PSN Academy

In this video, you will get the idea of Central Tendency, Median & Mode, Frequency and Grouped & Ungrouped data. ----------------------------------------------------------------------------- Median: Value of the variable which divides the whole distribution into two equal parts Median class: The class in which the (𝑛/2)^𝑡ℎ cum. frequency falls. If 𝑛/2 is not present, 𝑛/2+1 will be considered to determine median class. ----------------------------------------------------------------------------- Property 1: Number of the observations below and above the median is same. Property 2: Median is not affected by extremely large or extremely small values or by open end class intervals. When median is useful: Useful when the distribution is skewed (asymmetric) Merits of Median 1. Not affected by extreme values 2. Can be calculated for open end classes 3. Can be calculated even if the other classes are of unequal width Demerits of Median Not rigidly defined (a distribution may have more than one mode) ----------------------------------------------------------------------------- 1. For Ungrouped Data (Method 1) If 𝑥_1,𝑥_2,…,𝑥_𝑛 are the 𝑛 observations then the arithmetic mean 𝑥 ̅=(〖𝑥_1+𝑥〗_2+…+𝑥_𝑛)/𝑛=(∑_(𝑖=1)^𝑛▒𝑥_𝑖 )/𝑛 If 𝑓_𝑖 is the frequency of 𝑢_𝑖their mean is 𝑥 ̅=(〖〖𝑓_1 𝑢〗_1+𝑓_2 𝑢〗_2+…+〖𝑓_𝑘 𝑢〗_𝑘)/(𝑓_1+𝑓_2+…+𝑓_𝑘 )=(∑_(𝑖=1)^𝑘▒〖𝑓_𝑖 𝑢〗_𝑖 )/(∑_(𝑖=1)^𝑘▒𝑓_𝑖 ), 𝑘 = Distinct observation count 1. For Ungrouped Data (Method 2) Take deviations from any arbitrary point “A”. Then 𝑥 ̅=𝐴+(∑_(𝑖=1)^𝑛▒𝑑_𝑖 )/𝑛 𝑑_𝑖=𝑥_𝑖−𝐴 If 𝑓_𝑖 is the frequency of 𝑢_𝑖 then 𝑥 ̅=𝐴+(∑_(𝑖=1)^𝑘▒〖𝑓_𝑖 𝑑〗_𝑖 )/(∑_(𝑖=1)^𝑘▒𝑓_𝑖 )=(∑_(𝑖=1)^𝑘▒〖𝑓_𝑖 𝑢〗_𝑖 )/(∑_(𝑖=1)^𝑘▒𝑓_𝑖 ) 2. For Grouped Data (Method 1) If 𝑓_𝑖 is the frequency of 𝑚_𝑖 (the mid value of the 𝑖^𝑡ℎ class interval) their mean 𝑥 ̅=(〖〖𝑓_1 𝑚〗_1+𝑓_2 𝑚〗_2+…+〖𝑓_𝑘 𝑚〗_𝑘)/(𝑓_1+𝑓_2+…+𝑓_𝑘 )=(∑_(𝑖=1)^𝑘▒〖𝑓_𝑖 𝑚〗_𝑖 )/(∑_(𝑖=1)^𝑘▒𝑓_𝑖 ), 𝑘 = Distinct observation count. 2. For Grouped Data (Method 2) If 𝐴 is an arbitrary point and 𝑓_𝑖 is the frequency of 𝑚_𝑖 (the mid value of the 𝑖^𝑡ℎ class interval) their mean 𝑥 ̅=𝐴+(〖〖𝑓_1 𝑑〗_1+𝑓_2 𝑑〗_2+…+〖𝑓_𝑘 𝑑〗_𝑘)/(𝑓_1+𝑓_2+…+𝑓_𝑘 )=𝐴+(∑_(𝑖=1)^𝑘▒〖𝑓_𝑖 𝑑〗_𝑖 )/(∑_(𝑖=1)^𝑘▒𝑓_𝑖 ), 𝑘 = Distinct observation count, 𝑑_𝑖=𝑚_𝑖−𝐴. N.B.: It is illeagal to download, distribute or display in public any part of this video production.

In this video, you will get the idea of Binomial Theorem expansion with Positive Integral Index, Negative Intetgral Index and Rational Index. 1 term (monomial) : 16, 4𝑥, 3𝑥^7, 𝑎^4 𝑏 2 terms (binomial) : 𝑥+𝑦, 𝑝−𝑞, 𝑥^2−4𝑦, 7𝑎+3𝑏^2 3 terms (trinomial) : 𝑥+2𝑦−3𝑧, 𝑎−𝑏−𝑐, 2𝑥+3𝑦^4+6𝑧 4 or more terms (multinomial) : 𝑤+𝑥+5𝑦+𝑧, 2𝑎^3−𝑏−𝑐−𝑑^2+3𝑒 Binomial Theorem is a statement that describes this expansion of a binomial. 1. (_^𝑛)𝐶_0 , (_^𝑛)𝐶_1 . (_^𝑛)𝐶_2 ... (_^𝑛)𝐶_𝑛 are known as binomial coefficients. 2. Exponents of 𝑎: exponent of 𝑎 in the term with binomial coefficient (_^𝑛)𝐶_𝑟 will be 𝑛−𝑟. 3. Exponents of 𝑏: exponent of 𝑏 in the term with binomial coefficient (_^𝑛)𝐶_𝑟 will be 𝑟. 4. Sum of the exponents of 𝑎 and 𝑏 in each term: 𝑛 5. Total terms after expansion: 𝑛+1 6. Index: MUST be positive integer. When 𝒏 goes negative OR 𝒏 is rational, then this will not work. N.B.: It is illeagal to download, distribute or display in public any part of this video production.

In this video, you will get the idea of factorial and its notations. Also you get the logic of fundamental principles of addition and multiplication. There are two fundamental principles of counting. These two principles solve the problems of counting. According to Grinstead and Snell (2006) counting is defined as: “Consider an experiment that takes place in several stages and is such that the number of outcomes m at the nth stage is independent of the outcomes of the previous stages. The number m may be different for different stages. We want to count the number of ways that the entire experiment can be carried out.” Fundamental Principle of Multiplication (FPM) -------------------------------------------------------------------------- Situation: Two or more jobs to be done sequentially. When the first job can be done in 𝑚 distinct ways and the second job can be done in 𝑛 distinct ways, then both jobs can take place (one followed by other) in 𝑚×𝑛 distinct ways. Fundamental Principle of Addition (FPA) ---------------------------------------------------------------- Situation: One of the two or more jobs to be done. When the first job can be done in 𝑚 distinct ways and the second job can be done in 𝑛 distinct ways, then one of the two jobs can take place in 𝑚+𝑛 distinct ways.

In this video, you will get the idea of Permutation and Combination. Permutations: Unique arrangements of items 1 of 2. Linear Permutation Number of unique possible ways a set of items can be arranged in a line taken some or all at a time. 2 of 2. Circular Permutation Clockwise and anti-clockwise direction produce DIFFERENT arrangements: (n-1)! permutations Clockwise and anti-clockwise direction produce SAME arrangements: (n-1)!/2 permutations. Combination: Grouping of items (Position has no effect in counting) taken some or all at a time