Score Distribution Analyzer

Score data:

Use one row per score. Labels remain hidden in results unless Show labels is enabled.

Input format:

Auto detect reports the resolved format in the summary and source status.

Maximum score:

Use 100 for percentages or the actual total points available.

points

Grade bands:

Example: A,90 then B,80. Include a final catch-all band such as F,0.

Pass threshold:

{{ formatNumber(pass_threshold, 0) }}%

Move the policy cutoff from 0% to 100%.

Outlier review:

IQR is robust for skewed sets; Z-score asks about distance from the mean.

Z-score threshold:

Common review thresholds are 2.0 to 3.0 standard deviations.

Histogram bin size:

Use 5 or 10 percentage points for most cohort views.

percentage points

Display precision:

Choose 0 to 3 decimal places for presentation.

Show labels:

Off keeps row-level identities out of the visible result and exports.

Header preview:

This changes presentation only and does not alter parsing.

{{ keep_header_preview ? 'Parser details shown' : 'Parser details hidden' }}

Measure	Value	Context	Copy
{{ row.label }}	{{ row.value }}	{{ row.context }}

The distribution chart could not load.

Band	Minimum	Scores	Share	Copy
{{ row.label }}	{{ formatPercent(row.minimum_percent) }}	{{ row.count }}	{{ formatPercent(row.share_percent) }}

Line	Status	Label	Raw score	Percent	Band or reason	Copy
{{ row.line_number }}	{{ row.status }}	{{ row.label }}	{{ row.raw_display }}	{{ row.percent_display }}	{{ row.detail }}

Two groups can earn the same average score and still tell very different stories. One may cluster tightly around the middle. Another may split into high and low groups, or contain a long tail that pulls the mean away from most learners. A distribution keeps center, spread, shape, and policy thresholds visible at the same time.

Scores first need a common scale. Raw marks out of 40 and percentages out of 100 become comparable only after each mark is divided by its own maximum. Grade bands and a pass threshold then add policy decisions to the statistical summary. Changing those cutoffs can change pass and band counts without changing any learner's underlying percentage.

Main score-distribution measures and their uses
Measure	What it describes	Common misread
Mean and median	The arithmetic center and the middle ordered score.	Assuming one center describes tails or separate clusters.
Standard deviation	Spread around the mean for the entered set.	Comparing values from differently scaled scores.
Interquartile range	The width of the middle 50% of scores.	Treating values outside its fences as automatic errors.
Histogram	Counts inside fixed percentage intervals.	Changing bin width and reading the new shape as changed data.

Outlier flags are prompts for review. An unusual score may be a data-entry error, an accommodation issue, a genuinely exceptional performance, or evidence that one part of an assessment behaved differently. Removing it without checking the source can erase the most informative observation in the set.

Descriptive statistics do not decide whether an assessment was fair, valid, or well taught. Those judgments require the question design, marking method, cohort size, missing work, accommodations, and comparable prior evidence. Small groups deserve particular care because one score can noticeably move every summary.

How to Use This Tool:

Set the score scale before interpreting the distribution, then resolve rejected rows before sharing the summary.

Paste one score per line or load CSV-style rows. Keep Input format on Auto detect unless comma-separated and line input are ambiguous.
Set Maximum score to the points available. Use 100 only when the source values are already percentages.
Enter Grade bands as a label and inclusive minimum percentage on each line, with a final catch-all such as F,0.
Choose the inclusive Pass threshold and an Outlier review method. Set a Z-score threshold only when using population Z-scores.
Check Score review for rejected rows and flags, then compare the class snapshot, histogram, and grade-band counts. Keep Show labels off when row identities are unnecessary.

Interpreting Results:

Read the mean beside the median. A lower mean can indicate a low-score tail, while a higher mean can indicate a high-score tail. Then check the interquartile range, population standard deviation, and histogram before accepting that one difference describes the whole shape.

A score exactly at the pass threshold counts as passing. A score exactly at a grade-band minimum enters that band.
IQR flags use strict outer comparisons, so a score exactly on an inner fence is not flagged. Z-score flags use an inclusive absolute threshold.
Rejected rows do not contribute to any statistic. Review their reasons before assuming the accepted set represents the intended cohort.
Shape labels are compact descriptive cues, not statistical tests and not evidence of a normal distribution.

Technical Details:

Accepted marks are converted to percentages before any statistic, threshold, grade band, outlier rule, or histogram count is calculated. Negative marks, nonnumeric values, and marks above the selected maximum are rejected rather than clamped. Blank rows are ignored.

Formula Core:

The central statistics use full-precision percentages. Population standard deviation divides by the number of accepted scores because the entered set is treated as the complete group being summarized.

\begin{array}{lcl} p_{i} & = & \frac{x_{i}}{x_{max}} \times 100 \\ \bar{p} & = & \frac{\sum_{i = 1}^{n} p_{i}}{n} \\ σ & = & \sqrt{\frac{\sum_{i = 1}^{n} {(p_{i} - \bar{p})}^{2}}{n}} \\ z & = & \frac{p - \bar{p}}{σ} \\ g_{1} & = & \frac{n}{(n - 1) (n - 2)} \sum_{i = 1}^{n} {(\frac{p_{i} - \bar{p}}{σ})}^{3} \end{array}

x is a raw score, x_max is the maximum score, p is a percentage, and n is the accepted-row count. Bias-corrected skewness g₁ is calculated only with at least three scores and a nonzero population standard deviation; otherwise it is 0. Display precision of 0 to 3 decimal places changes presentation only.

Quartiles use linear interpolation at position (n - 1)q in the sorted percentages, with q equal to 0.25 or 0.75. The interquartile range is Q3 minus Q1. The mode groups percentages rounded to three decimals for frequency matching; a tie selects the lower numeric value.

Rule Core:

Score band, outlier, histogram, and shape rules
Result	Exact rule
Grade band	Assign the first band in descending threshold order where score ≥ minimum; otherwise Unbanded.
Pass	Percentage ≥ selected pass threshold.
IQR outlier	With at least four scores and IQR > 0, flag below Q1 − 1.5×IQR or above Q3 + 1.5×IQR.
Z-score outlier	When population standard deviation > 0, flag \|z\| ≥ selected threshold.
Histogram bucket	Include the lower edge and exclude the upper edge; the final bucket also includes 100%.

Shape cues are applied in order. Fewer than five scores becomes Small sample. Otherwise a mean-median gap within 2 percentage points, IQR at most 12, and standard deviation at most 10 becomes Tight cluster. A gap of −4 or less or skewness below −0.45 becomes Low-tail skew; a gap of 4 or more or skewness above 0.45 becomes High-tail skew. IQR of at least 22 or standard deviation of at least 16 becomes Wide spread; remaining sets become Balanced spread.

Input text is limited to 1 MiB. Maximum score must be 0.01 to 1,000,000 points, histogram bins must be 1 to 50 percentage points wide, and the Z-score threshold must be 0.5 to 5.

Privacy and Interpretation Notes:

Score text and loaded files are processed in the current browser. Labels are hidden from visible row results and exports unless Show labels is enabled.

Remove names or identifiers before sharing when row-level identity is unnecessary.
Use the same maximum score, pass threshold, grade bands, and cohort rules when comparing groups.
Investigate flagged and rejected rows; do not delete them automatically.

Worked Examples:

Three scores at a 90% pass line

Scores of 80, 90, and 100 out of 100 have a mean and median of 90%, a population standard deviation of about 8.165 percentage points, and a pass rate of 2 out of 3 because the 90% boundary is inclusive. With A at 90 and F at 0, two scores enter A. The set remains too small for a confident shape judgment.

References:

Measures of Location, NIST/SEMATECH e-Handbook of Statistical Methods.
Measures of Scale, NIST/SEMATECH e-Handbook of Statistical Methods.
What are outliers in the data?, NIST/SEMATECH e-Handbook of Statistical Methods.
Histogram, NIST/SEMATECH e-Handbook of Statistical Methods.