7.4 Basic Statistics: Frequency Distributions, Central Tendency, Risk Ratios & Odds Ratios
Key Takeaways
A frequency distribution lists each category with its count, percent (relative frequency) and often a cumulative percent; the percents must sum to 100.
The mean is the arithmetic average, the median is the middle value of ordered data, and the mode is the most frequent value; the median is preferred for skewed data such as days from diagnosis to treatment.
Relative risk (risk ratio) = risk in the exposed ÷ risk in the unexposed, calculated from cohort data; a value of 1 means no association.
Odds ratio = (a × d) ÷ (b × c) from a 2×2 table, the measure used in case-control studies; for rare diseases it approximates the relative risk.
A 95% confidence interval for a ratio that includes 1.0 means the association is not statistically significant at the 0.05 level.
Why registrars need basic statistics
Registrars answer data requests, prepare cancer committee and annual reports, and support quality and research studies. You do not need advanced biostatistics for the exam, but you must be able to summarize registry data correctly and interpret the risk measures that appear in the studies your data support.
Types of data
| Type | Definition | Registry examples |
|---|---|---|
| Nominal (categorical) | Named categories with no order | Primary site, sex, race, Class of Case |
| Ordinal | Ordered categories without equal spacing | Stage group (I–IV), grade (1–3), performance status |
| Discrete numeric | Counts | Number of positive lymph nodes |
| Continuous numeric | Measurements on a scale | Age, tumor size in mm, days from diagnosis to treatment |
The data type decides the summary. Categorical data are summarized with counts and percents, and numeric data with measures of central tendency and spread.
Frequency distributions
A frequency distribution lists each value or category with the number of cases (frequency), the relative frequency (percent of the total), and often the cumulative percent.
Example: stage at diagnosis for 400 analytic breast cases
| AJCC stage group | Frequency | Percent | Cumulative percent |
|---|---|---|---|
| 0 | 60 | 15.0% | 15.0% |
| I | 180 | 45.0% | 60.0% |
| II | 100 | 25.0% | 85.0% |
| III | 36 | 9.0% | 94.0% |
| IV | 12 | 3.0% | 97.0% |
| Unknown | 12 | 3.0% | 100.0% |
| Total | 400 | 100.0% | — |
Rules of thumb:
- Always report the denominator. "45% stage I" means nothing without "n = 400."
- Show unknowns. Hiding them inflates the other percentages and conceals a data quality problem.
- Group continuous data into mutually exclusive, exhaustive intervals, such as age groups 0–14, 15–39, 40–64 and 65+, with no gaps or overlaps.
- Choose the right chart. Use bar charts for categories, histograms for grouped continuous data, line graphs for trends over time, and pie charts only for a few parts of one whole.
Measures of central tendency
| Measure | Definition | Best use |
|---|---|---|
| Mean | Sum of the values ÷ number of values | Roughly symmetric numeric data |
| Median | The middle value when the data are ordered (the average of the two middle values when n is even) | Skewed data or data with outliers |
| Mode | The most frequent value | Categorical data; finding the most common category |
Worked example: days from diagnosis to first treatment for 9 patients
Ordered values: 12, 14, 15, 18, 21, 22, 22, 30, 160.
- Mean = (12 + 14 + 15 + 18 + 21 + 22 + 22 + 30 + 160) ÷ 9 = 314 ÷ 9 ≈ 34.9 days
- Median = the 5th value = 21 days
- Mode = 22 days, the only value that appears twice
The one 160-day outlier pulls the mean up by about 14 days. The median better represents a typical patient, which is why time-to-treatment reports usually show the median.
Measures of spread
- Range: maximum − minimum (160 − 12 = 148 days).
- Interquartile range (IQR): the 75th percentile − the 25th percentile. It covers the middle 50% of the data and resists outliers.
- Standard deviation: the typical distance of values from the mean. It is most useful with roughly bell-shaped data.
Ratios, proportions and rates
- Ratio: one quantity divided by another, where the two may be unrelated (male-to-female cases, 1.2:1).
- Proportion: a part divided by the whole, where the numerator is included in the denominator (the percent of cases diagnosed at stage I).
- Rate: a proportion over a period of time in a population at risk (incidence per 100,000 per year; see Section 7.3).
Risk and the risk ratio (relative risk)
In a cohort study, exposed and unexposed people are followed forward in time. Arrange the results in a 2×2 table:
| Disease | No disease | Total | |
|---|---|---|---|
| Exposed | a | b | a + b |
| Unexposed | c | d | c + d |
- Risk (cumulative incidence) in the exposed = a ÷ (a + b); in the unexposed = c ÷ (c + d).
- Relative risk (RR), also called the risk ratio = [a ÷ (a + b)] ÷ [c ÷ (c + d)].
- Risk difference (attributable risk) = the risk in the exposed − the risk in the unexposed.
Worked example: 1,000 smokers and 2,000 non-smokers are followed for 10 years.
| Lung cancer | No lung cancer | Total | |
|---|---|---|---|
| Smokers | 30 | 970 | 1,000 |
| Non-smokers | 6 | 1,994 | 2,000 |
- The risk in smokers is 30/1,000 = 0.030, and in non-smokers it is 6/2,000 = 0.003.
- RR = 0.030 ÷ 0.003 = 10. Smokers had 10 times the 10-year risk.
- The risk difference is 0.030 − 0.003 = 0.027, or 27 excess cases per 1,000 smokers.
The odds ratio
In a case-control study, people are selected because they have the disease (cases) or do not (controls), and past exposure is compared. The design fixes how many people have the disease, so risk cannot be calculated. The study uses the odds ratio (OR) instead:
- OR = (a × d) ÷ (b × c), the cross-product of the 2×2 table.
Worked example: 200 mesothelioma cases and 400 controls are asked about asbestos exposure.
| Cases | Controls | |
|---|---|---|
| Exposed | 120 (a) | 100 (b) |
| Unexposed | 80 (c) | 300 (d) |
OR = (120 × 300) ÷ (100 × 80) = 36,000 ÷ 8,000 = 4.5. The odds of past asbestos exposure were 4.5 times as high among cases as among controls. When a disease is rare, the OR approximates the RR.
Interpreting ratios
| Value of RR or OR | Interpretation |
|---|---|
| 1.0 | No association |
| Greater than 1.0 | Exposure associated with higher risk (a possible risk factor) |
| Less than 1.0 | Exposure associated with lower risk (a possible protective factor) |
- 95% confidence interval: if the interval for an RR or OR includes 1.0 (for example, 0.8–2.3), the association is not statistically significant at the 0.05 level. If it excludes 1.0 (for example, 1.4–3.9), it is.
- Association is not causation. Bias, confounding (such as age or smoking) and chance must be considered before concluding that an exposure causes cancer.
In a case-control study, 90 of 150 cases and 60 of 300 controls report a particular exposure. What is the odds ratio?
6.0
3.0
1.5
0.17
A registrar reports the number of days from diagnosis to surgery for 51 patients. Most waited 20–35 days, but three waited more than 200 days because of delayed referrals. Which summary best represents the typical wait?
The mean, because it uses every value
The median, because it is not pulled toward the extreme values
The mode, because it is always the most accurate average
The range, because it shows the longest wait
In a cohort study, 40 of 2,000 exposed workers and 10 of 2,000 unexposed workers develop bladder cancer. What is the relative risk, and what does it mean?
4.0; exposed workers had four times the risk of the unexposed workers
0.25; exposure was protective
30; there were 30 more cases in the exposed group
4.0; the result proves the exposure causes bladder cancer
Sections you finish are checked off in the contents.