatives and integrations. Limit laws calculus means by using the laws of limits we have to sole various derivative function. In this process we have to know all laws of limits because when we solve derivative function may be one law or more than one law is used for same derivative function.  In Limit laws calculus which law should simplify calculus more accurately, for this purpose we must have clear understanding of law.

Limit Laws Examples
There are various limit laws examples such as if we take power of (x) is 2 separately with limit in left hand side. Then by using power limit law we get in right hand side power in both limit as well as (x). This is very simple limit laws example.

Derivatives of inverse trigonometric functions : derivative of arc tan function

The tan function:
The trigonometric function tan which is the acronym of tangent function is defined as the ratio of the opposite side to the adjacent side of the angle of which the tan is to be found. Alternatively, if we have a point P on the unit circle such that the co-ordinates of the point are (x,y), then tan theta of angle subtended at the centre by the arc AP would be y/x. See picture below:

The inverse of tan function:
Inverse of the tan function is also called the arctan function. We know that the tangent function is a periodic function. So it is not a one to one function. However if we suitably restrict the domain, then the tangent function is a one to one function and then its inverse function i.e. arctan can also be defined.
The arctan function is: arctan (y) = {(x,y) | tan x = y}.
Thus, if tan (pi/4) = 1, then we say that arctan(1) = pi/4.

What is the derivative of arctan?
The derivative of arc tan can be calculated as follows:
Let y = arctan x,
Then, x = tan y
So, dx/dy = sec^2 (y)
So, dx/dy = 1 + tan^2(y)
So, dx/dy = 1+x^2
So, dy/dx = 1/(1+x^2)
Thus the derivative of arctan is 1/(1+x^2)

The derivatives of other arc tan function such as derivative of arctan x 2, derivative of arctan 2 ^x, arctan(x^2) etc, can all be found using the chain rule along with the formula for derivative of arc tan. The derivative of arctan function represents the rate of change of the function y with respect to change in x. In other words it is the slope of the tangent at any point x. The graph of the derivative of arc tan can be shown as follows:

In the above picture, the red curve is the graph of arc tan (x) and the blue curve is the graph of derivative of arc tan (x). Note that at the point x=0, the graph of arc tan (x) changes from concave up to concave down. Therefore that is the point of inflection. We see that at that point the graph of the derivative of arc tan (x) has a maximum. Also the graph of the derivative is always positive, therefore arc tan is an always increasing function.