Denominators, distributions and model fit: choose the right denominator, describe the data honestly, and check whether a model fits before trusting an average.
Pull your team's data from the dashboard or the SAFE app.
Evidence Dashboard ↗ SAFE App ↗Choose your name to see your seat, your team and your team's questions.
A large event, such as a UN Day or CSW70 forum, produces impressive-looking numbers: thousands of attendees, many countries, hundreds of takeaways. The STAT 251 question is never "how big is the number?" It is "what is the observational unit, what is the right denominator, and what does this evidence actually support?"
| Step | What you do here | Chapter |
|---|---|---|
| QUESTION | State the business or policy question in one sentence. Is it a description of these data, or a claim about a larger population? | Ch 1 |
| DEFINE | Name the observational unit (a country, a participant, a survey, a takeaway), the population and sample, and each variable's type. Write an operational definition: what exactly counts as "attended" or "contributed"? | Ch 1–2 |
| DESCRIBE | Use a frequency table or chart; report rates, not just counts; give the center (mean, median), the variation (SD, CV), position (quartiles, IQR, z score) and outliers. | Ch 3–4 |
| MODEL | Use probability with the correct conditional denominator; check whether a Poisson model fits the counts or a Normal model fits the measurements. | Ch 5–7 |
| INFER | Ask whether these data can support a claim about a population. Often the honest answer is "describe only" because the data form a convenience sample. | Ch 1 (limits); later chapters |
| DECIDE / REVIEW | Separate the statistical evidence from managerial judgment. State what the evidence cannot support and what should be measured next. | All |
Observational unit: the "who or what" one row of data describes. The same event can be described with different units, and each unit needs a different denominator: one country (N = 139 nations), one participant survey (N = 1,027), one takeaway (N = 230), or one attendee (about 15,000).
Population vs sample. A population is the set of all subjects or elements we want to make inferences about. A sample is the subset we actually observed. A parameter describes a population; a statistic describes a sample.
| Variable (UN Day / CSW70 context) | Type | Fitting summaries | Fitting displays |
|---|---|---|---|
| Country of delegate | Qualitative · nominal | Frequency, relative frequency (%), mode | Bar chart, Pareto chart |
| Region (e.g., Africa, Asia-Pacific) | Qualitative · nominal | Counts and rates by region; cross tabulation with another variable | Bar chart, side-by-side bars |
| Session type (plenary, side event, workshop) | Qualitative · nominal | Frequency table; rates per session type | Bar chart |
| Satisfaction rating (1–5 survey scale) | Qualitative · ordinal | Median, percentages per level | Ordered bar chart |
| Contributed a takeaway? (yes/no) | Qualitative · nominal | Proportion; conditional proportions by group | Contingency table, bar chart |
| Delegates per country (attendance count) | Quantitative · discrete | Median and IQR (if skewed), mean and SD, CV | Histogram, box plot |
| Takeaways submitted per session | Quantitative · discrete (count per interval) | Mean count λ, variance; Poisson plausibility | Histogram, dot plot |
| Minutes in a session / waiting time | Quantitative · continuous | Mean, median, SD; z scores; Normal model check | Histogram, box plot |
A country code or seat number is a label. Averaging it is meaningless. Ask whether arithmetic differences make sense. If they don't, treat the variable as qualitative.
"Attendee" could mean registered, badge-scanned, or present in a session. "Takeaway" could mean any comment or only a completed form. Two teams using different definitions will get different counts. That gap comes from how the variable was measured, not from a real difference.
Paste your team's numeric series (for example, delegates per country). Separate values with commas, spaces or new lines.
Empirical Rule: if the distribution is bell-shaped, about 68% of the data lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. An exact Normal model gives 68.2% and 95.4%.
Data being continuous, or n being large, doesn't make the data Normal. Check the shape. A large z score tells you a value is unusual relative to this distribution; it doesn't tell you the process failed or that the value is an error. Investigate before you conclude.
Enter counts of participant surveys by region and by whether the respondent contributed a takeaway. Each cell counts one observational unit: a survey. EXAMPLE DATA The pre-filled counts are made up and add to 1,027 only so the totals echo the survey N. Replace them with your team's real counts.
| Region (row) | Contributed: Yes | Contributed: No | Row total |
|---|---|---|---|
| Column total | – | – | – |
Two events are independent if the occurrence of one does not affect the occurrence of the other, i.e. P(Yes | Region) = P(Yes) for every region. In this table they are dependent if the conditional rates differ from the overall rate. In a sample, small differences can come from chance alone. Deciding whether a difference is bigger than chance is an inference question for later chapters, so describe the size of the gap and avoid calling it proof.
P(Yes | Region A) and P(Region A | Yes) answer different questions. Bayes reasoning converts one into the other using the base rate, the overall share of each region. Natural frequencies make this concrete: "Of every 100 surveys from Region A, about ___ contributed; of every 100 contributors, about ___ came from Region A." A region can supply most of the takeaways simply because it is most of the room (a high base rate) even if its members are less likely to contribute.
For a continuous random variable, probability is area under the density curve. The total area is 1, and the probability of any single exact value is 0: P(X = x) = 0, so P(X < x) and P(X ≤ x) are the same.
X is uniform on [a, b] when every interval of the same width inside [a, b] is equally likely. The density is a flat rectangle of height 1/(b − a), so P(c < X < d) = (d − c)/(b − a) for a ≤ c < d ≤ b. Mean μ = (a + b)/2, standard deviation σ = (b − a)/√12.
EXAMPLE A side-event speaker's start time is equally likely to be any moment between 10:00 and 10:20 (X = minutes after 10:00, so a = 0, b = 20). What is the probability the talk starts between 10:05 and 10:12?
If X is Normal with mean μ and standard deviation σ, standardize with z = (x − μ)/σ. Z is the Standard Normal (mean 0, SD 1), and probabilities are areas under its curve. Only use a Normal model after checking that the data's shape fits (tab 2).
EXAMPLE Minutes a participant stays in a 60-minute side event, modeled as Normal with μ = 42 and σ = 9 (made-up values).
"What stay length marks the 90th percentile?" Find z with P(Z < z) = p, then un-standardize: x = μ + z·σ.
This uses the x̄ and s from your last Run Diagnostics in tab 2. It asks the Normal model what share of values should be negative and compares that with what you actually observed.
First decide whether the question is descriptive (collect, organize, analyze and present these data) or inferential (draw a conclusion about a population or parameter from a sample). Then pick the tool that fits the variable type.
| If the question is… | Kind | Tool | Ch | Answers we might find |
|---|---|---|---|---|
| How many countries / which regions were represented? | Descriptive | Frequency table, relative frequency, bar or Pareto chart | 3 | A few countries account for a large share of delegates |
| What is a typical delegation size? | Descriptive | Mean vs median; SD, CV; box plot | 4 | Skewed to the right: median well below mean; report the median as "typical" |
| Is one country's delegation unusually large? | Descriptive (relative position) | z score; 1.5·IQR outlier fences | 4, 7 | Outlier by the fences; a large z, which is unusual relative to these data, not an error |
| Does the Empirical Rule / a Normal model describe these data? | Model fit | Percent within ±1s, ±2s; sign of x̄ − 2s; shape | 4, 7 | x̄ − 2s < 0 for counts: the Normal model is a poor fit |
| What share of surveys included a takeaway? | Descriptive | Proportion with the correct denominator | 3–5 | 230 per 1,027 surveys, not per 15,000 attendees |
| Are respondents from some regions more likely to contribute? | Descriptive probability | Contingency table; P(Yes | Region) vs P(Yes) | 5 | Dependence between region and participation, i.e. participation friction |
| Most takeaways came from Region A, so is Region A most engaged? | Probability reasoning | Compare P(A | Yes) with P(Yes | A); base rate | 5 | Region A may simply be the largest group (a base-rate effect) |
| How many takeaways should we expect per session? | Model | Discrete random variable; expected value; Poisson if conditions hold | 6 | λ estimated by x̄; check s² ≈ x̄ and the Poisson conditions |
| How long do people stay in a session? | Model | Continuous model; Normal only if shape fits; z scores | 7 | Continuous ≠ Normal: check the histogram first |
| When will a speaker start if any moment in a 20-minute window is equally likely? | Model | Uniform distribution: area = width × height | 7 | P(c < X < d) = (d − c)/(b − a); P(exactly 10:10) = 0 |
| What share of participants stay under 30 minutes, or what is the 90th percentile? | Model | Normal probabilities via z = (x − μ)/σ; inverse Normal | 7 | Only valid if the stay-time distribution looks bell-shaped |
| Do all 15,000 attendees agree with the takeaways? | Inferential | Needs a probability sample of attendees | 1 (limits) | Not supported: describe only and state the limitation |
| Untrained thinker says… | STAT 251 thinker reframes… | Ch |
|---|---|---|
| "230 takeaways from 15,000 attendees: only 1.5% care." | Wrong denominator. Only people who were surveyed (1,027) could contribute. Match the denominator to the observational unit and the question. | 1, 3, 5 |
| "The average delegation is 48, so a typical country sends about 48." | In data skewed to the right, a few large values pull the mean up. The median is resistant to outliers; report it and the IQR. | 4 |
| "It's continuous / n is large, so it's Normal." | Continuous ≠ Normal, and a large n doesn't change the shape of the data. Check the histogram, the ±1s/±2s percentages and the sign of x̄ − 2s. | 4, 7 |
| "Most contributors were from Region A, so Region A members are most likely to contribute." | That confuses P(A | Yes) with P(Yes | A). Use the row total for "likely to contribute." | 5 |
| "Region A had the most takeaways, so it's the most engaged." | Counts vs rates. Divide by each region's number of surveys before comparing. | 3, 5 |
| "The flagged group is mostly bad actors; the screen is accurate." | Ignoring base rates. When a group is rare, even a good screen produces many false alarms. Work it in natural frequencies. | 5 |
| "1,027 surveys is a lot, so it represents everyone." | Volunteers who chose to respond form a convenience sample. Many observations don't make a sample representative. Describe it, and state who is missing. | 1 |
| "This country is 4 SD above the mean, so something went wrong." | Unusual ≠ failure. A large z score means a value is far from the center of this distribution. In skewed data, large values are expected. | 4, 7 |
| "The speaker starts at exactly 10:10 with probability 1/20." | For a continuous random variable, P(X = x) = 0. Probability is area over an interval, e.g. P(10:09 < X < 10:11) = 2/20. | 7 |
| "A stay of 60 minutes and a delegation of 60 people are equally extreme: both are 60." | Raw values on different scales can't be compared. Standardize first: z = (x − μ)/σ gives each value's relative position in its own distribution. | 4, 7 |
| "Delegation counts are Normal with x̄ = 48, s = 117, so look up P(X > 300)." | Using a Normal model for counts skewed to the right gives impossible answers, such as a large P(X < 0). Check the fit first, or describe with the median and IQR. | 7 |
| "Regions with more side events had more takeaways, so side events cause engagement." | Correlation is not causation. Larger delegations may drive both. Look for causal factors and lurking variables; only a designed experiment isolates cause. | 2, 4 |
Focus: Country representation (N = 139 nations). Observational unit: one country.
Focus: Actionability denominator (N = 230 takeaways vs N = 1,027 participant surveys; about 15,000 attended).