If you’ve taken a measurement and evaluation course, you met these two statistics and possibly never used them again. They’re worth reviving, because between them they answer a question no class average can: is this exam producing scores I can defend?
This guide gives both formulas with a single worked example carried all the way through, the interpretation tables, and — the part usually left out — what to actually change when the numbers come back low.
The distinction that matters: KR-20 describes the whole exam. Point-biserial describes one item. A low KR-20 tells you the exam isn’t measuring consistently; the point-biserials tell you which items are responsible.
Part 1 — KR-20: is the exam internally consistent?
Reliability asks whether the exam would rank your students the same way if you gave it again. You obviously can’t give it again, so internal consistency substitutes: do the items agree with each other about who knows the material?
The Kuder-Richardson Formula 20 is the standard answer for dichotomously scored items (right/wrong, 1/0):
KR-20 = (k ÷ (k − 1)) × (1 − Σpq ÷ σ²)
Where:
- k = number of items
- p = proportion who answered that item correctly
- q = 1 − p
- σ² = variance of the total scores
Worked example
A 40-item midterm taken by 35 students. Total-score variance σ² = 42.5.
First, compute pq for every item and sum them:
| Item | p | q | pq |
|---|---|---|---|
| 1 | 0.85 | 0.15 | 0.1275 |
| 2 | 0.60 | 0.40 | 0.2400 |
| 3 | 0.35 | 0.65 | 0.2275 |
| … | … | … | … |
| Σpq (all 40) | 7.80 |
Then substitute:
KR-20 = (40 ÷ 39) × (1 − 7.80 ÷ 42.5)
= 1.0256 × (1 − 0.1835)
= 1.0256 × 0.8165
= 0.84
Interpreting the coefficient
| KR-20 | Interpretation | What to do |
|---|---|---|
| 0.90 and above | Excellent — suitable for high-stakes decisions | Nothing |
| 0.80 – 0.89 | Good — solid for a major exam | Nothing |
| 0.70 – 0.79 | Acceptable for classroom use | Review the weakest items |
| 0.60 – 0.69 | Questionable | Revise before reusing |
| Below 0.60 | Unreliable — scores are substantially noise | Do not reuse without rebuilding |
Our example exam at 0.84 is in good shape.
KR-21 — the shortcut, and its cost
If you don’t want to compute p for all 40 items, KR-21 needs only the mean and variance:
KR-21 = (k ÷ (k − 1)) × (1 − (M × (k − M)) ÷ (k × σ²))
Same exam, mean score M = 26:
= (40 ÷ 39) × (1 − (26 × 14) ÷ (40 × 42.5))
= 1.0256 × (1 − 364 ÷ 1700)
= 1.0256 × 0.7859 = 0.81
Lower than KR-20’s 0.84 — and that’s expected. KR-21 assumes every item is equally difficult, which is never true, so it systematically underestimates reliability. Treat it as a floor: if KR-21 clears 0.70, KR-20 certainly does.
What actually raises reliability
The three levers, in order of effect:
- Length. More items measuring the same construct means more reliability, and this dominates everything else. A 20-item exam struggles to reach 0.80 no matter how well written.
- Spread of scores. Reliability is a ratio involving score variance. A class where everyone scores 88–92 will produce a low coefficient even from excellent items — there’s nothing to be consistent about. This is why a low KR-20 on a small, homogeneous graduate section is often not a defect.
- Item quality. Items with weak discrimination drag the coefficient down. This is where point-biserial comes in.
Note what is not on the list: making the exam harder. Difficulty and reliability are different properties.
Part 2 — point-biserial: which items are pulling their weight?
The point-biserial correlation (r_pbis) measures how strongly performance on one item relates to performance on the whole exam. A good item is one that students with high total scores tend to get right.
r_pbis = ((M₁ − M₀) ÷ σ) × √(pq)
Where:
- M₁ = mean total score of students who got the item right
- M₀ = mean total score of students who got it wrong
- σ = standard deviation of total scores
- p, q = as before
Worked example — same exam, item 2
Item 2 had p = 0.60. Of the 35 students, the 21 who got it right averaged 29.4 overall; the 14 who missed it averaged 22.1. From σ² = 42.5, σ = 6.52.
r_pbis = ((29.4 − 22.1) ÷ 6.52) × √(0.60 × 0.40)
= (7.3 ÷ 6.52) × √0.24
= 1.120 × 0.490
= 0.55
Interpreting point-biserial
| r_pbis | Interpretation |
|---|---|
| 0.40 and above | Excellent discrimination |
| 0.30 – 0.39 | Good |
| 0.20 – 0.29 | Marginal — revise |
| 0.10 – 0.19 | Poor — rewrite |
| Below 0.10 | Not measuring the same thing as the rest of the exam |
| Negative | Strong students missed it more than weak ones — check the answer key |
Item 2 at 0.55 is excellent. It’s exactly the item you want: moderately difficult, and it sorts the class the same way the exam as a whole does.
A negative point-biserial remains the single most actionable number in item analysis. It is a miskey until you’ve proved otherwise.
How point-biserial differs from the upper-lower discrimination index
Both measure discrimination, and their interpretation bands are conventionally the same — which is why they’re easy to confuse. The difference:
| Upper-lower D | Point-biserial | |
|---|---|---|
| Uses | Top 27% vs bottom 27% | Every student |
| Discards data | Yes — the middle 46% | No |
| Computed by hand | Easily | Painfully |
| Stability in small groups | Lower | Higher |
| Ceiling | 1.00 | Below 1.00 in practice, because p constrains it |
Point-biserial is the better statistic; the upper-lower index survives because you can compute it with a pencil. If software is doing the work, prefer point-biserial. Our item analysis guide covers the upper-lower method in full, along with difficulty and distractor analysis.
One consequence of that last row is worth knowing: an item everyone gets right cannot discriminate. When p approaches 1.00, √(pq) approaches zero and so does r_pbis. A very easy item will always show weak discrimination — that’s arithmetic, not a defect, and it’s why you judge difficulty and discrimination together rather than separately.
Reading the two together
| KR-20 | Point-biserials | Diagnosis |
|---|---|---|
| High | Mostly high | Healthy exam — bank these items |
| Low | Mostly high, scores tightly clustered | Restricted range, not a bad exam. Check the score spread before rewriting anything. |
| Low | A cluster near zero or negative | Those specific items are the problem. Fix or drop them. |
| Low | Uniformly middling on a short exam | Too few items. Length is the fix. |
| High | One strongly negative | The exam is fine; that one item is miskeyed. |
Frequently asked questions
Does KR-20 work for partial-credit items?
No — KR-20 requires right/wrong scoring. For rubric-scored or partial-credit items, use Cronbach’s alpha, which is the general case (KR-20 is alpha applied to dichotomous data).
Is a low KR-20 my fault?
Not necessarily. Short exams and academically homogeneous classes both produce low coefficients from perfectly sound items. Check the number of items and the score spread before concluding the exam is bad.
What’s an acceptable reliability for a classroom exam?
0.70 is the usual floor for course-level decisions. Standardized and licensure instruments are held to 0.90+, but they have hundreds of items and thousands of examinees — that’s not the comparison to hold yourself to.
Should I drop items with negative point-biserial from the grades?
Fix the key if it’s a miskey and rescore — that’s standard practice and defensible. Dropping a merely weak item after the fact changes the scale for everyone and usually needs a stated policy. Decide your rule in advance and apply it uniformly.
Can I improve reliability by adding items after the fact?
Not to an exam already given. But it tells you the direction for next time: if a 25-item exam returns 0.65, a 40-item version of the same quality will land considerably higher.
Do I have to compute these by hand?
No, and the point-biserial in particular is unpleasant by hand across 40 items. Any tool that computes it from a score matrix will do — the value is in reading the output, not in the arithmetic.
Get both, automatically, from the same score sheet
Skoolari’s College/SUC mode produces a complete item report from your class record — per-item difficulty, discrimination, distractor pull, and KR-20 for the whole exam — with a Keep / Revise / Reject verdict on every item and the reasoning shown when you click it. Export to Excel for your department’s records. Want the same diagnosis before the exam? See our guide to pilot testing an exam.