INTRODUCTORY STATISTICS
MIDTERM EX AM CHEAT SHEET • FIRST-YEAR U.S. COLLEGE 2026
01 DATA & SAMPLING 04 FORMULA QUICK-LOOK
Population: Entire group being studied.
SAMPLE MEAN IQR
Sample: Subset of the population.
x̄ = Σx / n Q 3 − Q1
Parameter: Numerical value describing a population.
Statistic: Numerical value calculated from a sample. EXPECTED VALUE Z-SCORE
E(X) = Σx·P(x) z = (x − μ) / σ
Qualitative: Categorical / non-numerical data.
Quantitative: Numerical data. BINOMIAL MEAN BINOMIAL SD
• Discrete: Countable values. μ = np σ = √(np(1 − p))
• Continuous: Measurable on a continuum.
Sampling Methods: Simple random, stratified, cluster,
systematic. 05 BINOMIAL DISTRIBUTION
Watch Out: Convenience and voluntary-response Probability Formula:
sampling introduce severe bias.
P(X = x) = nCx · px · (1 − p)n − x
When to use:
02 DESCRIPTIVE STATISTICS 1. Fixed number of trials (n)
2. Two outcomes (Success / Failure)
Mean: x̄ = Σx / n
3. Constant probability of success (p)
Median: Middle value in an ordered dataset.
4. Independent trials
Mode: Most frequently occurring value.
Range: Range = max − min Exam Move: Always state and check the four binomial
conditions before calculating!
IQR: IQR = Q3 − Q1
Outlier Fences: Q1 − 1.5(IQR) and Q3 + 1.5(IQR)
06 NORMAL DISTRIBUTION
Skewness:
• Right-skew: Mean usually > Median Properties: Bell-shaped and symmetric, centered at μ.
• Left-skew: Mean usually < Median Standard Normal: Z ~ N(0, 1)
Standard Deviation (SD): Typical spread/distance of Empirical Rule (68-95-99.7):
data points around the mean.
• 68% of data within 1 SD (μ ± 1σ)
• 95% of data within 2 SD (μ ± 2σ)
03 PROBABILITY
• 99.7% of data within 3 SD (μ ± 3σ)
Range: 0 ≤ P(A) ≤ 1 z-Score Interpretation:
Complement Rule: P(Ac) = 1 − P(A) • z > 0: Above the mean
Addition Rule (OR): • z < 0: Below the mean
P(A ∪ B) = P(A) + P(B) − P(A ∩ B) • z = 0: Exactly at the mean
Conditional Probability: Probability: Represented by the area under the curve.
P(A | B) = P(A ∩ B) / P(B)
Exam Move: Sketch and shade the curve before
Independent Events: P(A ∩ B) = P(A) · P(B) calculating probabilities!
Mutually Exclusive: P(A ∩ B) = 0
MIDTERM EX AM CHEAT SHEET • FIRST-YEAR U.S. COLLEGE 2026
01 DATA & SAMPLING 04 FORMULA QUICK-LOOK
Population: Entire group being studied.
SAMPLE MEAN IQR
Sample: Subset of the population.
x̄ = Σx / n Q 3 − Q1
Parameter: Numerical value describing a population.
Statistic: Numerical value calculated from a sample. EXPECTED VALUE Z-SCORE
E(X) = Σx·P(x) z = (x − μ) / σ
Qualitative: Categorical / non-numerical data.
Quantitative: Numerical data. BINOMIAL MEAN BINOMIAL SD
• Discrete: Countable values. μ = np σ = √(np(1 − p))
• Continuous: Measurable on a continuum.
Sampling Methods: Simple random, stratified, cluster,
systematic. 05 BINOMIAL DISTRIBUTION
Watch Out: Convenience and voluntary-response Probability Formula:
sampling introduce severe bias.
P(X = x) = nCx · px · (1 − p)n − x
When to use:
02 DESCRIPTIVE STATISTICS 1. Fixed number of trials (n)
2. Two outcomes (Success / Failure)
Mean: x̄ = Σx / n
3. Constant probability of success (p)
Median: Middle value in an ordered dataset.
4. Independent trials
Mode: Most frequently occurring value.
Range: Range = max − min Exam Move: Always state and check the four binomial
conditions before calculating!
IQR: IQR = Q3 − Q1
Outlier Fences: Q1 − 1.5(IQR) and Q3 + 1.5(IQR)
06 NORMAL DISTRIBUTION
Skewness:
• Right-skew: Mean usually > Median Properties: Bell-shaped and symmetric, centered at μ.
• Left-skew: Mean usually < Median Standard Normal: Z ~ N(0, 1)
Standard Deviation (SD): Typical spread/distance of Empirical Rule (68-95-99.7):
data points around the mean.
• 68% of data within 1 SD (μ ± 1σ)
• 95% of data within 2 SD (μ ± 2σ)
03 PROBABILITY
• 99.7% of data within 3 SD (μ ± 3σ)
Range: 0 ≤ P(A) ≤ 1 z-Score Interpretation:
Complement Rule: P(Ac) = 1 − P(A) • z > 0: Above the mean
Addition Rule (OR): • z < 0: Below the mean
P(A ∪ B) = P(A) + P(B) − P(A ∩ B) • z = 0: Exactly at the mean
Conditional Probability: Probability: Represented by the area under the curve.
P(A | B) = P(A ∩ B) / P(B)
Exam Move: Sketch and shade the curve before
Independent Events: P(A ∩ B) = P(A) · P(B) calculating probabilities!
Mutually Exclusive: P(A ∩ B) = 0
