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Measurability and Variability

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If we are performing operation on any statically data it many happen that we obtain result after performing two operations. Then it can happen that we get two different results for the same data then it is called as Statistics variability. Variability statistics definition can be described as the variation in distribution from its Mean. If there will be a distribution which is lacking in variability statistics then it is called as homogeneous. There are three measures of variability, Range, standard deviation and variance of proportion. Firstly we will discuss range
Range of any data is the difference between the highest Point and lowest point. If we are having a data as 8, 5, 6, 2 then The Range will be 8 - 2 = 6. So the required range will be 6. Now we will move to variance of proportion,
If we talk about any population then variance of proportion is the average Square deviation from the mean of the population, we can find the variance with the help of formula given below,
σ2 = Σ (Xi - μ)2 / N,
Here σ2 is the variance of population; ‘μ’ is the mean of the population, ‘n’ is the number of element in the population and ‘Xi’ is the ith element of the sample.
Now we will discuss about the Standard Deviation,
We can find standard deviation of any population with the help of variance; if we do the square root of variance we will get the standard deviation. The formula for the standard deviation is given below,
σ = sqrt Σ ( Xi - μ )2 / N,
Here ‘σ’ is the standard deviation of population; ‘μ’ is the mean of the population, ‘n’ is the number of element in the population and ‘Xi’ is the ith element of the sample.
This is all about the three measures of variability.

The Range

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Smallest Interval length containing all Set of data is called as Range. It can be calculated by finding the smallest and greatest observations and then subtracting them. Units of the Statistics range is same as of the data. Its calculation is dependent on two sample observations, so it is considered as Weak Measurement of the Statistics specially when the sample size is large.
Mostly when the sample size is large range is greater or equal to the Standard Deviation. It is the difference between the smallest and largest so it is called as Crude range. Range helps us to calculate the spread of the data in a set of data and it is the practical use of the range. It is one of the properties of statistical distributions. It is basically calculated for that data sample where data is having Random Variable. If any value lies outside the range value, the function value at that Point will be automatically 0. Range statistics is calculated in the form of whole Numbers values , integers and Decimals.
Let us understand the concept of the range in statistics with the help of example
Example : Some diamond pieces of these detailed grams weight is given 321, 368, 421, 470 and 481. Find the range.
Solution : Range =Largest Sample – Smallest Sample
Range = 481-321
Range =160gms

Variance of a Random Variable

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Random variables are the variables which we will get after the experiment, that variable can be any numeric value. We always get a specific value after the experiment if this value is fixed then experiment is called as Random Variable and if it is not then it is called infinite random variable. For finding variance we need to find the Standard Deviation first then only we can find the Variance of a Random Variable, here we will discuss variance in detail and we will solve an example related to the variance. Variance is denoted by ‘σ2’. The variance is always defined for the large number of population and it is the only parameter that describes the actual Probability distribution. Now we will see how we can find the variance for the above problem for that we need to follow some steps given below, if we are given a series and we are asked to find the variance random variable then we can apply the simple method given below.
For finding random variable variance we need to find the Mean of the of the given series, mean is the average value of any given series, we can also define it as a middle value of any series and mean is related to every member of the given series, so we can easily find the mean as
6+3+4+6+8+9/6
=6
Six is the required mean for the given series now we can find the variance as,
We will subtract the mean from the given Numbers of series then we will Square all the given numbers and divide by the average again,
σ2= ((6 - 6)2 + (3 - 6)2 + (4 - 6)2 + (8 - 6)2 + (9 - 6)2 + (6-6)2)/6,
σ2= 0+9+4+4+9+0/6
σ2=26/6
σ2=4.33
σ= √4.33
This is the required answer in this way we can calculate variance, we can also calculate variance for the given population but for that we need to apply different methods.

Standard Deviation of a Random Variable

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We can define Random Variable as; if ‘X’ is a random variable and we are doing any experiment then we can assign any value to ‘X’ which we are getting from the given experiment. We are here to discuss about Standard Deviation of a Random Variable, before studying that we should have a good knowledge of random variable so we will discuss that in detail. There are two types of random variables Discrete Random Variable and continuous random variable. We can understand discrete random variable with the help of an example, if we toss four coins and we need to find the Probability of outcomes of number of heads then we know that there can be only five Probabilities (0, 1, 2, 3, 4). All the results are in discrete form so we can say that this is a discrete random variable experiment. And for continuous random variable we can take the example of temperature of the given city as it can change with time so we can call it as continuous random variable. They can take any value in a continuous interval. Now we will discuss some examples of finite random variable. If we are performing the experiment of roll of dices then we know that there can be only six outcomes (1, 2, 3, 4, 5, 6). So this type of experiment is called as finite random variable. Now we will discuss one example for more clarification about finite random variable.
If we toss three coins and we need to find the random variable when the outcome is T, H, H and we are finding the probability of number of tails, as we can see in our result that the Numbers of tails are, so the value of random variable ‘X’ is one. Now moving to infinite random variables, if we roll a die and we are asked to find the probability of getting of five then we don't know how much experiment we need to perform so it can happen that we need to perform infinite number of experiment so here the value of ‘x’ can be infinite. Now moving to an example how we can find the Standard Deviation of a Random Variable.
If we are given table in which probability of number of woman is given in particular villages and we are asked to find standard deviation of the table given below?
Number of woman probability
1 0.25
2 0.50
3 0.30
4 0.10
We can find standard deviation by multiplying number of woman to their probability and add them, so standard deviation of the given table is,
E(x) = 1*0.25+2*0.50 +3*0.30 + 4*0.10,
= 2.55,
This is the standard deviation of the above table.
We can also find the variance with the help of standard deviation as we just need subtract the number of woman in each case with the variance and then multiply it with probability of that particular event. In this we can find the standard deviation and variance of any random experiment.

Statistics Notations

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Mathematical objects and ideas are represented by the notation or in other word we can say that it is system of symbolic representation of many ideas and many objects. It also includes the relatively symbolic representation. Such as the number i.e. 0, 1, 2, it also include function symbols such as sin and +. Conceptual symbol also included in the notation such as limit, derivatives and equations and also some of the complex diagrammatic notations such as graphical notation and diagrams. Notation is also an important part of a communication. Statistical notations are essential to use the correct notation for statistical concepts. In Statistics, we cannot take another letter in place of standard notation. (a b n p q r s t x y z E F H P) these are the given letter which has a specific meaning in the statistics.

We can say that a mathematical notation is a writing system which is used for recording concept in the mathematics.
In the statistical notation the symbol and symbolic expression are uses which are intended that means the symbols and symbolic have their semantic meaning.
In the mathematics, these symbols are denoted by the number, shapes, pattern and change.
In the statistics notation the special letter are commonly used to denote the unknown parameters (population parameter).
A parameter is denoted by placing a caret i.e. (usually refers to the spacing symbol or we can say it uses for modifying the other letters) over the corresponding symbol. Some of the commonly used symbols are:
The samples Mean, the sample variance, the sample correlation coefficient ‘r’.
Some commonly used symbol for the population parameter i.e.
Mean for population, variance for population, the population correlation.

Probability and statistics notations table or Statistics Symbols Chart is shown below:

Symbols

Symbol Name

Meaning / definition

Example

Q (B)

Probability function

Probability event of B

Q (B) = 0.5

Q (S ∪ T)

Probability of events Intersection

Probability that of events S and T

Q (S ∪ T) = 0.5

Q (S | T)

Conditional Probability function

Probability of event S given event T occurred

Q (S | T) = 0.3

f (x)

Probability density function (pdf)

Q (a ≤ x ≤ b) = ∫f (x) dx

-----

F (x)

Cumulative distribution function (cdf)

F (x) = Q (X ≤ x)

----

μ

Population mean

Mean of population values

μ = 10

E (x)

Expectation value

Expected value of Random Variable X

E (X) = 10

E (A | B)

Conditional expectation

Expected value of random variable A given B

E (A | B = 2) = 5

Var (A)

Variance

Variance of random variable A

Var (A) = 4

Σ2

Variance

Variance of random variable X

Σ2 = 4

Std (X)

Standard deviation

Standard deviation of random variable X

Std (X) = 2

Σx

Standard deviation

Standard deviation value of random variable X

Σx = 2

X’

Median

Middle value of random variable X

X’ = 5

Cov (A, B)

Covariance

Covariance of random variable A and B

Cov (A, B) = 4

Corr (A, B)

Covariance

Correlation of random variable A and B

Corr (A, B) = 0.6

∇ A, B

Correlation

Correlation of random variable A and B

∇ A, B = 0.6

Σ

Summation

Summation – sum of all values in range of series

Σi = 14 xi = x1 + x2 +x3+x4

∑∑

double summation

double summation

Σi = 12 Σi = 18 xi, 1= Σi = 18 xi, 1 + Σi = 18 xi, 2

Mo

mode

value that occurs most frequently in population

------

MR

mid-range

MR = (xmax+xmin)/2

-------

Md

sample Median

half the population is below this value

--------

Q1

lower / first quartile

25% of population are below this value

-------

Q2

median / second quartile

50% of population are below this value = median of samples

------

Q3

upper / third quartile

75% of population are below this value

------

X’

sample mean

average / arithmetic mean

x = (2+5+9) / 3 = 5.333

s 2

sample variance

population samples variance estimator

s 2 = 4

s

sample Standard Deviation

population samples standard deviation estimator

s = 2

zx

standard score

zx = (x-x) / sx

------

X ~

distribution of X

distribution of random variable X

X ~ N(0,3)

N(μ,σ2)

normal distribution

gaussian distribution

X ~ N(0,3)

U(a,b)

uniform distribution

equal probability in range a,b

X ~ U(0,3)

exp(λ)

exponential distribution

f (x) = λe-λx , x≥0

------

gamma(c, λ)

gamma distribution

f (x) = λ c xc-1e-λx / Γ(c),x≥0

-----

χ 2(k)

chi-square distribution

f (x) = xk/2-1e-x/2 / ( 2k/2Γ(k/2) )

------

F (k1, k2)

F distribution

-----

-----

Bin(n,p)

binomial distribution

f (k) = nCk pk(1-p)n-k

-----

Poisson(λ)

Poisson distribution

f (k) = λke / k!

-----

Geom(p)

geometric distribution

f (k) = p (1-p) k

-----

HG(N,K,n)

hyper-geometric distribution

-----

-----

Bern(p)

Bernoulli distribution

------

-----

n!

factorial

n! = 1·2·3·...·n

5! = 1·2·3·4·5 = 120

nPk

Permutation

npk = n! / (n – k)!

5P3 = 5! / (5-3)! = 60

nCk

combination

npk = n! / k! (n – k)!

5C3 = 5!/[3!(5-3)!]=10


Variance of a Proportion

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We know that the variance of a proportion can be either direct variance or the inverse variance. We use the variance proportion to find the variance in the different areas. Sometimes we come across the problems where the value of one item increases then simultaneously the value of another variable also increases with it. Let us make it more clearly with the help of an example. If we say that the cost of the chair is $ 6, then the cost of 10 chairs will be 60 $. It indicates as the number of chairs increases, the money paid is also increasing. Such cases are the examples of the problems of direct variance.
In other type of situations, we come across the problems where the value of one unit increases, and then simultaneously the value of another unit decreases in the same proportion. We say these are the problems of inverse variance. Let us take an example of men and work. If 4 men can do a piece of work in 7 days, and if the number of men increases, we find that the number of days used to get the same piece of work decreases. Similarly we can take the example of the consumption of any articles by the number of people. If we say that 50 kg of rice is consumed by 14 persons in 30 days. If the number of person increases from 14 to 20 and the same amount of rice i.e. 50 kg of rice is consumed, then it will surely finish in less number of days. Thus we conclude that with the increase in the number of persons, the number of days in which the rice will be consumed will decrease, so these are the situations of inverse variance.
This is all about proportion of variance.