The Core Relationship
A t-statistic measures how many estimated standard errors an estimate sits away from the hypothesized value. In a one-sample mean test, the structure is:
t = (sample mean - hypothesized mean) / standard errorA p-value measures how unusual that test statistic would be if the null hypothesis were true. For a fixed test distribution and a fixed alternative hypothesis, the relationship is monotonic: a more extreme t-statistic gives a smaller p-value.
That is the anchor. If an exam item shows a very large absolute t-statistic and a large p-value for the same test, something in the setup needs checking before you interpret the result.
Why The Pair Can Look Wrong
The most common CFA trap is treating the t-statistic and p-value as independent facts. They are linked by the same distribution, same degrees of freedom, and same tail convention. If one of those inputs changes, the displayed p-value changes too.
One Statistic, Several Possible P-Values
Suppose Northbridge Analytics tests whether an average abnormal return is different from zero. Its sample estimate is 1.80 percent, the standard error is 0.75 percent, and the sample has 26 degrees of freedom.
t = 1.80 / 0.75 = 2.40For a two-tailed test, the p-value is the probability of a value at least as far from zero in either direction. For a one-tailed test in the positive direction, the p-value is roughly half the two-tailed p-value. For a one-tailed test in the negative direction, a positive t-statistic is evidence in the wrong direction, so the p-value is large.
That means the same t-statistic can support different decisions depending on the stated alternative:
| Alternative | Direction tested | Expected p-value pattern | ||
|---|---|---|---|---|
| Mean return is not zero | Both tails | Small if ` | t | ` is large |
| Mean return is greater than zero | Right tail | Smaller when t is positive and large | ||
| Mean return is less than zero | Left tail | Large when t is positive |
The statistic did not change. The tail being evaluated changed.
The Diagnostic Checklist
When the numbers seem inconsistent, move through the checks in order.
1. Confirm The Same Row
Regression output often places many estimates, t-statistics, and p-values close together. A coefficient for the market factor may have a t-statistic of 3.10 and a small p-value, while a separate size factor may have a t-statistic of 0.45 and a large p-value. Mixing rows creates a false contradiction.
2. Check Absolute Value Versus Sign
For a two-tailed test, extremeness is based on |t|, not the sign. A t-statistic of -2.60 is just as extreme as +2.60. For a one-tailed test, sign matters because the alternative has a direction.
3. Match The Tail Convention
A p-value printed by many regression packages is two-tailed unless otherwise stated. If the question asks for a one-tailed decision, the analyst may need to adjust the two-tailed p-value only when the test statistic points in the hypothesized direction.
4. Use The Correct Degrees Of Freedom
The t-distribution has thicker tails when degrees of freedom are low. A t-statistic near a cutoff can move from significant to not significant as degrees of freedom change. This effect is smaller for very large samples, where the t-distribution approaches the standard normal distribution.
5. Interpret The P-Value Correctly
A p-value is not the probability that the null hypothesis is true. It is the probability, assuming the null hypothesis is true, of observing a test statistic at least as extreme as the one calculated.
That wording matters because CFA questions often test decision discipline. The correct decision is based on p-value versus alpha, not on a statement that the null has been proven.
Worked Example: Output Alignment
Evergreen Factors estimates a factor model for monthly fund excess returns. The analyst reports the following simplified output:
| Variable | Coefficient | Standard error | t-statistic | p-value |
|---|---|---|---|---|
| Intercept | 0.10 | 0.20 | 0.50 | 0.620 |
| Momentum | 0.42 | 0.13 | 3.23 | 0.003 |
| Value | -0.18 | 0.16 | -1.13 | 0.266 |
If a question says "the momentum coefficient has a large t-statistic but a high p-value," the table does not support that statement. Momentum has the large t-statistic and small p-value. The high p-value belongs to the intercept.
The exam move is to slow down, match row to row, and then decide. At a 5 percent significance level, momentum is statistically significant because 0.003 <= 0.05. The intercept is not statistically significant because 0.620 > 0.05.
Exam Framing
On the CFA exam, the fastest path is not memorizing a single p-value cutoff. It is building a consistency map:
- Identify the null and alternative.
- Decide whether the test is one-tailed or two-tailed.
- Match the t-statistic to the correct p-value row.
- Compare the p-value with alpha.
- State the decision without overclaiming.
If a large absolute t-statistic and high p-value appear together for the same two-tailed test, treat the pairing as suspicious. If the alternative is one-tailed in the opposite direction, the high p-value can be perfectly reasonable. The difference is not the size of the statistic alone; it is the full test specification.