Statistics - Empirical Rule, Normal Distribution, and Z-Scores (Vocabulary Flashcards)

Empirical Rule (68-95-99.7) and Z-scores: Study Notes

• Administrative note: If you have accommodations and need to fill out the form, submit the DRC request form to the instructor ASAP.
• This material focuses on the empirical rule, areas under the normal curve, standardization, and reading probability statements using the standard normal distribution.

The Empirical Rule (68-95-99.7) and its interpretation

• The empirical rule describes approximate areas under a normal curve centered at the mean μ with standard deviation σ.
• Key percentages around the center:
  • About 68% of data lie within one standard deviation of the mean: P(-1 \,\le\, Z \le 1) \approx 0.68.
  • About 95% lie within two standard deviations: P(-2 \,\le\, Z \le 2) \approx 0.95.
  • About 99.7% lie within three standard deviations: P(-3 \,\le\, Z \le 3) \approx 0.997.
• Interpretation: When solving area-under-curve problems quickly, you can expect these approximate cutoffs to appear, especially for questions framed around “one, two, or three standard deviations.”
• Important caveat: The empirical rule is historical and approximate; exact results require more precise methods.
• Practical reminders:
  • Some problems ask for percentages (out of 100) and others for proportions (as fractions). Pay attention to what is requested.
  • Visualize the center and how far you move from μ in both directions to locate the relevant area.

Worked example: test scores (normal assumption)

• Given: mean μ = 288, standard deviation σ = 38, distribution ~ N(μ, σ^2).
• To capture about 95% of scores, use the ±2σ rule:
  • Interval: [\mu - 2\sigma, \mu + 2\sigma] = [288 - 2\cdot 38,\; 288 + 2\cdot 38] = [212, 364].
• Practical usage: This interval represents approximately 95% of students.
• In reverse problems, you may know a range and need to find the total number of students in that range: compute the percentage for the interval and multiply by the total number of students.
  • Example given: the range 212 to 364 corresponds to about 95% of students; if total students are N, expected in-range is roughly 0.95\,N.
• Another variant: finding how many scored higher than a given value, e.g., 326.
  • 326 is μ + σ (since 326 = 288 + 38), i.e., z = 1.
  • Area to the right of +1σ is about 16%: P(X > 326) = P(Z > 1) \approx 0.1587 \approx 16\%.
  • The speaker notes that shading the appropriate tail or segment is often easier than memorizing exact numbers.
• General strategy: know the center μ, the spread σ, and identify how many standard deviations away the cutoff is; shade the corresponding area.
• A rough check for class size example: using the empirical rule (and the above intervals) you may estimate counts like several thousand depending on the total population. The key idea is to connect area with counts via the percentage of the population in that area.

From empirical rule to standardization: the z-score and standard normal distribution

• Motivation: To compare different distributions or different data sets, standardize observations so they can be compared on a common scale.
• Standardization transform (z-score):
  • Define Z = \dfrac{X - \mu}{\sigma}.
  • The resulting variable Z ~ N(0, 1), the standard normal distribution.
• Consequences:
  • Shifts and scales do not change the shape; only location and scale are adjusted.
  • All normal distributions can be mapped to the standard normal distribution for comparison.
• Interpretation of z-scores:
  • A positive z-score means the value is above the mean; a negative z-score means it is below the mean.
  • A higher z-score indicates relatively better performance within its group (percentile rank relative to the group).
• Practical example: ACT vs SAT scores
  • Although raw scores differ, you can compare performance by computing z-scores for each test using their respective means and std deviations.
  • If both z-scores are known, you can say who performed relatively better within their own distribution.
  • The key is to measure relative standing, not absolute score equivalence.
• Takeaway: Standardizing to Z enables cross-domain comparisons and reduces multiple distributions to a single, interpretable framework.

The standard normal table (z-table): reading areas left of Z

• The z-table provides areas to the left of a given z-value: P(Z \le z) = \Phi(z).
• How to use the table:
  • If you need P(Z < a), look up Φ(a) in the table.
  • If you need P(Z > a), use the complement: P(Z > a) = 1 - \,\Phi(a).
  • If you need P(a < Z < b), compute: \Phi(b) - \Phi(a).
• Table layout and symmetry:
  • The first page typically covers negative z-values; the second page covers positive z-values.
  • By symmetry, Φ(-z) = 1 - Φ(z), and probabilities can be obtained using either page and these relationships.
• Examples from the transcript (illustrative values):
  • For z = -2.56, the area to the left is about \Phi(-2.56) \approx 0.0052. This is about 0.52% in the left tail.
  • For z = -1.67, the area to the left is about \Phi(-1.67) \approx 0.0475. This is about 4.75% in the left tail.
• Notes on extremes:
  • For z-values far in the tail beyond the table’s range, the table rounds to essentially zero; in practice, probabilities become negligibly small (often reported as 0.0000 when rounded to four decimal places).
• Practical tips:
  • When a problem asks for a tail probability (area to the left of a negative z or to the right of a positive z), convert using Φ(z) and 1 − Φ(z).
  • If the problem asks for probability between two z-values, use the difference of their left-tail areas: P(a < Z < b) = \Phi(b) - \Phi(a).

Interpreting and applying probability statements (in z terms)

• The six standard probability statements with Z generally involve z in them (the standard normal curve):
  • Example templates include: P(-1 ≤ Z ≤ 1), P(Z ≤ 1), P(Z ≥ -1), P(Z > 2.5), P(-∞ < Z ≤ z), etc.
• The empirical rule can be re-expressed in z terms, e.g.:
  • P(-1 ≤ Z ≤ 1) ≈ 0.68
  • P(-2 ≤ Z ≤ 2) ≈ 0.95
  • P(-3 ≤ Z ≤ 3) ≈ 0.997
• Reading a probability statement like "the probability that something is greater than 326" in the context of a raw-score distribution requires converting to z-scores first if you only have μ and σ.
  • Convert the cutoff to z: z = \dfrac{326 - \mu}{\sigma}.
  • Then use the z-table to find the corresponding tail probability and interpret in the correct direction (left or right tail).
• General strategy: translate every problem into the standard normal framework, read areas using the z-table, and then interpret results in the original units if needed.

Practical problem-solving strategies and common pitfalls

• Visualize and draw the distribution: shading the appropriate region helps avoid misinterpretation of the direction and boundary.
• Always check whether the problem asks for a percentage (out of 100) or a proportion (as a fraction). This affects how you report the answer.
• When solving for counts from a percent, multiply the percent by the total population: count = (percent) × (total).
• When a problem involves a cutoff point like 326, recognize whether that corresponds to within 1σ, 2σ, or 3σ, to determine the likely tail or central region.
• Be mindful of the difference between a normal distribution (any μ, σ) and the standard normal distribution N(0, 1). If a problem states "normal distribution" but provides μ and σ, you may need to standardize using Z.
• Boundary considerations in continuous distributions:
  • Whether you use < or ≤ changes nothing for continuous distributions, because the probability of hitting an exact point is zero.
• For area-between-two-values problems, you can compute the larger interval’s area and subtract the smaller interval’s area, or simply use Φ(b) − Φ(a) when you have left-tail areas.
• Numerical precision and table use:
  • Normal tables are typically tabulated to four decimal places in probabilities; rounding is common in practice, especially for tail areas.
• Practical caution about empirical rule:
  • It is a convenient rough guide, not a precise calculation. For accurate results, use the standard normal table or software to compute Φ(z) values.

Quick reference: key formulas and conversions

• Z-score transformation:
  Z = \dfrac{X - \mu}{\sigma}.
• Standard normal distribution: Z \sim N(0,1).
• Area relations:
  • P(Z \le z) = \Phi(z).
  • P(Z > z) = 1 - \Phi(z).
  • P(a < Z < b) = \Phi(b) - \Phi(a).
• Empirical rule in z terms:
  • P(-1 \le Z \le 1) \approx 0.68,
  • P(-2 \le Z \le 2) \approx 0.95,
  • P(-3 \le Z \le 3) \approx 0.997.$
• Interval in raw scores corresponding to ±2σ:
  • [\mu - 2\sigma, \mu + 2\sigma].
  • Example: with μ = 288, σ = 38, the 95% interval is [212, 364].
• Rough estimate of σ from a range (normal assumption):
  • If the data range is from a to b, rough estimate:
  • \sigma \approx \dfrac{b - a}{6}.
• Numerical tail probabilities in the tails (example values):
  • \Phi(-2.56) \approx 0.0052, approximately 0.52% in the far left tail.
  • \Phi(-1.67) \approx 0.0475,$$ approximately 4.75% in the far left tail.
• Interpretation reminder: A higher z-score indicates a value that is relatively higher within its distribution; negative z-scores indicate below-average performance within the distribution.

Connections to prior concepts and real-world relevance

• The empirical rule and z-scores connect to the broader idea of standardization used across statistics and data analysis to compare disparate datasets.
• Standardization enables meaningful comparisons across different tests (e.g., ACT vs SAT) by placing scores on a common scale, highlighting relative performance rather than absolute scores.
• Understanding symmetry and area properties reduces computational burden and aids intuition when solving probability problems by hand, especially in contexts without calculators or software.
• The historical note about tabulated functions (logarithms, trigonometric tables) underscores the evolution of how we compute areas under curves and highlights why modern methods rely on standard normal tables and software.

Ethical, philosophical, and practical implications

• Interpreting standardized scores responsibly involves recognizing that z-scores measure relative standing, not absolute ability. Communicators should avoid implying that higher z-scores mean universally better outcomes across contexts without considering the underlying distribution and sample.
• When comparing groups (e.g., different classes or tests), ensure that the underlying distributions are appropriate for standardization; differences in population characteristics can affect comparability.
• Relying on empirical rules for precise decisions can be risky; use exact calculations when precision matters (e.g., high-stakes testing, policy decisions).

Summary of takeaways

• The empirical rule provides quick, approximate areas around the mean for a normal distribution: ~68% within ±1σ, ~95% within ±2σ, ~99.7% within ±3σ.
• Z-scores standardize any normal distribution to N(0,1), enabling direct comparison across different datasets and problems.
• The z-table gives areas to the left of a given z; use 1 − Φ(z) for areas to the right and Φ(b) − Φ(a) for areas between two z-values.
• In real problems, practice translating raw-score questions into z-scores, reading the z-table, and interpreting results in terms of the original units and context.
• Mastery comes from combining visual intuition (drawing the curve and shading) with the formal z-table rules, while avoiding overreliance on any single method or tool.