Source: Moore & Notz, Statistics: Concepts and Controversies (10th ed.), Chs. 11–13
Tags: normal distribution, bell curve, Gaussian, z-score, standardized score, empirical rule, 68-95-99.7, percentile, standard normal, normal table, STAT 101, intro statistics
Difficulty: Beginner to Intermediate | Prerequisites: Descriptive Statistics, Variables, and Data Visualisation (Document 1). You need to be comfortable with mean and standard deviation before starting here.
Many natural and social phenomena produce data that clusters symmetrically around a centre, with progressively fewer observations the further you move from that centre. The normal distribution (bell curve) is the mathematical model for that pattern. Once you know a variable follows a normal distribution, you can answer questions like "what percentage of values fall above 85?" or "how unusual is this observation?" using just two numbers: the mean and the standard deviation. The z-score is the tool that converts any normal distribution into a single standard scale, which is why it appears in nearly every subsequent topic in the course.
The normal distribution is a symmetric, bell-shaped model described entirely by its mean and standard deviation. Z-scores let you translate any value on any normal distribution into a common scale, making it possible to compare observations across different datasets and to look up probabilities in a single table.
Normal distribution (Gaussian distribution)
A continuous probability distribution that is symmetric about its mean, with the shape of a bell curve. The mean, median, and mode are all equal. In simple terms, it is the classic bell-shaped curve where most values cluster near the middle and extreme values are rare on both sides.
Standard normal distribution
A special case of the normal distribution with a mean of 0 and a standard deviation of 1. This is the reference distribution used in z-score tables. Think of it as the "universal ruler" version of the bell curve.
Z-score (standard score, standardized score)
The number of standard deviations an observation sits above or below the mean. A z-score of +2 means the value is two standard deviations above the mean; a z-score of −1.5 means it is one and a half standard deviations below. In simple terms, the z-score answers "how far from average is this, measured in standard-deviation units?"
Empirical rule (68-95-99.7 rule)
For any normal distribution, approximately 68% of observations fall within 1 SD of the mean, about 95% within 2 SDs, and about 99.7% within 3 SDs. Think of it as a quick mental shortcut for estimating how unusual a value is without doing a full calculation.
Percentile
The value below which a given percentage of observations fall. If your score is at the 90th percentile, 90% of all scores are at or below yours. In simple terms, percentiles tell you where you stand relative to everyone else.
Density curve
A smooth curve that describes the overall shape of a distribution. The total area under the curve equals 1 (or 100%). For a normal distribution, the density curve is the bell shape. In simple terms, it is the idealised version of a histogram when you have infinitely many observations.
Parameters (μ and σ)
μ (mu) is the population mean; σ (sigma) is the population standard deviation. These two numbers completely define a specific normal distribution. Think of μ as the centre of the bell and σ as how wide or narrow it is.
Perfectly symmetric about the mean.
The tails extend infinitely in both directions but get very close to zero.
The shape is entirely determined by μ and σ. Changing μ shifts the curve left or right. Changing σ makes it wider (larger σ) or narrower (smaller σ).
Real data are never perfectly normal, but many variables are close enough that the model is useful.
Within 1 SD (μ ± σ): roughly 68% of data.
Within 2 SDs (μ ± 2σ): roughly 95% of data.
Within 3 SDs (μ ± 3σ): roughly 99.7% of data.
This means only about 5% of observations lie more than 2 SDs from the mean, and only 0.3% lie more than 3 SDs away. An observation beyond 3 SDs is genuinely unusual.
The z-score formula converts a raw value x into standard-deviation units.
Once you have a z-score, you can use a standard normal table (or calculator) to find the proportion of observations below (or above) that value.
Z-scores also let you compare values from different distributions. If Student A scores 82 on an exam with mean 70 and SD 8 (z = 1.5), and Student B scores 90 on a different exam with mean 78 and SD 10 (z = 1.2), Student A's score is relatively more impressive despite the lower raw number.
Most tables give the area to the left of a z-score (the cumulative probability).
To find the area to the right, subtract the table value from 1.
To find the area between two z-scores, look up both and subtract the smaller from the larger.
Some tables cover only positive z-values. Use symmetry: the area to the left of z = −1.5 equals the area to the right of z = +1.5.
Sometimes the question gives you the percentile and asks for the raw value.
Find the z-score that corresponds to the given percentile in the table.
Plug into the rearranged formula: x = μ + zσ.
Look at a histogram: is it roughly symmetric and bell-shaped?
A normal quantile plot (Q-Q plot) plots each observation against its expected position under a normal distribution. If the points roughly follow a straight line, the data are approximately normal.
Small departures from normality are common and usually not a problem. Large departures (strong skew, heavy tails, multiple peaks) mean the normal model should not be applied.
Z-score
z = (x − μ) / σ
where x is the individual observation, μ is the population mean, and σ is the population standard deviation. For sample data, replace μ with x̄ and σ with s.
Raw value from z-score
x = μ + zσ
Use this when working backwards from a percentile or probability.
Empirical rule intervals
68% interval: μ − σ to μ + σ
95% interval: μ − 2σ to μ + 2σ
99.7% interval: μ − 3σ to μ + 3σ
Standardized test scores (SAT, GRE, IQ) are designed to follow a normal distribution, which is why score reports include percentile ranks alongside raw scores. In manufacturing, quality control uses the normal model and the empirical rule to set tolerance limits: a part more than 3 standard deviations from the target specification is flagged as defective. In medicine, reference ranges for blood tests (e.g. cholesterol, blood glucose) are typically built from the middle 95% of a healthy population, which is just the empirical rule in action.
Students often believe that all data follow a normal distribution. They do not. Many real-world variables (income, house prices, waiting times) are skewed, and the normal model would be misleading for them.
Students sometimes think a z-score of 0 means "no data" or "nothing." A z-score of 0 means the observation is exactly at the mean, which is the most typical value.
Students mix up "within 2 standard deviations" (which captures about 95%) with "within 2 standard deviations on one side" (which captures about 47.5%). The empirical rule is always two-sided.
Students forget that the z-score formula requires the standard deviation, not the variance. If you are given the variance, take the square root first.
⚠️ Z-score calculations are among the most frequently tested items in STAT 101. Practise until the formula is automatic.
⚠️ Expect questions asking you to find the percentage of observations above, below, or between two values using the standard normal table.
⚠️ The empirical rule is a favourite for quick-answer and true/false questions. Know 68, 95, and 99.7 cold.
⚠️ Comparison questions ("Who performed better relative to their class?") test whether you understand that z-scores, not raw scores, are the right basis for comparison.
True or false: A z-score of −2.0 means the observation is two standard deviations below the mean.
Fill in the blank: Approximately ______% of values in a normal distribution fall within one standard deviation of the mean.
True or false: If two datasets have different means and standard deviations, you cannot compare individual observations across them.
Fill in the blank: The standard normal distribution has a mean of ______ and a standard deviation of ______.
True or false: The area under the entire normal curve equals 0.5.
Answers: 1. True. 2. 68. 3. False (z-scores make the comparison possible). 4. 0 and 1. 5. False (it equals 1, or 100%).
Q: Exam scores are normally distributed with a mean of 74 and a standard deviation of 8. What is the z-score for a student who scored 90?
A: z = (90 − 74) / 8 = 16 / 8 = 2.0. The student scored exactly 2 standard deviations above the mean.
Q: Using the empirical rule, approximately what percentage of students scored between 66 and 82 on the exam described above?
A: 66 is one SD below the mean (74 − 8) and 82 is one SD above (74 + 8). By the empirical rule, about 68% of students scored in this range.
Q: A standard normal table shows that the area to the left of z = 1.25 is 0.8944. What percentage of observations fall above z = 1.25?
A: 1 − 0.8944 = 0.1056, so about 10.56% of observations fall above z = 1.25.
Q: A student's score is at the 95th percentile. The mean is 500 and the standard deviation is 100. What is the approximate raw score? (Use z = 1.645 for the 95th percentile.)
A: x = 500 + (1.645)(100) = 500 + 164.5 = 664.5.
Q: Two students take different exams. Student A scores 78 on an exam with mean 70, SD 5. Student B scores 85 on an exam with mean 75, SD 8. Who performed better relative to their class?
A: Student A: z = (78 − 70) / 5 = 1.6. Student B: z = (85 − 75) / 8 = 1.25. Student A performed better relative to their class because their z-score is higher.
The normal distribution is the bridge between descriptive statistics (Document 1) and inferential statistics (Document 3). Confidence intervals and hypothesis tests rely on the assumption that sampling distributions are approximately normal, a result guaranteed by the Central Limit Theorem for large enough samples. Understanding z-scores here sets up the logic of test statistics later: a test statistic is, at its core, a z-score (or a close relative) that measures how far a sample result sits from what you would expect under the null hypothesis.
normal distribution, Gaussian distribution, bell curve, z-score, standard score, standardized score, empirical rule, 68-95-99.7 rule, percentile, standard normal distribution, density curve, mu, sigma, parameters, Q-Q plot, normal quantile plot, cumulative probability, standard normal table, z-table, area under the curve, STAT 101, intro to statistics, principles of statistics