Statistics and Advanced Algebra Pointers

Measures of Central Tendency: The Mean

• The mean is the most popular and widely used measure of central tendency. It is the value most people refer to when discussing averages. It is described as the "center of gravity" of a distribution and is generally the most convenient measure, though it has limitations as a representative value for certain sets of measurements.

• Formula for Ungrouped Data Sample Mean: $χ = \frac{\sum X}{n}$

• Formula for Ungrouped Data Population Mean: $μ = \frac{\sum X}{N}$

• Example: Family Sizes in a Slum Area:

  • Data for a sample of $10$ households: $2$, $2$, $3$, $3$, $3$, $4$, $4$, $5$, $7$, $7$.
  • Calculation: $\sum X = 2 + 2 + 3 + 3 + 3 + 4 + 4 + 5 + 7 + 7 = 40$.
  • $n = 10$.
  • Mean: $\u03C7 = \frac{40}{10} = 4$.

• Weighted Mean (Facilitated Computation):

  • When scores occur multiple times, the mean is computed by multiplying each value ($X$) by its frequency ($f$), summing those products, and dividing by the total number of values ($n$).
  • Formula: $\u03C7 = \frac{\sum fX}{n}$.
  • Example Table:
  • $X = 2, f = 2, fX = 4$
  • $X = 3, f = 3, fX = 9$
  • $X = 4, f = 2, fX = 8$
  • $X = 5, f = 1, fX = 5$
  • $X = 7, f = 2, fX = 14$
  • Total: $n = 10, \sum fX = 40$.
  • Result: $\u03C7 = \frac{40}{10} = 4$.

Measures of Central Tendency: The Median

• The Median is the score point that divides a ranked distribution into two equal parts; exactly $50\%$ of the data lies below this value.

• Characteristics:

  • Unlike the mean, the median is not sensitive to extreme scores (outliers).
  • It is appropriate when there are relatively large or small values compared to the rest of the data.
  • It is useful for grouped frequency distributions with open-ended intervals.

• Formula for Median Position (Ungrouped Data):

  • Sample: $\text{mdn} = \left( \frac{n + 1}{2} \right)\text{th score from the lowest}$.
  • Population: $\text{Mdn} = \left( \frac{N + 1}{2} \right)\text{th score from the lowest}$.

• Example 1: Median of Sample Data:

  • Data: $6, 8, 15, 18, 23, 24, 42$.
  • $n = 7$, so position is $\left( \frac{7 + 1}{2} \right) = 4\text{th score}$.
  • Median: $18$.

• Example 2: Median of Population Data:

  • Data: $121, 108, 120, 98, 132, 100, 92, 140, 102, 98$.
  • Arranged in ascending order: $92, 98, 98, 100, 102, 108, 120, 121, 132, 140$.
  • $N = 10$, so position is $\left( \frac{10 + 1}{2} \right) = 5.5\text{th score}$.
  • Referencing the array, the values at the 5th and 6th positions are listed as $102$ and $108$ (the transcript notes $103$ and $108$).
  • Median calculation: $\text{Mdn} = \frac{102 + 108}{2} = 105$.

Measures of Central Tendency: The Mode

• The Mode is the value with the highest frequency in a data set. It is rarely used because it does not provide information about values other than the most frequent one.

• Classifications:

  • Bimodal: Data sets with two modes.
  • Multimodal: Data sets with more than two modes.
  • A set can have no mode if each score appears only once.

• Characteristics:

  • Like the median, it is not influenced by extreme scores.
  • It does not indicate how many values are greater or less than the most frequent value.

• Example:

  • Data: $4, 5, 8, 8, 9, 12, 20$.
  • Result: $mo = 8$.

Measures of Position: Quartiles

• Quartiles divide a distribution into four equal parts:

  • First Quartile ($Q_1$): $25\%$ of data falls below this point.
  • Second Quartile ($Q_2$): $50\%$ of data falls below this point (equal to the median).
  • Third Quartile ($Q_3$): $75\%$ of data falls below this point.

• Calculation Sequence:

  • Multiply $n$ by $\frac{1}{4}$ for $Q_1$, by $\frac{1}{2}$ for $Q_2$, and by $\frac{3}{4}$ for $Q_3$.

• Formulas for Grouped Data:

  • $Q_1 = X_{LB} + \left( \frac{\frac{n}{4} - cf_p}{f_q} \right) \times i$
  • $Q_3 = X_{LB} + \left( \frac{\frac{3n}{4} - cf_p}{f_q} \right) \times i$
  • $X_{LB}$: Lower boundary of the quartile class.
  • $cf_p$: Cumulative frequency of the class preceding the quartile class.
  • $f_q$: Frequency of the quartile class.
  • $i$: Class interval size.

• Example (Table 3.1):

  • Data table: $20-24 (f=2, cf=2)$, $25-29 (6, 8)$, $30-34 (9, 17)$, $35-39 (10, 27)$, $40-44 (12, 39)$, $45-49 (7, 46)$, $50-54 (4, 50)$.
  • $i = 5$, $n = 50$.
  • First Quartile ($Q_1$):
  • $\frac{1}{4}n = \frac{50}{4} = 12.5$. $Q_1$ class is $30-34$.
  • $X_{LB} = 29.5$, $cf_p = 8$, $f_q = 9$.
  • $Q_1 = 29.5 + \left( \frac{12.5 - 8}{9} \right) \times 5 = 32.00$.
  • Third Quartile ($Q_3$):
  • $\frac{3}{4}n = \frac{150}{4} = 37.5$. $Q_3$ class is $40-44$.
  • $X_{LB} = 39.5$, $cf_p = 27$, $f_q = 12$.
  • $Q_3 = 39.5 + \left( \frac{37.5 - 27}{12} \right) \times 5 = 43.88$.

Measures of Position: Deciles and Percentiles

• Deciles:

  • Nine score points ($D_1, D_2, \dots, D_9$) divide a distribution into ten equal parts.
  • Formulas follow the median/quartile structure using multiples of $\frac{n}{10}$.
  • Examples: $D_1 = X_{LB} + \left( \frac{\frac{1n}{10} - cf_p}{f_d} \right) \times i$; $D_9 = X_{LB} + \left( \frac{\frac{9n}{10} - cf_p}{f_d} \right) \times i$.

• Percentiles:

  • Ninety-nine score points ($P_1, P_2, \dots, P_{99}$) divide a distribution into one hundred equal parts.
  • The $k^{\text{th}}$ percentile ($P_k$) marks the point below which $k\%$ of data falls.
  • General Formula: $P_k = X_{LB} + \left( \frac{\frac{kn}{100} - cf_p}{f_p} \right) \times i$.
  • Percentile Ranks (PR): Percentages of cases below a given score.
  • Example: "$40\%$ are less than $90$" means $90$ is the percentile and $40$ is the percentile rank.
  • Example: "$70\%$ are greater than $110$" means the percentile is $110$ and the percentile rank is $30$.
  • Relationship values:
    • $Q_1, Q_2, Q_3$ ranks are $25, 50, 75$ respectively.
    • $D_1, D_2, \dots, D_9$ ranks are $10, 20, \dots, 90$.
    • The percentile rank of $P_k$ is $k$.

Sequences: Fundamentals

• Definition: A sequence function has a domain of consecutive integers, usually starting from $1$. Independent variable is $n$, function name is $a$, and values are denoted $a_n$ ("a sub n").

• Types of Sequences:

  • Infinite Sequence: The domain is the entire set of positive integers ${1, 2, 3, \dots}$.
  • Finite Sequence: The domain is restricted to integers from $1$ to a fixed integer $k$. Example: $a_n = 5n$ for $n = {1, 2, 3, 4, 5}$ identifies the sequence $5, 10, 15, 20, 25$.

• General Term ($a_n$): The formula used to find any term in the sequence.

  • Example: Write the first five terms of $a_n = n^2(2n - 1)$.
  • $a_1 = 1^2[2(1) - 1] = 1$
  • $a_2 = 2^2[2(2) - 1] = 4(3) = 12$
  • $a_3 = 3^2[2(3) - 1] = 9(5) = 45$
  • $a_4 = 4^2[2(4) - 1] = 16(7) = 112$
  • $a_5 = 5^2[2(5) - 1] = 25(9) = 225$

Arithmetic Sequences

• Definition: A sequence where the difference between consecutive terms is constant, known as the common difference ($d$).

• $n^{\text{th}}$ Term Formula: $a_n = a_1 + (n - 1)d$.

  • Example 1: Find the 30th term where $a_1 = 3$ and $d = 2$.
  • $a_{30} = 3 + (30 - 1)(2) = 3 + 58 = 61$.
  • Example 2: Find the 10th term of sequence $3, 7, 11$.
  • $d = 7 - 3 = 4$.
  • $a_{10} = 3 + (10 - 1)(4) = 3 + 36 = 39$.

• Sum of the First $n$ Terms ($S_n$):

  • Formula 1: $S_n = \frac{n}{2}(a_1 + a_n)$.
  • Formula 2: $S_n = \frac{n}{2}[2a_1 + (n - 1)d]$.
  • Example: If $a_1 = -2$, $a_{12} = 42$, and $n = 12$, find $d$ and $S_{12}$.
  • Finding $d$: $42 = -2 + 11d \rightarrow d = 4$.
  • Finding $S_{12}$: $S_{12} = \frac{12}{2}(-2 + 42) = 6(40) = 240$.

Geometric Sequences

• Definition: A sequence where the ratio between consecutive terms is constant, known as the common ratio ($r$).

• $n^{\text{th}}$ Term Formula: $a_n = a_1 r^{n-1}$.

  • Example 1: Find the 8th term of $64, -32, 16, -8, \dots$.
  • Common ratio: $r = \frac{-32}{64} = -\frac{1}{2}$.
  • $a_8 = 64 \times (-\frac{1}{2})^{8-1} = 64 \times (-\frac{1}{128}) = -\frac{1}{2}$.
  • Example 2: Find common ratio $r$ if $a_1 = 2$ and $a_4 = \frac{1}{4}$.
  • $\frac{1}{4} = 2 \times r^3 \rightarrow r^3 = \frac{1}{8} \rightarrow r = \frac{1}{2}$.

• Sum of the First $n$ Terms ($S_n$):

  • Formula 1: $S_n = \frac{a_1(1 - r^n)}{1 - r}$.
  • Formula 2: $S_n = \frac{a_1 - a_n r}{1 - r}$.
  • Condition: $r \neq 1$.

Fundamental Principle of Counting

• Principle: If a first task can be done in $n_1$ ways, a second independently in $n_2$ ways, up to a $k^{\text{th}}$ task in $n_k$ ways, the total number of ways the tasks can be done in sequence is $n_1 \times n_2 \times n_3 \times \dots \times n_k$.

• Examples:

  • Coins: Three coins fall in $(2)(2)(2) = 8$ ways ($HHH, HHT, HTH, THH, TTH, THT, HTT, TTT$).
  • Dice: Two dice fall in $(6)(6) = 36$ ways (1,1$ to $6,6).

• Digit-Based Counting (Digits: $0, 2, 5, 6, 9$):

  • 3-digit numbers (no repeats):
  • Hundreds digit: $4$ choices (cannot be $0$).
  • Tens digit: $4$ choices (includes $0$).
  • Units digit: $3$ choices.
  • Total: $4 \times 4 \times 3 = 48$.
  • Even Numbers (no repeats): Ends in $0, 2, 6$.
  • Ends in $2$ or $6$: $(3)(3)(2) = 18$.
  • Ends in $0$: $(4)(3)(1) = 12$.
  • Total: $18 + 12 = 30$.
  • Odd Numbers (no repeats): Ends in $5$ or $9$.
  • $(3)(3)(2) = 18$.
  • Greater than 600: Hundreds digit must be $6$ or $9$.
  • $(2)(4)(3) = 24$.
  • Repeats Allowed: $(4)(5)(5) = 100$ total numbers.

Permutations

• Definition: An arrangement of $n$ different objects where order matters. The number of permutations of $n$ objects is $n!$ ($n$ factorial).
• $n$ objects taken $r$ at a time: ${_n}P_r = n(n-1)(n-2) \dots (n-r+1)$.
• Examples:
  • Seating $4$ boys and $3$ girls in $5$ chairs: ${_7}P_5 = \frac{7!}{2!} = 2,520$.
  • Ranking top $3$ students out of $10$: ${_{10}}P_3 = 10 \times 9 \times 8 = 720$.
  • Permutations with Identical Objects: To find distinct permutations for the word "STATISTICS" ($S=3, T=3, I=2, A=1, C=1$):
  • $P = \frac{10!}{3! 3! 2! 1! 1!} = 50,400$.
  • Circular Permutations: Arranging objects in a circle. One object is fixed. Formula: $(n - 1)!$.
  • Example: $8$ people at a round table: $(8 - 1)! = 7! = 5,040$.

Combinations

• Definition: Selections and arrangements of objects where order does NOT matter.
• Formula: ${_n}C_r = \frac{n!}{r!(n - r)!}$.
• Examples:
  • Committee of $4$ chosen from $8$ people: ${_8}C_4 = \frac{8!}{4!4!} = 70$.
  • Selecting $2$ spades and $3$ diamonds from a deck ($13$ of each):
  • Spades: ${_{13}}C_2 = 78$.
  • Diamonds: ${_{13}}C_3 = 286$.
  • Total combinations: $78 \times 286 = 22,308$.

Probability Concepts

• Types of Probability:

  1. Subjective Approach: Personal assessment based on available evidence and intuition. Useful when there is little evidence or only one opportunity (e.g., survival of a patient, economic recession).
  2. Empirical (Relative Frequency) Approach: Proportion of times an outcome occurs in a very large number of observations.
  • Example: If $1$ in $100$ students has an IQ of $150$, the probability is $\frac{1}{100} = 0.01$.
  • Example: If $3$ of $10$ customers make a purchase, the probability is $\frac{3}{10} = 0.3$.
  1. Classical (A Priori) Approach: If an experiment has $N$ equally likely outcomes and event $E$ can result in $n$ outcomes, $P(E) = \frac{n}{N}$.

• Complementary Events: The probability of $A$ not occurring ($A'$) is $1 - P(A)$.

  • Example: Probability Candidate A wins is $0.35$. Probability Candidate B wins: $1 - 0.35 = 0.65$.

• Simple Probabilities:

  • Odd even number on one die toss: Sample Space=${1,2,3,4,5,6}$, N=$6$, Event=${2,4,6}$, n=$3$. $P = \frac{3}{6} = \frac{1}{2}$.
  • Sum of $7$ on two dice: Total outcomes $N=36$. Sum of 7 pairs: $(1,6), (6,1), (2,5), (5,2), (3,4), (4,3)$. $P = \frac{6}{36} = \frac{1}{6}$.
  • Colorful Balls: Box has $5$ red, $4$ blue, $3$ white ($N=12$).
  • Prob(not red): $1 - \frac{5}{12} = \frac{7}{12}$.
  • Prob(not white): $1 - \frac{3}{12} = \frac{9}{12} = \frac{3}{4}$.

Rules of Probability

• Mutually Exclusive vs. Non-mutually Exclusive:

  • Mutually Exclusive: Events cannot occur simultaneously (joint probability is zero). Example: Drawing an ace AND a king.
  • Non-mutually Exclusive: Events share at least one sample point. Example: Drawing an ace AND drawing a spade (the Ace of Spades).

• The Addition Rule:

  • For any events: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$.
  • Example (Card): Probability of Spade ($13$ cards) or Face Card ($12$ cards):
  • $P = \frac{13}{52} + \frac{12}{52} - \frac{3}{52} = \frac{22}{52} = \frac{11}{26}$.
  • Example (Students): Prob pass History ($0.62$), pass Sociology ($0.50$), pass both ($0.40$):
  • Prob pass at least one: $0.62 + 0.50 - 0.40 = 0.72$.

• Conditional Probability:

  • Probability that $B$ occurs given $A$ has occurred: $P(B/A) = \frac{n(A \text{ and } B)}{n(A)}$ or $P(B/A) = \frac{P(A \text{ and } B)}{P(A)}$.
  • Example (Die): Prob even ($E$) given number is $>3$ ($G$). $G={4,5,6}$. $E \cap G = {4,6}$. $P(E/G) = \frac{2}{3}$.

• Dependent and Independent Events:

  • Independent: Occurrence of one does not affect the other. $P(A/B) = P(A)$. Example: Tossing a coin and a die.
  • Dependent: One event affects the other.

• The Multiplication Rule:

  • Dependent: $P(A \text{ and } B) = P(A) \times P(B/A)$.
  • Independent: $P(A \text{ and } B) = P(A) \times P(B)$.
  • Example (Materials): Prob materials arrive on time ($0.6$). Prob finish orders if materials on time ($0.8$). Prob both: $0.6 \times 0.8 = 0.48$.

Mathematical Functions

• Definition: A central idea in mathematics where one quantity corresponds to another according to a definite rule ($y = f(x)$). Variable $x$ is the independent variable (input) and $y$ is the dependent variable (output).

• Notation Example: $f(D) = \pi D$ where $D$ is diameter and result is circumference $C$.

• Evaluation Examples:

  • Given $f(x) = 2x + 3$:
  • $f(\frac{1}{2}) = 2(\frac{1}{2}) + 3 = 4$.
  • $f(2) = 2(2) + 3 = 7$.
  • Given $f(x) = x^2 - 2x + 1$:
  • $f(-2) = (-2)^2 - 2(-2) + 1 = 4 + 4 + 1 = 9$.
  • $f(0) = 0 - 0 + 1 = 1$.
  • $f(a) = a^2 - 2a + 1$.

• Operations on Functions:

  • Sum: $(f + h)(x) = f(x) + h(x)$. Domain is intersection of domain $f$ and domain $h$.
  • Difference: $(f - h)(x) = f(x) - h(x)$.
  • Product: $(f \cdot h)(x) = f(x) \cdot h(x)$.
  • Quotient: $(\frac{f}{h})(x) = \frac{f(x)}{h(x)}$ where $h(x) \neq 0$.

• Example Comparison ($f(x) = x+1, h(x) = 2x^2$):

  1. $(f+h)(x) = 2x^2 + x + 1$. Domain: all real numbers.
  2. $(f-h)(x) = -2x^2 + x + 1$. Domain: all real numbers.
  3. $(f \cdot h)(x) = (x+1)(2x^2) = 2x^3 + 2x^2$. Domain: all real numbers.
  4. $(\frac{f}{h})(x) = \frac{x+1}{2x^2}$. Domain: all real numbers except $x = 0$.