Why Sample Composition Belongs in a Decision Report
A model result is easier to interpret when readers know what observations produced it. A report that says “the model was fitted on 1,200 records” gives a sample size, but not a picture of the sample. Those records might represent a balanced mix of business types, or mostly one type; they might span the intended population, or cover only a narrow part of it.
A composition summary should answer a practical question: who or what is represented, and are there important concentrations or gaps? The goal is not to list every field in the data. It is to select a small number of counts and descriptive summaries that help readers judge the scope of the results. This tutorial is aligned with the SOA Exam PA (Predictive Analytics) syllabus.
Keep three quantities distinct. The target population is the group the analysis is intended to inform. The eligible sample is the set of records meeting stated inclusion rules. The analysis sample is the set actually used for the reported analysis. These groups can differ. A composition table describes the particular sample named in its caption; it does not, by itself, establish that the sample represents the target population.
Choose a Small, Useful Set of Summaries
A concise report typically starts with the number of observations and a brief description of the unit—for example, policies, claim records, or applicants. It then reports a few characteristics that matter for interpreting the analysis. Which characteristics matter depends on the decision and the prediction scope. Potential choices include product, region, time period, customer type, or a meaningful range of a quantitative variable.
For a categorical characteristic, counts and percentages are usually easy to scan together. For a quantitative characteristic, a few interpretable summaries may be more useful than a long list of individual values. For a skewed variable, such as claim amount, a median and interquartile range can describe its centre and spread without allowing a few extreme observations to dominate the summary. State units and, where needed, how a category or summary was defined.
If categories are mutually exclusive and collectively cover all \(n\) observations, the percentage in category \(j\), with count \(n_j\), is
The denominator \(n\) must be named or made unambiguous. A percentage of all records answers a different question from a percentage within a subgroup. If categories do not cover every record—for example, because some values are unrecorded—say whether the denominator includes those records and show how they were handled. Never let a table appear to describe the full sample when it describes only records with known values.
Why Counts and Denominators Matter
The percentage formula follows directly from the meaning of a proportion: category \(j\) contributes \(n_j\) of the \(n\) observations, so its fraction of the sample is \(n_j/n\). Multiplication by 100 expresses that fraction as a percentage. If the categories form a complete partition, their counts add to the sample size, so
This check is useful: percentages for a complete, non-overlapping set of categories should add to 100%, apart from rounding. If the total differs materially, check for a missing category, overlapping definitions, or a mismatched denominator. If it differs only slightly, explain that displayed percentages are rounded.
A combined percentage across subgroups is also a weighted result, not usually the simple average of subgroup percentages. If subgroup \(g\) has \(n_g\) observations and a category percentage \(p_g\), the combined percentage is
where the \(p_g\) values are expressed on the same percentage scale. The weights are the subgroup sizes because larger groups contribute more observations to the combined sample. This distinction matters whenever the report compares subgroup composition with overall composition.
Worked Example: Summarizing Business Type in a Sample
A hypothetical insurer’s analysis sample contains 1,200 distinct commercial policies. The sample includes 720 microbusiness policies, 360 small-business policies, and 120 midsize-business policies. Prepare a concise composition statement and identify what the figures do not establish.
Method. First check the unit and sample size: the unit is a distinct policy, and \(n=1{,}200\). The three category counts sum to \(720+360+120=1{,}200\), so the stated categories cover the sample.
Calculate the shares. Microbusiness: \(720/1{,}200=0.60\), or 60%. Small business: \(360/1{,}200=0.30\), or 30%. Midsize business: \(120/1{,}200=0.10\), or 10%. The shares total \(60\%+30\%+10\%=100\%\), which checks the calculation.
Report. “The analysis sample comprised 1,200 distinct commercial policies: 720 (60%) microbusiness, 360 (30%) small-business, and 120 (10%) midsize-business policies.” This compact summary shows that the sample is concentrated in microbusiness policies.
Interpretation and trap. The figures describe the analysis sample, not the insurer’s entire portfolio or any broader target population. A common mistake is to call the sample “representative” merely because all three business types appear. Presence is not evidence that their sample shares match the population shares. The correction is to report the observed composition and make no representativeness claim without an appropriate comparison.
Use Cross-Tabulations When Margins Hide a Relevant Imbalance
A single characteristic at a time may conceal how characteristics occur together. A compact cross-tabulation is useful when the joint pattern affects what the sample represents. It should answer a real question, not multiply the number of rows and columns simply because the data allow it.
When a table shows both counts and percentages, label whether percentages are calculated within rows, within columns, or across the full sample. Row percentages answer “among records in this row group, what share falls in each column category?” Percentages of the full sample answer “what share of all records is in this cell?” The two views can both be useful, but they are not interchangeable.
Worked Example: Showing Channel Mix Within Business Size
A hypothetical sample contains 960 business insurance applications: 600 arrived through brokers and 360 through a direct channel. The cross-tabulation below gives counts by channel and business size. Calculate overall size shares and size shares within each channel, then state what each set of percentages says.
| Channel | Microbusiness | Small business | Medium business | Total |
|---|---|---|---|---|
| Broker | 420 | 150 | 30 | 600 |
| Direct | 156 | 138 | 66 | 360 |
| Total | 576 | 288 | 96 | 960 |
Step 1: Check the margins. The broker row sums to \(420+150+30=600\); the direct row sums to \(156+138+66=360\). The channel totals sum to \(600+360=960\). The size totals also sum to \(576+288+96=960\).
Step 2: Calculate overall size shares. Microbusiness: \(576/960=0.60\), or 60%. Small business: \(288/960=0.30\), or 30%. Medium business: \(96/960=0.10\), or 10%. These percentages describe the whole sample.
Step 3: Calculate shares within each channel. Among broker applications, the shares are \(420/600=70\%\), \(150/600=25\%\), and \(30/600=5\%\). Among direct applications, they are \(156/360=43.333\ldots\%\), \(138/360=38.333\ldots\%\), and \(66/360=18.333\ldots\%\). Rounded to one decimal place, the direct shares are 43.3%, 38.3%, and 18.3%; the unrounded shares sum to 100%.
Step 4: Interpret the two views. The overall shares show that 60% of the full sample is microbusiness. The within-channel shares show that the broker and direct groups have different size mixes: 70% of broker applications are microbusiness, compared with about 43.3% of direct applications.
Common denominator trap. Dividing 420 by 960 gives 43.75%, but that is the share of the whole sample that consists of broker-channel microbusiness applications. It is not the share of broker applications that are microbusiness. For that question, the correct denominator is 600, giving \(420/600=70\%\). Label each percentage by its denominator and the question it answers.
Report Material Changes Between Cohorts
Sometimes a report needs to describe more than one sample—for example, an eligible cohort and the smaller analysis cohort. In that case, show their sizes and compare composition using a consistent set of categories. Counts reveal how many observations each group contains; percentages make the mixes easier to compare. A difference in composition is descriptive: it identifies a change between cohorts but does not, by itself, explain why the change occurred.
A percentage-point difference is the subtraction of two percentages. For example, a change from 30% to 25% is a decrease of 5 percentage points, not 5%. Stating percentage points avoids ambiguity between an absolute difference and a relative percentage change.
Worked Example: Comparing an Eligible Cohort With the Analysis Sample
A hypothetical study starts with 1,200 eligible applications and uses 900 in its final analysis. Their age-group counts are shown below. Describe how the sample composition changed and calculate the inclusion rate within each age group.
| Age group | Eligible cohort | Analysis sample |
|---|---|---|
| Under 35 | 360 | 225 |
| 35 to 54 | 600 | 495 |
| 55 and over | 240 | 180 |
| Total | 1,200 | 900 |
Step 1: Check the counts. The eligible counts sum to \(360+600+240=1{,}200\). The analysis counts sum to \(225+495+180=900\). Thus the totals agree with the stated cohort sizes.
Step 2: Calculate the eligible-cohort composition. Under 35: \(360/1{,}200=30\%\). Ages 35 to 54: \(600/1{,}200=50\%\). Ages 55 and over: \(240/1{,}200=20\%\).
Step 3: Calculate the analysis-sample composition. Under 35: \(225/900=25\%\). Ages 35 to 54: \(495/900=55\%\). Ages 55 and over: \(180/900=20\%\). Each cohort’s percentages sum to 100%.
Step 4: Compare the mixes. The under-35 share decreases from 30% to 25%, a decrease of 5 percentage points. The 35-to-54 share increases from 50% to 55%, an increase of 5 percentage points. The share aged 55 and over remains at 20%.
Step 5: Calculate inclusion rates by age group. Divide each analysis count by its corresponding eligible count: under 35, \(225/360=62.5\%\); ages 35 to 54, \(495/600=82.5\%\); ages 55 and over, \(180/240=75\%\). As a check, the included counts \(225+495+180=900\), and the excluded counts are \(360-225=135\), \(600-495=105\), and \(240-180=60\), which sum to \(300=1{,}200-900\).
Report and interpretation. State both cohort sizes and the composition changes. The analysis sample has a larger share of applicants aged 35 to 54 and a smaller share under 35 than the eligible cohort. The group-specific inclusion rates describe that difference; they do not establish its cause. A common mistake is to compare counts alone and conclude that the largest group was most strongly represented. Counts depend on group size, so compare percentages for composition and use the appropriate group denominator for inclusion rates.
Keep the Summary Decision-Useful
A table is not automatically informative because it contains many numbers. The following reporting sequence helps keep the emphasis on interpretation:
Specify whether the count refers to policies, people, claims, or another unit, and give the sample size.
Include a small set that helps readers judge the scope of the analysis or understand material concentrations.
Make clear whether a percentage refers to the full sample, a row, a column, or another specified group.
Point out important imbalances or differences between named cohorts, without treating a descriptive pattern as an explanation.
Avoid unnecessary decimal places: they can suggest more precision than a composition summary needs. Use a consistent rounding rule and say so when rounding affects totals. If a group is small, retain its count alongside its percentage; a seemingly large percentage can be based on very few observations. Do not suppress a relevant group merely because it complicates the narrative.
The purpose is not to claim that a sample is balanced or representative by appearance. It is to give readers enough context to assess which observations are present, which groups dominate, and how far the results can be interpreted without making unsupported claims. A concise, well-labelled summary is more useful than a large table whose denominators and relevance are unclear.
Check Your Understanding
For each problem, identify the relevant denominator and check that the final summary matches the stated sample.
- A hypothetical sample contains 500 applications: 125 direct and 375 broker-submitted. What percentage came through each channel? (Answer: Direct 25%; broker-submitted 75%.)
- Of 300 renewal policies, 90 are in a particular region. What is the region’s share of renewals? If the full sample contains 600 policies, what share of the full sample is made up of those regional renewals? (Answer: 30% of renewals; 15% of the full sample.)
- A cohort has 80 records in category A, 20 in category B, and 10 with category unrecorded. Report each share using all eligible records as the denominator, and give category A’s share among records with a recorded category. (Answer: 72.7% category A, 18.2% category B, 9.1% unrecorded; among recorded categories, A is 80.0%.)
- An eligible cohort has 400 observations, of which 120 are in group X. An analysis sample contains 280 observations, of which 70 are in group X. Compare group X’s shares in the two cohorts in percentage points. (Answer: Eligible cohort 30%; analysis sample 25%; decrease of 5 percentage points.)
- Two channels each contain 100 applications. In channel A, 60 are small businesses; in channel B, 30 are small businesses. What is the overall small-business share, and what are the within-channel shares? (Answer: Overall, \(90/200=45\%\); within channel A, 60%; within channel B, 30%.)