At a Glance
Most indicator reports reduce a column of numbers to one figure, usually an average. For skewed measures such as spending, income, distance or time, that figure can hide the very thing the program cares about. This page helps you choose which summaries to report for a numeric indicator, and shows what each one hides. You need the raw values for each unit, the unit itself (household, person) and, if the sample was not equal-probability, the weights. All numbers are made up.
At a Glance
| Summary | Answers | Misleads When |
|---|---|---|
| Mean | What is the total divided by the number of units? | A few very large values pull it away from most units |
| Median | What does the middle unit look like? | You care about the total, or about the tails |
| Quartiles or percentiles | Where do the bottom and top groups start? | The sample is small, or the software definition is unknown |
| Share below a line | How many units are under a threshold that matters? | The line is arbitrary, or it hides how far below units fall |
| Average shortfall below the line | How deep is the gap, across all units? | It is read without the share, or the line differs between groups |
| Spread (standard deviation, interquartile range) | How different are the units? | The distribution is skewed and a standard deviation is read as normal |
Worked Case
Worked Case: Two Villages With the Same Average
A fictional program has a weekly spending line of 60 local currency units per household, and wants to describe two villages. In each, 20 households were drawn by simple random sampling. The values are weekly spending per household, sorted. The villages are different places, so this is a description of each and a comparison, not an effect of anything.
| Village | Weekly Spending per Household |
|---|---|
| A | 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 80, 90, 100, 120, 150, 200, 300, 500, 965 |
| B | 50, 60, 75, 85, 95, 105, 115, 120, 130, 140, 150, 160, 170, 180, 190, 200, 210, 230, 260, 275 |
Step 1: Mean and Median
Both villages sum to 3,000, so both have a mean of 150. A report that said "average weekly spending is 150 in both villages" would be arithmetically right and practically misleading.
| Village A | Village B | |
|---|---|---|
| Mean | 150.0 | 150.0 |
| Median | 67.5 | 145.0 |
| Largest household as share of total | 32.2% | 9.2% |
| Three largest as share of total | 58.8% | 25.5% |
| Mean without the largest household | 107.1 | 143.4 |
In village A the median is less than half of the mean, and one household accounts for nearly a third of all spending. Remove it and the mean drops by 43 points while the median barely moves, from 67.5 to 65.0. The mean answers "what is total spending divided by households?", which is useful for budgets. The median answers "what does the middle household spend?", which is closer to what the typical household experiences. Neither is wrong; each answers a different question. Lumley and others (2002) show that tests and intervals for means remain valid in large samples even for very skewed data, and that the real limit is whether the mean answers the question. Here, with 20 households, one value moves it.
Step 2: Spread and Shape
The sample standard deviation is 223.3 in village A and 64.4 in village B. In a skewed distribution that number is dominated by the largest values and should not be read as if the data were symmetric. Quartiles describe the shape better.
| Village A | Village B | |
|---|---|---|
| First quartile | 43.75 | 102.5 |
| Third quartile | 127.5 | 192.5 |
| Interquartile range | 83.75 | 90.0 |
| Smallest and largest | 20 and 965 | 50 and 275 |
These quartiles use linear interpolation between sorted values, the default in R and NumPy. Packages disagree: Hyndman and Fan (1996) describe nine definitions. Excel's PERCENTILE.INC returns the values in the table, but PERCENTILE.EXC, which interpolates between positions at multiples of 1/(n + 1) instead of 1/(n - 1), gives a first quartile of 41.25 and a third of 142.5 for village A. With 20 values the choice matters; with thousands it rarely does. Name the definition.
A grouped count shows the shape without a definition:
| Weekly Spending | Households in A | Households in B |
|---|---|---|
| Under 60 | 8 | 1 |
| 60 to 99 | 5 | 4 |
| 100 to 199 | 3 | 10 |
| 200 to 499 | 2 | 5 |
| 500 or more | 2 | 0 |
Step 3: Shares Below a Line and Depth of Shortfall
For a program targeting low-spending households, the useful summary may be the share below the line of 60. In village A, 8 of 20 households (40%) are below it. In village B, 1 of 20 (5%) is.
A share does not show how far below households fall. The average shortfall measures that, as a proportion of the line (the poverty gap measure described in Haughton and Khandker 2009 and belonging to the family of Foster, Greer and Thorbecke 1984). Each household's shortfall is (60 - spending) / 60 if below the line, and zero otherwise, averaged over all 20.
In village A the shortfalls sum to (40 + 35 + 30 + 25 + 20 + 15 + 10 + 5) / 60 = 3.0, so the average is 3.0 / 20 = 0.15, or 15% of the line per household. Among the eight below the line, the average shortfall is 37.5% of the line. In village B, one household is 10 below, so the average is 0.0083, or 0.8%. The share counts how many are below; the shortfall shows how far.
With 20 households each, uncertainty is large. A 95% Wilson interval for 8 of 20 is 21.9% to 61.3%, and for 1 of 20 it is 0.9% to 23.6%. The two intervals overlap slightly, from 21.9% to 23.6%, so with 20 households each the data do not firmly separate the villages, even though the point estimates are eight times apart. A confidence interval should accompany every share. A design with clusters or weights needs the methods in reading a survey estimate.
Step 4: What Weights Change
If households were not selected with equal probability, summaries must use the weights. A tiny fictional example: stratum 1 has 40 households and stratum 2 has 120. Four households were sampled in each, so each stratum 1 household has a weight of 10 and each stratum 2 household a weight of 30. The total weight is 160.
| Stratum | Weekly Spending (Weight) |
|---|---|
| 1 | 200, 250, 300, 500 (weight 10 each) |
| 2 | 30, 40, 50, 100 (weight 30 each) |
| Unweighted | Weighted | |
|---|---|---|
| Mean | 183.75 | 119.375 |
| Median | 150 | 50 |
| Share below 60 | 37.5% | 56.25% |
The weighted mean is (10 x 1,250 + 30 x 220) / 160 = 119.375. For the weighted median, sort the values and add the weights: 30 (30), 40 (60), 50 (90). Half of 160 is 80, and the cumulative weight first passes 80 at the value 50. Dropping the weights would have reported a median three times too high and a share below the line that is too low. A weighted quantile has its own definitional choices, and its interval needs survey methods. See sampling weights.
What to Report
The unit and the line, with the source of the line.
A typical value and a measure of the total, with the definition, such as "median 67.5; mean 150.0".
The share below the line, with an interval, and the average shortfall if depth matters.
The shape: quartiles or a grouped count, and the sample size.
Whether weights were used, and the quantile definition if the sample is small.
How extreme values were handled: kept, checked or corrected, never silently dropped. See how to clean your dataset.
Limits and Checks
What the Summaries Cannot Show
A summary describes the people measured. It does not show whether the measure is valid, whether the sample represents the population, or why villages differ. Spending reported by a respondent can be misreported, and extreme values might be errors or real. Checking a 965 against the source record is more useful than any choice of summary. A fixed line also divides the population arbitrarily: households at 61 and 59 are not very different, so show a range of lines when decisions depend on the share. Disaggregated summaries have smaller samples; see disaggregation.
Check an AI-Generated Summary
A fictional AI draft says: "Both villages have an average weekly spending of 150, so they are similar." Checking it:
Ask for the code or formulas, and run them on the raw values. Confirm 3,000 / 20 = 150 for each village, then compute the median, quartiles and the share below 60 yourself.
Compare the mean and median. Where they differ by a factor of two, ask the tool to explain, and check the largest values against source records.
Check the quantile definition, by comparing the tool's quartiles with the spreadsheet functions PERCENTILE.INC and PERCENTILE.EXC. If neither matches, the tool used something else.
Test the weights: run the summary with and without them. If the numbers do not change, check the weight variable was applied. A weighted median is not the unweighted one.
Look at the data. A sorted list or a grouped count would have shown the gap in two lines.
Record the model, version and date, with any human changes.
Which summary matters for the decision, and whether the line is meaningful, stay with the people who know the program.
Common Mistakes
Reporting only the mean for a skewed indicator. Add the median and the share below the line.
Reading a standard deviation as if the data were normal. In a skewed distribution it is dominated by the largest values.
Dropping the biggest value because it looks wrong. Check it against the record; a real extreme is data.
Giving a share without an interval, or without the sample size. 40% of 20 households is a very different statement than 40% of 2,000.
Ignoring weights when summarizing. Medians and shares need them, not only means.
Reporting a quantile without its definition. For small samples, the definition changes the answer.