Normal Distribution and Z-Scores: Worked Examples
A focused walkthrough of the normal distribution and z-scores: properties of the bell curve, the empirical rule, computing z-scores, looking up probabilities, and worked examples covering exam scores, manufacturing tolerances, and quality control charts.
What You'll Learn
- ✓State the properties of the normal distribution
- ✓Apply the empirical rule (68-95-99.7)
- ✓Compute z-scores for any normal variable
- ✓Look up probabilities using z-tables or software
- ✓Apply to exam scores, manufacturing, and quality control
1. Properties of the Normal Distribution
The normal distribution is the famous bell curve — symmetric, unimodal, and asymptotic to the x-axis. It is fully characterized by two parameters: mean (mu) and standard deviation (sigma). The PDF is f(x) = (1 / (sigma × sqrt(2pi))) × e^(-(x-mu)^2 / (2sigma^2)). Key properties. (1) Symmetric around the mean — mean = median = mode. (2) Total area under the curve = 1. (3) Approximately 68% of values lie within 1 SD of mean; 95% within 2 SD; 99.7% within 3 SD (the "empirical rule"). (4) Asymptotic to x-axis — values can theoretically range from -infinity to +infinity, but probabilities beyond 4-5 SD are vanishingly small. (5) Z-distribution is the standard normal: mean = 0, SD = 1. Why normal matters in statistics. (1) Many natural phenomena approximate normal (heights, weights, exam scores). (2) The Central Limit Theorem produces approximately normal sampling distributions for means even when source data is non-normal. (3) Many inferential tests assume normality of residuals or sampling distributions. When normal fits poorly. Skewed data (income, time-to-event), heavy tails (financial returns, extreme events), bounded data (proportions near 0 or 1), and multi-modal distributions. Many transformations (log, square root, Box-Cox) can move non-normal data toward approximate normality.
Key Points
- •Symmetric, unimodal, asymptotic (bell curve)
- •Two parameters: mean (mu), standard deviation (sigma)
- •Empirical rule: 68-95-99.7 within 1, 2, 3 SD
- •Standard normal (z): mean = 0, SD = 1
- •Central Limit Theorem produces normal sampling distributions
2. The Z-Score Formula
A z-score (standard score) converts any value from a normal distribution into a standardized value on the standard normal scale. Formula: z = (X - mu) / sigma. Where X is the observed value, mu is the population mean, and sigma is the population standard deviation. Interpretation. Z = 0 means the value equals the mean. Z = 1 means the value is 1 SD above the mean. Z = -2 means 2 SD below the mean. Z = 2.5 means 2.5 SD above the mean (in the right tail, typically rare). Why convert to z-scores. (1) Compare values from different distributions. A student scoring 85 on a test with mean 75, SD 10 has z = 1.0. A student scoring 920 on the SAT with mean 1000, SD 100 has z = -0.8. The first student is above average; the second is below. Comparing raw scores directly is meaningless across different scales. (2) Look up probabilities. The z-table gives the probability of z being less than a specified value. (3) Identify outliers. Common rule: |z| > 3 indicates a potential outlier. Worked z-score example. Student scores 88 on an exam with mean 75 and SD 8. Z = (88 - 75) / 8 = 1.625. This student is 1.625 standard deviations above the mean. From the z-table, P(Z < 1.625) = 0.948. The student outperformed approximately 94.8% of test-takers.
Key Points
- •z = (X - mu) / sigma
- •Z = 0 at mean; Z = 1 at +1 SD; Z = -1 at -1 SD
- •Used for cross-distribution comparison
- •Used for probability lookup via z-table
- •|z| > 3 typically flags potential outlier
3. Empirical Rule and Common Probability Lookups
The empirical rule provides quick estimates without needing a z-table. | Range | Probability | |---|---| | Within 1 SD (z between -1 and 1) | 68.27% | | Within 2 SD (z between -2 and 2) | 95.45% | | Within 3 SD (z between -3 and 3) | 99.73% | | Beyond 1 SD (|z| > 1) | 31.73% | | Beyond 2 SD (|z| > 2) | 4.55% | | Beyond 3 SD (|z| > 3) | 0.27% | For more precise lookups, the z-table gives P(Z < z) for any z value. Modern software (R: pnorm(); Python: scipy.stats.norm.cdf(); Excel: NORM.S.DIST()) computes exact probabilities. Common thresholds for inference. P(Z > 1.96) = 0.025 (used for two-tailed alpha = 0.05). P(Z > 1.645) = 0.05 (used for one-tailed alpha = 0.05). P(Z > 2.576) = 0.005 (used for two-tailed alpha = 0.01). These thresholds appear constantly in hypothesis testing and confidence interval construction.
Key Points
- •Empirical rule: 68-95-99.7 within 1, 2, 3 SD
- •P(Z < 1.96) = 0.975 (critical value for two-tailed alpha = 0.05)
- •P(Z < 1.645) = 0.95 (critical value for one-tailed alpha = 0.05)
- •P(Z < 2.576) = 0.995 (critical value for two-tailed alpha = 0.01)
- •Software functions: pnorm (R), norm.cdf (Python), NORM.S.DIST (Excel)
4. Three Worked Examples
Example 1: Exam Scores. A standardized test has mean 500 and SD 100, approximately normal. A student scores 650. What percentile is this score? Z = (650 - 500) / 100 = 1.5. From z-table, P(Z < 1.5) = 0.933. The score is at the 93rd percentile. Probability of randomly drawing a higher score: 1 - 0.933 = 6.7%. Example 2: Manufacturing Tolerances. A machine produces parts with mean diameter 10 mm and SD 0.1 mm, approximately normal. Specifications require diameters between 9.85 and 10.15 mm. What proportion of parts meet spec? Z for 9.85: (9.85 - 10) / 0.1 = -1.5. Z for 10.15: (10.15 - 10) / 0.1 = 1.5. P(-1.5 < Z < 1.5) = P(Z < 1.5) - P(Z < -1.5) = 0.933 - 0.067 = 0.866. About 86.6% of parts meet spec; 13.4% fail. If the machine can be tightened to SD = 0.05, the new z values become ±3, yielding 99.73% within spec — a major reduction in defect rate. Example 3: Quality Control Chart. A QC chart monitors the mean of 25-unit samples. Process parameters: mu = 100, sigma = 5. By CLT, sample mean has approximately Normal(100, 5/sqrt(25)) = Normal(100, 1). Control limits at ±3 SE from process mean: 97 to 103. Each new sample mean computed and plotted. If a sample mean falls outside 97-103, the process is flagged for investigation. Probability of false alarm per sample = 0.27%, so a sample falling outside ±3 SE strongly suggests the process has shifted (not just random variation).
Key Points
- •Exam scores: convert to percentile via z-table
- •Manufacturing: compute proportion within specification limits
- •QC charts: control limits at ±3 SE from process mean
- •Tightening SD reduces defect rate dramatically
- •CLT applies sampling distribution of mean is approximately normal
5. How StatsIQ Helps With Normal Distribution Problems
Snap a photo of any normal distribution problem (probability, percentile, threshold, sample size) and StatsIQ produces the z-score, probability lookup, and worked solution. For non-normal data, the app runs normality tests (Shapiro-Wilk, Q-Q plots) and recommends appropriate transformations. For exam prep, StatsIQ generates practice problems at all levels of complexity, including problems requiring the empirical rule and problems requiring exact z-table lookups. This content is for educational purposes only.
Key Points
- •Computes z-scores and probabilities
- •Runs normality tests on data
- •Recommends transformations for non-normal data
- •Practice problems at varying difficulty
- •Works from photographed problem statements
Key Takeaways
- ★Normal distribution: symmetric bell curve with mean mu, SD sigma
- ★Empirical rule: 68% within 1 SD, 95% within 2, 99.7% within 3
- ★Standard normal (z): mean = 0, SD = 1
- ★Z-score formula: z = (X - mu) / sigma
- ★Z-table gives P(Z < z) for any z value
- ★P(Z < 1.96) = 0.975 (two-tailed alpha = 0.05 critical value)
- ★P(Z < 1.645) = 0.95 (one-tailed alpha = 0.05 critical value)
- ★P(Z < 2.576) = 0.995 (two-tailed alpha = 0.01 critical value)
- ★Central Limit Theorem: sample means approach normal for n ≥ 30
- ★Sample mean SD = population SD / sqrt(n)
- ★|z| > 3 typically flags outlier
- ★Software: pnorm (R), norm.cdf (Python), NORM.S.DIST (Excel)
Practice Questions
1. A test has mean 80 and SD 10. A student scores 95. What is the z-score and approximate percentile?
2. A normally distributed variable has mean 50 and SD 5. What proportion of values lie between 45 and 55?
3. A manufacturing process has mean 100 mm and SD 2 mm. Specifications require values between 96 and 104 mm. What proportion of parts meet spec?
4. In a normal distribution, what percentage of values fall above z = 1.96?
5. A population has mean 100, SD 20. A sample of n = 25 is taken. What is the standard error of the sample mean?
FAQs
Common questions about this topic
These are the integrated areas under the normal PDF between -1 and 1 SD, -2 and 2 SD, and -3 and 3 SD. The exact values are 68.27%, 95.45%, and 99.73%, rounded for memorization. These are properties of the standard normal distribution and apply to any normal distribution after standardization. They are extremely useful for quick mental estimates without needing a z-table.
You can compute a z-score for any value (z = (X - mean) / SD), but the probability interpretation requires normality. For non-normal data, a z-score of 2 does not necessarily mean 97.5th percentile — it could be much higher or lower depending on the distribution shape. For symmetric but non-normal distributions, Chebyshev's inequality gives a bound: at least 1 - 1/k² values lie within k SD of mean. So 75% within 2 SD, 89% within 3 SD — much looser than empirical rule.
The CLT states that the sampling distribution of the sample mean approaches normal as sample size grows, regardless of the source population distribution shape (with mild conditions). For most distributions, n ≥ 30 produces a near-normal sampling distribution. This is why we can use normal-based inference even when the underlying data is non-normal — we are working with the sampling distribution of the mean, which is normal. The CLT is the bridge that makes normal-distribution techniques applicable to a wide range of real-world data.
Standard deviation describes variability of individual values around the mean. Standard error describes variability of the sample mean across repeated samples. SE = SD / sqrt(n). As sample size grows, SE shrinks while SD stays the same. SE is what we use for confidence intervals and hypothesis tests about means, NOT SD. Confusing the two is a common error in inference problems.
Use z when the population SD is known. Use t when the population SD is unknown (which is almost always — we substitute the sample SD). The t-distribution accounts for the additional uncertainty from estimating SD. For large samples (n ≥ 30), the t-distribution is very close to z. For small samples, t is meaningfully wider with heavier tails, giving wider confidence intervals and more conservative tests. Software automatically uses t in real data analysis.
Snap a photo of any normal distribution problem and StatsIQ computes z-scores, looks up probabilities, and produces worked solutions. For data, StatsIQ runs normality tests (Shapiro-Wilk, Q-Q plots) and recommends transformations when normality fails. For exam prep, the app generates problems at all complexity levels, including problems requiring the empirical rule and exact z-table lookups. This content is for educational purposes only.