Source: MATH 399N Statistics Discussions, Weeks 5–6 | Arizona State University
Tags: z-score, standard score, normal distribution, bell curve, confidence interval, margin of error, confidence level, hazard ratio, sample mean, standard deviation, MATH 399N
Difficulty: Intermediate | Prerequisites: Weeks 1–2 notes (mean, standard deviation)
Week 5 introduces the z-score, a way to express any data point as "this many standard deviations above or below the mean." This standardises values so you can compare observations from entirely different datasets on the same scale. The empirical rule (68-95-99.7) tells you what proportion of data falls within 1, 2, or 3 standard deviations of the mean in a normal distribution. Week 6 introduces confidence intervals, which give a range of plausible values for a population parameter based on sample data. Together, z-scores and confidence intervals are the tools that connect descriptive statistics to inferential statistics.
A z-score tells you how many standard deviations a value is from the mean: positive means above, negative means below. Values within 2 standard deviations are considered normal; beyond that, they are unusual. A confidence interval gives a range within which the true population value is likely to fall, typically at the 95% confidence level.
Z-score (standard score)
The number of standard deviations a data point is from the mean: z = (x − μ) / σ. A z-score of 0 means the value equals the mean. Think of it as a universal ruler for comparing values across different datasets.
Standard normal distribution
A normal distribution with a mean of 0 and a standard deviation of 1. Every z-score lives on this scale.
Empirical rule (68-95-99.7 rule)
In a normal distribution, approximately 68% of data falls within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs. In simple terms, almost everything is within 3 standard deviations of the average.
Unusual value
A data point more than 2 standard deviations from the mean (|z| > 2). A value with |z| > 3 is sometimes called "very unusual."
Confidence interval (CI)
A range of values, calculated from sample data, that is likely to contain the true population parameter. A 95% CI means that if you repeated the sampling process many times, about 95% of the intervals produced would contain the true value.
Confidence level
The percentage of intervals that would capture the true parameter if the sampling were repeated indefinitely. Common levels are 90%, 95%, and 99%.
Margin of error
The half-width of a confidence interval. It represents the maximum expected difference between the sample estimate and the true population value at the chosen confidence level.
Hazard ratio
A measure used in survival analysis to compare the rate of an event (such as death) in two groups. A hazard ratio of 1.46 means the event is 46% more likely in the treatment group than in the comparison group.
Formula: z = (x − x̄) / s, where x is the individual value, x̄ is the sample mean, and s is the sample standard deviation.
A positive z-score means the value is above the mean. A negative z-score means it is below.
A z-score of 0 means the value is exactly at the mean.
Example: with a mean water bill of $64.69 and a standard deviation of $23.41, a bill of $31.42 gives z = (31.42 − 64.69) / 23.41 = −1.42. This bill is 1.42 standard deviations below the mean.
A bill of $97.50 gives z = (97.50 − 64.69) / 23.41 = +1.40. This bill is 1.40 standard deviations above the mean.
Within 2 standard deviations of the mean (z between −2 and +2) is considered "normal" or typical.
Example: normal water bill range = mean ± 2 × SD = $64.69 ± 2(23.41) = $17.86 to $111.51. Any bill in this range is not unusual.
A bill outside this range would be flagged as unusual. Beyond 3 standard deviations is very unusual.
A confidence interval gives a range of values that is likely to contain the true population parameter.
Example: a study of 11,779 men examined aspirin use and prostate cancer mortality. The hazard ratio for post-diagnostic aspirin use was 1.46 with a 95% CI of (1.29, 1.65). Because the entire interval is above 1.0, the researchers concluded the association was statistically significant.
The 95% confidence level means: if this study were repeated many times, about 95% of the resulting intervals would contain the true hazard ratio.
When a CI for a hazard ratio or relative risk includes 1.0, the result is not statistically significant at that confidence level (because 1.0 represents "no difference").
Larger sample sizes produce narrower confidence intervals, giving more precise estimates.
z = \frac{x - \bar{x}}{s}Where x is the value, x̄ is the sample mean, and s is the sample standard deviation.
\text{Normal range} = \bar{x} \pm 2sValues outside this range are considered unusual.
\text{Confidence interval} = \bar{x} \pm z_{\alpha/2} \cdot \frac{s}{\sqrt{n}}Where z(α/2) is the critical z-value for the chosen confidence level (1.96 for 95%), s is the sample standard deviation, and n is the sample size.
Z-scores are used in clinical settings to assess whether a patient's lab result is within the normal range. Growth charts for children express height and weight as z-scores relative to the population mean for that age and sex. In finance, z-scores (Altman Z-score) are used to predict corporate bankruptcy. Confidence intervals appear in virtually every published clinical trial: regulators will not approve a drug if the confidence interval for its effectiveness includes zero (or, for a hazard ratio, includes 1.0).
Students often confuse a confidence level with a probability statement about one specific interval. A 95% CI does not mean there is a 95% chance the true value is in this particular interval. It means that 95% of intervals constructed this way, across many samples, would contain the true value.
A negative z-score does not mean something is wrong with the data. It simply means the value is below the mean.
Students sometimes think that any value outside 1 standard deviation is unusual. The threshold for "unusual" is 2 standard deviations, not 1. About 32% of values naturally fall outside 1 SD in a normal distribution.
Confusing sample standard deviation with standard error of the mean is common. Standard deviation measures spread in the data; standard error measures how precisely the sample mean estimates the population mean.
⚠️ You will be asked to calculate z-scores from raw data and interpret them. Know the formula cold.
⚠️ Be able to determine the "normal" range (mean ± 2 SD) and identify which data points fall outside it.
⚠️ Understand what a confidence interval communicates. If a CI for a hazard ratio includes 1.0, the result is not statistically significant.
⚠️ Know the critical z-values: 1.645 for 90% CI, 1.96 for 95% CI, 2.576 for 99% CI.
True or False: A z-score of −2.5 means the value is 2.5 standard deviations above the mean. Answer: False. It is 2.5 standard deviations below the mean.
Fill in the blank: Values within ______ standard deviations of the mean are considered normal. Answer: 2.
True or False: A 95% confidence interval means there is a 95% probability the true value is in this particular interval. Answer: False. It means 95% of similarly constructed intervals would contain the true value.
Fill in the blank: The critical z-value for a 95% confidence interval is ______. Answer: 1.96.
Q: Your monthly electricity bills have a mean of $120 and a standard deviation of $30. Last month's bill was $185. Calculate the z-score and determine whether this bill is unusual.
A: z = (185 − 120) / 30 = 65 / 30 = 2.17. Because |z| > 2, this bill is unusual, falling more than 2 standard deviations above the mean.
Q: What range of electricity bills would be considered normal using the data above?
A: Normal range = mean ± 2 SD = 120 ± 2(30) = $60 to $180. Bills within this range are not unusual.
Q: A study reports a hazard ratio of 1.84 with a 95% CI of (1.59, 2.12). Is this result statistically significant? Why?
A: Yes. The entire confidence interval is above 1.0, meaning the increased risk is unlikely to be due to chance at the 95% confidence level.
Q: If a study's 95% CI for a risk ratio is (0.85, 1.20), what can you conclude?
A: The result is not statistically significant at the 95% level, because the interval includes 1.0, which represents no difference between the groups.
Z-scores are the bridge between descriptive and inferential statistics. The z-score formula reappears in hypothesis testing (Week 7), where you compare a test statistic to a critical value. Confidence intervals extend naturally into Week 7's hypothesis tests: if a hypothesised value falls outside the CI, you reject the null hypothesis. Correlation and regression (Week 8) also rely on the normal distribution assumptions covered here.
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