Notes on Normal Distribution, Z-scores, and Calculator Techniques

Normal Distribution, Z-scores, and Calculator Techniques

• Goal: understand how to work with the normal model, compute areas (probabilities) between z-scores or values, and recover values from given areas using calculators or inverse norm tables.

• Notation and key concepts:

  • Normal model: a random variable Y with mean er c mean μ and standard deviation  = σ, denoted Y  Normal(μ, σ^2).
  • Standard normal: Z  Normal(0, 1). Relationship: z = (y - μ)/σ and y = μ + z σ.
  • Cumulative distribution function (CDF) of the standard normal: Φ(z) = P(Z  z) for Z  N(0,1).
  • To find probabilities for a non-standard normal, standardize or use NormalCDF with mean μ and sd σ.
  • Inverse normal: given a left-tail area p, z = Φ^{-1}(p)  the z-score at which Φ(z) = p. Some calculators call this invNorm.
  • Percentiles and quartiles: the p-th percentile has area p/100 to the left of the value.
  • The empirical rule (68-95-99.7):
  • About 68% within [-1σ, +1σ], 95% within [-2σ, +2σ], 99.7% within [-3σ, +3σ].
  • Practical advice: use sketches, keep track of tails, and report probabilities with consistent decimal precision (often four decimals). Round cautiously; rounding can affect results slightly.
  • Be mindful: mean and standard deviation are not robust to outliers. If there are outliers or skewness, consider other models or use a picture before applying the normal model.

• Calculator workflow (Normal CDF vs. Inverse Norm):

  • To compute area between two z-scores (standard normal): use Normal CDF with lower bound L and upper bound U, and default μ = 0, σ = 1:
  • Area = P(L  Z  U) = Φ(U) - Φ(L).
  • If your calculator has a wizard, you typically select Normal CDF (often option 2).
  • For standard normal: lower bound and upper bound are z-scores; mean μ = 0, sd σ = 1.
  • If your calculator is older and lacks a wizard, you type: NormalCDF(lower, upper, mean, sd).
  • Infinity substitutes on calculators: use a very large magnitude value for effectively infinite bounds, e.g., -10^3, -10^4, etc. The limit is effectively reached long before these stand out values change the result.
  • Example interpretation: Normal CDF between -1.43 and 0.87 (with μ=0, σ=1) gives the area under the curve in that interval, i.e., the probability a standard normal variable falls in that range (roughly 0.73).

• Working with actual scores vs. z-scores:

  • If you have raw data (mean μ and sd σ) and you want the probability that Y lies between a and b, you can either:
  • Standardize a and b to z-values: za = (a - μ)/σ, zb = (b - μ)/σ, then compute Φ(zb) - Φ(za).
  • Or use NormalCDF with μ and σ directly: NormalCDF(a, b, μ, σ).
  • If the question asks for a z-score corresponding to a percentile, use Inverse Norm (invNorm) with the left-tail area p (or equivalently use the right-tail with 1-p, depending on calculator tail option). If the calculator only returns left-tail areas, always think in terms of the left tail.

• Practical tips when solving problems:

  • Always draw a quick sketch of the normal curve and shade the region of interest.
  • Use the standard normal to reason about rough percentages, then compute exactly via calculator or tables.
  • When converting between z-scores and actual scores (unstandardizing): y = μ + z σ.
  • If a percentile or area is given (e.g., 25th percentile), convert to a z-score using invNorm on the left-tail (or equivalently 1 - p for a right-tail, depending on your calculator).
  • When reporting results:
  • If asked for a percentage, convert the probability to a percentage (multiply by 100).
  • If asked for a z-score, give the z value (not the raw score).
  • If asked for an area between two values, report the probability (as a decimal or percentage as requested).

• Quick note on data contexts:

  • The normal model is appropriate only if data are approximately normal: unimodal, symmetric, and not bounded on one side in an unrealistic way.
  • Income distributions or other data with obvious lower bounds or skew may violate normality; be cautious and check with a picture.

• Common denotations and formulas you should memorize:

  • z-score:  z = rac{y - {}{}}{{}{}}
  • Standardization: Z = rac{Y - }{{}{}}
  • Area between two z-scores: P(a  Z  b) = (1)(b) - (1)(a) = \,\Phi(b) - \,\Phi(a)
  • Inverse normal (to find z from area): $z = \Phi^{-1}(p)$
  • Unstandardization: $y = \mu + z \sigma$
  • If you use the standard normal with μ=0, σ=1, you can simplify to: P(a  Z  b) = \Phi(b) - \Phi(a)

• Notes on precision and rounding:

  • In example problems, different decimal places in z or probability can yield slightly different results. Always show your exact inputs to allow others to reproduce your work.
  • For many problems, four decimal places in probabilities are typical in education settings.

• Quick practice prompts (to reinforce the ideas):

  • Compute P(-2.0 ≤ Z ≤ 2.0).
  • Find the z-score corresponding to the 25th percentile (Q1).
  • A test score is normally distributed with μ = 500 and σ = 100. What score is needed to be in the top 10%?
  • A company reports mean income μ = 1200 and σ = 600. What proportion earns between 1400 and 2250?
  • If 70% of data lie within [-1σ, 1σ] of the mean, what is the corresponding central interval in raw units for μ and σ? (Invert the idea to find y-values).

• Practical exam-oriented notes from the session:

  • The exam will likely include a mix of questions asking for z-scores, normal CDF computations (areas between values), and inverse norm (finding scores or z-scores from areas).
  • You may be asked to produce your own sketched diagrams and show work steps (not just final answers).
  • Handwritten two-sided note sheets (front and back on 8.5"x11" paper) are allowed/required; include essential formulas, typical problems, and quick methods.

• Relationship to other topics:

  • Five-number summary and box plots relate to distributions and can be used to assess normality visually.
  • Histograms and dot plots help visualize quantitative variables and guide whether the normal model is appropriate.
  • Contingency tables and marginal/conditional distributions cover categorical data, offering a contrast to the continuous normal model.

• Summary of key ideas to remember:

  • Use Normal CDF to find area between two z-scores or between a lower and an upper bound, with μ and σ if not standard normal.
  • Use Inverse Norm to convert an area (percentile) to the corresponding z-score (and then optionally to a raw score using y = μ + z σ).
  • Always start with a picture and identify which tail you are using (left, right, or between).
  • Check whether the normal model is appropriate for the data you are modeling; otherwise, consider alternative distributions or non-parametric methods.

• Reference examples cited in the transcript for practice:

  • Example A: Height distribution with μ = 70 inches and σ = 3 inches. Proportion taller than 72 inches (6 feet): compute z = (72 - 70)/3 = 0.6667 and area P(Z > 0.6667) ≈ 1 - Φ(0.6667) ≈ 0.2525 (about 25%). Alternatively, compute via NormalCDF with lower bound 72, upper bound ∞, μ = 70, σ = 3.
  • Example B: SAT verbal scores approximately Normal(μ = 505, σ = 110). Top 10% cutoff: area to the left = 0.9; z ≈ Φ^{-1}(0.9) ≈ 1.2816; cutoff score y = 505 + 1.2816 × 110 ≈ 646.0.
  • Example C: College acceptance with top 60% criterion (i.e., the score above which 40% lie to the left). Solve Φ(z) = 0.4; z ≈ -0.2533; raw cutoff y ≈ μ + z σ with μ = 500, σ = 100 → y ≈ 475.
  • Example D: Income problem with μ = 1200, σ = 600; between 1400 and 2250. z1 = (1400 - 1200)/600 = 0.3333; z2 = (2250 - 1200)/600 = 1.75; Probability ≈ Φ(1.75) - Φ(0.3333) ≈ 0.9599 - 0.6306 ≈ 0.3293 (about 32.9%). Depending on rounding, you might see values around 33.4% in practice.
  • Example E: Quartiles and z-scores: Q1 corresponds to left-tail area 0.25; z{0.25} ≈ Φ^{-1}(0.25) ≈ -0.6745; Q3 corresponds to left-tail area 0.75; z{0.75} ≈ Φ^{-1}(0.75) ≈ 0.6745. The symmetric relation yields z{Q1} = -0.6745 and z{Q3} = +0.6745.
  • Example F: Using inverse norm with left-tail area 0.25 to estimate Q1 and converting back to raw scores using y = μ + z σ.

Calculations and examples (worked steps with formulas)

• Probability between z-scores: $P(a \,\le\, Z \,\le\, b) = \Phi(b) - \Phi(a).$

• Infinite bounds approximation on calculators:

  • Lower bound: use a very large negative number, e.g., $-10^3 \quad \text{or} \quad -10^4.$
  • Upper bound: use a very large positive number, e.g., $10^3 \quad \text{or} \quad 10^4.$

• Example: Between -1.43 and 0.87 (standard normal):

  • Lower bound L = -1.43, upper bound U = 0.87, μ = 0, σ = 1.
  • Area ≈ $\text{NormalCDF}(-1.43, 0.87, 0, 1) \approx 0.73.$

• Example: Z between -2.35 and 1.52 (standard normal):

  • Area ≈ $\text{NormalCDF}(-2.35, 1.52, 0, 1) \approx 0.9264.$

• Example: Z between 0.85 and 1.56 (standard normal):

  • Area ≈ $\text{NormalCDF}(0.85, 1.56, 0, 1) \approx 0.1383.$

• Example: Z between 0 and 1 (standard normal):

  • Area ≈ Φ(1) - Φ(0) = 0.8413 - 0.5 = 0.3413 (34.13%).

• Example: P(Z < 1.63):

  • Area ≈ Φ(1.63) ≈ 0.9484.

• Example: Inverse norm for Q1 (left tail area 0.25):

  • z ≈ Φ^{-1}(0.25) ≈ -0.6745; Q3 z ≈ +0.6745 (due to symmetry).

• Example: SAT verbal top 10% (μ = 505, σ = 110):

  • Left-tail area for top 10% is 0.90; z ≈ Φ^{-1}(0.90) ≈ 1.2816.
  • Cutoff score y = μ + z σ ≈ 505 + 1.2816 × 110 ≈ 646.

• Example: University acceptance top 60% cutoff (μ = 500, σ = 100):

  • Left-tail area for top 60% is 0.40; z ≈ Φ^{-1}(0.40) ≈ -0.2533.
  • Cutoff score y ≈ μ + z σ ≈ 500 + (-0.2533) × 100 ≈ 474.7  475.

• Example: Incomes (μ = 1200, σ = 600) between 1400 and 2250:

  • Lower bound z1 = (1400 - 1200)/600 = 0.3333; Upper bound z2 = (2250 - 1200)/600 = 1.75.
  • Probability ≈ Φ(1.75) - Φ(0.3333) ≈ 0.9599 - 0.6306 ≈ 0.3293 (approximately 32.93%).

• Percentiles and areas in practice:

  • If you know an area to the left (e.g., 0.25, 0.90), use invNorm to find the corresponding z; then convert back to a raw score if needed.
  • If you know a percentile and you are using a standard normal model with μ, σ, you can use z and then y = μ + z σ.

• Common pitfalls to avoid:

  • Mixing up left-tail vs. right-tail areas; always check whether you want P(Z ≤ z) or P(Z ≥ z).
  • Rounding differences in z-values can affect probability values; ensure inputs are consistent.
  • When the data are not well-modeled by a normal distribution (e.g., incomes with a hard lower bound at 0), be cautious about applying the normal model; consider transformations or different models.
  • Ensure that any required decimal precision aligns with the problem's instructions (e.g., report to four decimal places for probabilities).

• Quick study advice from the session:

  • Build a personal practice set that mixes problems about areas and inverse areas.
  • Practice calculating both using the standard normal and directly with μ, σ in NormalCDF.
  • Practice creating and reading five-number summaries and box plots to assess normality visually.
  • Prepare for a mix of multiple-choice and show-your-work problems.
  • Handwrite your one-page note sheet and bring it to the exam; ensure it contains essential formulas, steps, and common problem templates.

• Summary of formulas to memorize (handy quick reference):

  • Standardization: $z = \frac{y - \mu}{\sigma}.$
  • Area between bounds (standard normal): $P(a \le Z \le b) = \Phi(b) - \Phi(a).$
  • Raw-score from z-score: $y = \mu + z\sigma.$
  • Inverse normal (percentile to z): $z = \Phi^{-1}(p).$
  • Probability with raw bounds: $P(a \le Y \le b) = \text{NormalCDF}(a, b, \mu, \sigma).$
  • Infinite bounds approximation: use large negatives/positives for -∞ and ∞ when using calculators.

End of summary