Data Management and Logic Statement

Flashcards to help you study key vocabulary terms from lecture notes, designed for exam preparation.

Data Management

Organizing, storing, and handling data efficiently.

Statistics

Collecting, analyzing, interpreting, and presenting data.

Mean (Average)

Sum of values divided by number of values

=AVERAGE(range)

MODE → 2: STAT → 1: 1-VAR → Enter data → SHIFT → 1 → 4: Var → 2: x̄

Median

Middle value of an ordered set

=MEDIAN(range)

Manually sort → If odd: middle; If even: avg of 2 middle

Mode

Most frequent value(s)

=MODE(range) or =MODE.MULT(range)

Identify frequent value manually

Percentile

Divides data into 100 parts

=PERCENTILE(range, k)

Quartile

Divides data into 4 parts

=QUARTILE(range, k)

Range

Max – Min

=MAX(range) - MIN(range)

Sample Variance (sx^2)

Avg squared deviation (sample)

=VAR.S(range)

MODE → STAT → 1-VAR → Enter data → SHIFT → 1 → 4: Var → 4: sx → square it

Population Variance

Avg squared deviation (population)

=VAR.P(range)

SHIFT → 1 → 4: Var → 3: σx → square it

Sample Standard Deviation (sx)

Square root of sample variance

=STDEV.S(range) or =STDEV(range)

SHIFT → 1 → 4: Var → 4: sx

Population Standard Deviation (σx)

Square root of population variance

=STDEV.P(range)

SHIFT → 1 → 4: Var → 3: σx

Mean Absolute Deviation (MAD)

Avg absolute difference from mean

=AVEDEV(range)

Find the mean of the data. Subtract the mean from each data point. Take the absolute value of each difference. Sum all the absolute values. Divide the sum by the number of data points (n).

Interquartile Range (IQR)

Q3 - Q1

=QUARTILE(range,3) - QUARTILE(range,1)

Sort the Data: Arrange the data in ascending order. Find the Median (Q2):For an odd number of values: middle value. For an even number of values: average of the two middle values. Find Q1 (Lower Quartile): Median of the lower half of the data (exclude Q2). Find Q3 (Upper Quartile): Median of the upper half of the data (exclude Q2). Calculate IQR: IQR = Q3−Q1

Coefficient of variation (CV)

(Standard deviation / Mean) × 100

Coefficient of Correlation (r)

Measures linear relation of X and Y

MODE → STAT → 2: A+BX → Enter (X=, Y=) → SHIFT → 1 → 6: Reg → 3: r

Coefficient of Determination (r²)

% of Y explained by X

Square the r manually

Press Ans → x²

Slope (b)

Rate of change

SHIFT → 1 → 6: Reg → 2: b

Y-intercept (a)

Value of Y when X = 0

SHIFT → 1 → 6: Reg → 1: a

x is missing

x = y - a/b

Correlation

=CORREL(variable 1, variable 2)

Correlation Analysis

Statistical techniques to measure the association of variables.

• Measures relationship between two variables (X and Y).

• Correlation Coefficient (r):

  • Ranges from -1 to +1

    • Perfect Positive = +1

    • Perfect Negative = -1

  • No correlation = 0

Strength	Range
Strong	±0.70 to ±0.99
Moderate	±0.30 to ±0.69
Weak	±0.01 to ±0.29

Positive Correlation

As X ↑, Y ↑

See r value (0.01 to 1)

Negative Correlation

As X ↑, Y ↓

See r value (-1 to -0.01)

Perfect Correlation

r = ±1

SHIFT → 1 → 6: Reg → 3: r

No Correlation

r = 0

SHIFT → 1 → 6: Reg → 3: r

Rounding Rules

Measure	Decimal Places
Mean, Median, Mode	Same as given data
Variance, SD, Range	2 decimal places
Correlation & Regression	4 decimal places

Regression Equation

y = a + bx

SHIFT → 1 → 6: Reg → 1: a, 2: b

Regression Analysis

Used for significant correlation only. Develops a model to predict the values of the dependent variable.

Dependent Variable

Denoted by y.

Independent Variable

Denoted by x.

Logic Statements

Logic statement that is either true or false, but not both simultaneously

Imperative Sentences

Commands or requests (e.g., "Close the door." )

Exclamatory Sentences

Express strong emotions (e.g., "What a beautiful day!")

Interrogative Sentences

Questions (e.g., "Is it raining?")

Propositional Variables

A propositional variable is a symbol, typically a lowercase or uppercase letter (e.g., p, q, R), used to represent an arbitrary proposition whose truth value is unspecified.

Simple Proposition

A statement containing only one propositional variable

Compound Proposition

A statement formed by combining two or more simple propositions using logical connectives

Logical Connectives Used in Compound Statements

• Negation (¬p): "Not p" – Inverts the truth value of p.

• Conjunction (p ∧ q): "p and q" – True only if both p and q are true.

• Disjunction (p ∨ q): "p or q" – True if at least one of p or q is true.

• Implication (p → q): "If p, then q" – False only when p is true and q is false.

• Biconditional (p ↔ q): "p if and only if q" – True when p and q have the same truth value.

Propositional Form (Symbolic Form)

A sequence of symbols containing at least one propositional variable and at least one logical operator

Truth Tables

Systematically display the truth values of propositions based on their components

 Negation (¬p)

p	¬p
T	F
F	T

Translation: "Not p", "It is not the case that p", "It is false that p", "It is not true that p"

Conjunction (p ∧ q)

p	q	p ∧ q
T	T	T
T	F	F
F	T	F
F	F	F

Translation: "p and q", "p moreover q", "p although q", "p still q", "p furthermore q", "p also q", "p nevertheless q", "p however q", "p yet q", "p but q"

Disjunction (p ∨ q)

p	q	p ∨ q
T	T	T
T	F	T
F	T	T
F	F	F

Translation: "p or q"

Note: Disjunction can be inclusive (at least one is true) or exclusive (exactly one is true).

Implication (p → q)

p	q	p → q
T	T	T
T	F	F
F	T	T
F	F	T

Translation: "If p, then q", "p implies q", "p is a sufficient condition for q", "p only if q", "q is a necessary condition for p", "q if p", "q follows from p", "q provided p", "q whenever p", "q is a logical consequence of p"

Related Forms:

• Converse: q → p

• Inverse: ¬p → ¬q

• Contrapositive: ¬q → ¬p

Biconditional (p ↔ q)

p	q	p ↔ q
T	T	T
T	F	F
F	T	F
F	F	T

Translation: "p if and only if q", "p is equivalent to q", "p is a necessary and sufficient condition for q"

Order of Operations in Logical Expressions

1. Parentheses: Used to group expressions, especially with "both...and" and "either...or".

   • Both p or q, and r: (p ∨ q) ∧ r

   • p, or both q and r: p ∨ (q ∧ r)

   • Either p and q, or r: (p ∧ q) ∨ r

   • p, and either q or r: p ∧ (q ∨ r)
     
     

2. Negation of Compound Statements:

   • "Neither p nor q": ¬(p ∨ q)

   • "p and q are not both": ¬(p ∧ q)

   • "p and q are both not": ¬p ∧ ¬q

Logical Equivalences

Negation of a Conditional

¬(p → q) ≡ p ∧ ¬q

Interpretation: "It is not true that if p, then q" is equivalent to "p is true and q is false".

De Morgan's Laws

1. ¬(p ∨ q) ≡ ¬p ∧ ¬q

2. ¬(p ∧ q) ≡ ¬p ∨ ¬q

Explanation: These laws provide rules for distributing negation over conjunctions and disjunctions.

Classification of Propositions

• Tautology: A proposition that is always true.

  • Example: p ∨ ¬p

• Contradiction: A proposition that is always false.

  • Example: p ∧ ¬p

• Contingency: A proposition that is neither always true nor always false; its truth value depends on the truth values of its components.

  • Example: p ∧ q

Arguments

A collection of propositions where one proposition (the conclusion) is claimed to logically follow from the others (the premises)

Premises

Statements that provide support or reasons

Conclusion

The statement that follows from the premises, often introduced by "therefore" (∴).

Universal Quantifiers

Words like "all", "every", "no", "none" indicate that a statement applies to all elements in a domain

Existential Quantifiers

Words like "some", "there exists", "at least one" indicate that a statement applies to at least one element in a domain

Validity of Arguments

An argument is valid if, whenever all the premises are true, the conclusion is also true. It is invalid if it's possible for all premises to be true while the conclusion is false.

Determining Validity Using Truth Tables

Step 1: Symbolize each premise and the conclusion.

Step 2: Construct a truth table with columns for each premise and the conclusion.

Step 3: Identify any row where all premises are true, but the conclusion

Method A: Tautology Check

1. Create a truth table with all variables.

2. Form the conjunction (AND) of all premises.

3. Use this as the antecedent (if-part) and the conclusion as the consequent (then-part) of a conditional statement.

4. Check if the entire statement is a tautology (always true).

If it is a tautology → Argument is valid.
If not → Argument is invalid.

Method B: Direct Row Analysis

1. Make a table with columns for:

   • Premise 1

   • Premise 2

   • Conclusion

2. Look for a row where all premises are true but the conclusion is false.

If such a row exists → Argument is invalid.
Example: “The argument is invalid as shown in row 3.”

If no such row exists → Argument is valid.
Example: “The argument is valid because there isn’t any row that has both true premises but a false conclusion.”