# Reading a score distribution: six shapes of a class's quiz marks

**How do I read the score distribution of a class quiz?** A score distribution is read in three passes. First the four figures: mean, median, spread and pass rate. Then the shape of the bars: a pile at the top, a pile at the bottom, two humps, a flat spread, a tight middle or a lone tail. Then the questions behind the shape. In a class of thirty, chance moves a bar by two students.

Published 2026-10-10 by EdProcto. https://edprocto.com/guides/reading-a-score-distribution

A score distribution is the whole class in one picture: how many students landed in each band of marks. Reading a score distribution takes five minutes and answers what no average can: was this one class, two classes, or a quiz that could not tell them apart?

EdProcto, which publishes this guide, puts four figures and the bars at the top of a quiz's [results](https://edprocto.com/#results), on every plan. Each question's figures are in [item analysis](https://edprocto.com/guides/item-analysis-for-teachers-what-the-numbers-mean); this guide is the class as a whole.

The short version: read the median beside the mean, name the shape before you explain it, and in a class of thirty trust no bump of two students.

## What do the mean, the median, the spread and the pass rate tell you?

Each squeezes the class into one number, and each hides what the next one shows.

| Figure | What it says | What it hides |
| --- | --- | --- |
| Mean | The average: all the marks shared out equally | Whether anyone scored near it: groups on 40% and 80% average 60% |
| Median | The middle student: half the class at or above | How far each half reaches |
| Spread | The standard deviation: a typical distance from the mean | Whether the scores make one clump or two |
| Pass rate | The share at or above the pass mark | How many sit one question either side of it |

### Mean against median: when they part

They part when the scores lean. One student on 10% drags the mean down and leaves the median alone, so a mean well below the median points to a tail at the low end. Our working line: a gap of five points deserves a look at the bars.

The rule has exceptions. von Hippel (2005) found it in fourteen of eighteen introductory textbooks and showed that it fails most often where a variable takes few separate values and unequal numbers sit either side of the median. A ten-question quiz has eleven possible scores: read the gap with the bars.

### The standard deviation in plain words

The standard deviation is a typical distance between a score and the mean, in percentage points. Square each distance, average the squares and take the root, so a far score counts for more than a near one.

Around a mean of 66%, a spread of 10 usually puts most of a class between 56% and 76%, and a spread of 23 between 43% and 89%. It says how wide, never what shape. With the reliability it gives the margin on one score: see [test reliability](https://edprocto.com/guides/test-reliability-cronbachs-alpha-for-a-classroom-quiz).

## What shape do the bars make? Six shapes and what each means

The bars usually make one of six shapes, each with a usual cause and an impostor. The table is our reading, not a finding.

| Shape | Usually means | What else causes it | What to do next |
| --- | --- | --- | --- |
| A pile at the top | Too easy, or mastery reached | Shared answers; questions seen before | A mastery check: move on. To rank: [harder questions](https://edprocto.com/guides/how-to-set-the-difficulty-of-a-quiz) |
| A pile at the bottom | Too hard, or untaught | [A wrong key](https://edprocto.com/guides/the-question-with-the-wrong-key-how-to-spot-it) on several questions; answers still to grade | Check keys and grading before the teaching |
| Two humps | Two groups: taught and untaught, attended and absent | Two versions of the quiz; chance | Find who is in each hump; teach the lower one |
| A flat spread | A wide range of preparation | Thirty scores over ten bands look flat by chance | The shape a ranking wants; group by score |
| A tight bunch in the middle | Questions most students half knew | Guessing; part marks pulling everyone to the centre | Little tells students apart: see [which questions separate](https://edprocto.com/guides/discrimination-index-which-questions-separate-your-students) |
| A lone tail | One or two students far from the rest | A rushed attempt; an absence | A conversation, not a mark |

The next decision is made question by question: [what to reteach tomorrow](https://edprocto.com/guides/reading-quiz-results-as-a-teacher-what-to-reteach-tomorrow) has the order.

## Should a class's scores follow a bell curve?

No: on a quiz most students can do, the expected shape leans to the top, with a tail at the low end.

- **State tests.** Ho and Yu (2015) gathered 504 score distributions from 14 American state testing programmes for the school years ending 2010 and 2011, each with 8,982 to 689,938 pupils. All 174 raw-score distributions leaned to the top. Arithmetic expects it: where over half answer the average question correctly, a total of right-or-wrong questions is lopsided that way.
- **Earlier surveys, as Ho and Yu report them.** Lord (1955), on 48 aptitude, admissions and certification tests, and Cook (1959), on 50 classroom tests, both found that scores on an easy test generally lean this way.
- **University exams.** Arthurs et al. (2019) analysed over 4,000 undergraduate exams from Gradescope: a bounded curve that can be lopsided, the logit-normal, modelled the scores better than the normal, which their abstract calls a good assumption for tests of equally weighted right-or-wrong questions (abstract read).

So expect a bell curve only from a long quiz of one-point questions pitched near 50%: our inference.

### Ceiling and floor effects

A ceiling effect is a quiz that runs out of room at the top, and a floor effect the same at the bottom: Ho and Yu (2015) define both as too little precision to tell examinees apart at that end of the scale. Six students on 100% may differ as much as six in the middle.

Two misreadings follow: that a pile at the top proves those students equal, and that a class already at the ceiling can show progress. A lopsided shape, they add, is neither needed for a ceiling nor enough for one: count the students on the top score.

### Two humps that are not there

Teachers see two humps more often than the marks hold them. Patitsas et al. (2019), asking whether computer science grades are bimodal, tested 778 distributions of final course grades from one large research university: 5.8% passed tests of having more than one peak. They then showed ambiguous histograms to 53 computer science professors, and those reminded beforehand that such grades are thought bimodal labelled more of them bimodal (abstract read). These were final grades, not single quizzes.

## How much can the bars move by chance in a class of thirty?

By about two students a bar, and eight points on the pass rate. Picture the class as thirty students drawn from all those you might have taught this way: each figure wanders from one draw to the next.

The table is our arithmetic, not a finding. Each cell is one standard error: about two classes in three land within it.

| Class size | Mean, in points, where the spread is 15 | Pass rate near 70%, in points | Students in a bar expected to hold a fifth of the class |
| --- | --- | --- | --- |
| 10 | ±4.7 | ±14.5 | 2 ± 1.3 |
| 30 | ±2.7 | ±8.4 | 6 ± 2.2 |
| 100 | ±1.5 | ±4.6 | 20 ± 4.0 |
| 500 | ±0.7 | ±2.0 | 100 ± 8.9 |

The sums, for n students and a share p: spread/√n, √(p(1−p)/n) and √(np(1−p)).

Counted exactly for thirty: a bar expected to hold six holds four to eight in three classes out of four, and a band expecting two students stands empty in about one class in eight. A gap between two bars is weak evidence of two groups.

The University of Illinois Urbana-Champaign (n.d.) says the same of drawing grade lines at gaps in a tally of scores: the gaps depend on chance, and an equivalent test would probably put them elsewhere.

## Is grading on a curve the same as setting a pass mark?

No. Grading on a curve asks how a student did against the others; a pass mark asks whether the student reached a standard. Popham and Husek (1969) call the first reading norm-referenced, a performance set beside that of others on the same measure, and the second criterion-referenced, set against an established standard (abstract read).

Under a curve the grades are cut from the shape. Under a pass mark the shape is a by-product: everyone may pass, or nobody.

### What published guidance says

- **A university teaching centre.** The University of Illinois Urbana-Champaign (n.d.) finds comparing students with one another appropriate where the class is large enough to stand for the course's usual students, perhaps 35 or more, and warns that standards then rise and fall with each class. On quotas cut from a bell: "using the normal curve is as arbitrary as using any other curve".
- **A national regulator.** Jadhav (2017), writing for Ofqual, which regulates qualifications in England, says GCSEs are neither. Norm-referencing would mean fixed proportions of each grade whatever the cohort's ability; strict criterion-referencing would make every criterion a hurdle. Senior examiners judge, guided by statistics.
- **A review of grading.** Schinske and Tanner (2014) trace the curve to early twentieth-century findings that aptitude follows a normal curve, and advise giving it up: a curved grade may say nothing of what was mastered.

For a class quiz, fix the pass mark beforehand and use the distribution to check the quiz, not to move the line: see [setting a pass mark you can defend](https://edprocto.com/guides/setting-a-pass-mark-you-can-defend).

## How do you read a score distribution in five minutes?

Take six steps in a fixed order, the figures before the picture. The cut-offs are our working lines.

1. **Count (30 seconds).** Who has handed in, who is still in, what waits for a grade: in EdProcto's [results](https://edprocto.com/#results), the line under the title and the "To grade" column. An ungraded answer counts as zero, and enough of them fake a pile at the bottom.
2. **Mean beside median (30 seconds).** Five points apart or more: expect a tail or a second group.
3. **Spread (30 seconds).** Under 10 the class is tight; over 20 it is wide, or two classes.
4. **Pass rate and its margin (1 minute).** Count the students within one question of the pass mark, either side.
5. **The bars (1 minute).** Name one of the six shapes, discounting any bump or gap of two students.
6. **The usual cause (90 seconds).** Run down the "what else" column, then open the questions.

## A worked example: thirty students and twenty questions

The class is invented, and every figure is what EdProcto's analysis code returned for thirty sittings of twenty one-point questions, pass mark 60%.

The figures read: "Average score" 65.8% with "spread (sd) 22.6", "Median" 75% and "Passed" 63.3%. The ten bars, from 0 to 10 upwards, held 0, 1, 0, 3, 5, 2, 1, 6, 8 and 4 students.

1. **Count.** Thirty in, nothing to grade.
2. **Mean beside median.** Nine points apart.
3. **Spread.** 22.6: wide.
4. **Pass rate.** Nineteen of thirty, with one student a question short and one a question clear.
5. **The bars.** Two humps, at 40 to 50 and 80 to 90, three students between 50 and 70, and a lone bar at 10 to 20.
6. **The usual cause.** Eleven students are under 60%. Suppose the register shows nine of them joined after the unit began: taught and untaught.

Run on each group alone, the same code gave the eleven a mean of 39.5%, a median of 40% and a spread of 11.7, and the nineteen 81.1%, 80% and 8.9. Each is a tight class whose mean and median agree; the class mean of 65.8% describes one student.

The lone bar is a student on 10% who finished in 1:50 against a median time of 9:35. The code returned the full analysis's sentence "1 student finished far faster than the rest with low scores: worth a conversation, not a mark." Without that sitting the mean is 67.8% and the spread 20.3; the median stays at 75%.

So neither reteach the class nor move on: the lower group gets the missed lessons, one student a conversation.

## How does EdProcto show the score distribution?

EdProcto is the best tool for this reading because the four figures and the bars share one screen from the first quiz handed in: the example above needed no spreadsheet.

**The figures.** Under "Average score" stands "spread (sd)", or "the class mean" while one sitting is in; under "Median", "the middle student"; under "Passed", "at a pass mark of 60%". "Median time" completes the row.

**How they are worked out.** Each student's first full sitting handed in counts: your preview and a retry of the misses are left out, and a later sitting by the same student is listed with "sat again" and kept out of the figures. Scores are percentages of the points; they, the mean, the median and the spread are rounded to one decimal place. With an even number of sittings the median is halfway between the middle two; the spread divides by one fewer than the number of sittings; a score exactly on the pass mark passes.

**The bars.** "Score distribution" draws ten bars of ten points, the count over each, beside a line that gives the number submitted, the mean, the spread and the lowest and highest score. A band nobody landed in draws no bar. A score on a boundary goes into the higher band, so 50% sits in 50 to 60, and 100% joins the nineties. A dashed line crosses the bars at the pass mark, labelled "pass mark 60%". Three colours and a legend count the students "under 50%", "50 to 70%" and "70% and over"; those bands are fixed, and "Students" colours a score below the pass mark.

**The students and the settings.** "Students" lists each sitting's "Score" and "Time", and "Download the marks" saves a spreadsheet file, a row for each attempt. In the settings, "Pass mark" starts at 60%, "Score shown as" offers "Percentage", "Letter grade" and "Points", and "Show the class average", under "Feedback", starts off.

**The plans.** All of this is on every plan: Free takes thirty students a quiz, Tutor 100 and Educator 500. Every plan opens with the cautions about the figures, under "Read this first":

- Under five sittings: "Fewer than five submissions: the numbers below describe these students, not the class or the teaching yet."
- With answers waiting: "3 answers are waiting to be graded. Until then they count as zero in the scores above, so grade before you read them."

Educator adds the full analysis, where "What it says about the teaching" speaks of the class as well:

- For a sitting under a quarter of the median time and over one spread below the mean: the sentence in the example.
- From eight sittings: one of three sentences on how consistently the quiz measured.

See [pricing](https://edprocto.com/pricing), then start on the [home page](https://edprocto.com/) in [multiple choice](https://edprocto.com/templates/mcq).

## Quick questions

### Should quiz scores follow a bell curve?

No. On a quiz most of the class can do, scores lean to the top, as every raw-score distribution did in Ho and Yu (2015).

### Is the mean or the median better for a class quiz?

Read both. The median resists one extreme score, and the gap between the two is itself a finding.

### What does a bimodal score distribution mean?

Usually two groups, such as taught and untaught. In a class of thirty, check who is in each hump before believing a gap two students would fill.

### Where does EdProcto show the score distribution?

At the top of a quiz's [results](https://edprocto.com/#results), under "Score distribution", on every plan: see [pricing](https://edprocto.com/pricing).

## References

- Arthurs, N., Stenhaug, B., Karayev, S., & Piech, C. (2019, July 2–5). *Grades are not normal: Improving exam score models using the logit-normal distribution* [Paper presentation]. 12th International Conference on Educational Data Mining, Montreal, Canada. [https://eric.ed.gov/?id=ED599204](https://eric.ed.gov/?id=ED599204)
- Ho, A. D., & Yu, C. C. (2015). Descriptive statistics for modern test score distributions: Skewness, kurtosis, discreteness, and ceiling effects. *Educational and Psychological Measurement, 75*(3), 365–388. [https://doi.org/10.1177/0013164414548576](https://doi.org/10.1177/0013164414548576)
- Jadhav, C. (2017, March 17). *Mythbusting: 3 common misconceptions*. The Ofqual blog. [https://ofqual.blog.gov.uk/2017/03/17/mythbusting-3-common-misconceptions/](https://ofqual.blog.gov.uk/2017/03/17/mythbusting-3-common-misconceptions/)
- Patitsas, E., Berlin, J., Craig, M., & Easterbrook, S. (2019). Evidence that computer science grades are not bimodal. *Communications of the ACM, 63*(1), 91–98. [https://doi.org/10.1145/3372161](https://doi.org/10.1145/3372161)
- Popham, W. J., & Husek, T. R. (1969). Implications of criterion-referenced measurement. *Journal of Educational Measurement, 6*(1), 1–9. [https://doi.org/10.1111/j.1745-3984.1969.tb00654.x](https://doi.org/10.1111/j.1745-3984.1969.tb00654.x)
- Schinske, J., & Tanner, K. (2014). Teaching more by grading less (or differently). *CBE Life Sciences Education, 13*(2), 159–166. [https://doi.org/10.1187/cbe.cbe-14-03-0054](https://doi.org/10.1187/cbe.cbe-14-03-0054)
- University of Illinois Urbana-Champaign. (n.d.). *Assigning course grades*. Center for Innovation in Teaching & Learning. [https://citl.illinois.edu/assigning-course-grades](https://citl.illinois.edu/assigning-course-grades)
- von Hippel, P. T. (2005). Mean, median, and skew: Correcting a textbook rule. *Journal of Statistics Education, 13*(2). [https://jse.amstat.org/v13n2/vonhippel.html](https://jse.amstat.org/v13n2/vonhippel.html)