What it means
The NIST Engineering Statistics Handbook describes the t distribution as symmetric, with a single shape parameter called degrees of freedom. The plots for different values look alike, and the difference is the heaviness of the tails.
With one degree of freedom, it becomes a Cauchy distribution. NIST says the t distribution approaches a normal distribution as the degrees of freedom become large, and the approximation is quite good for values above 30.
This is why analysts use the normal curve for large samples and the t-distribution for small ones. The heavier tails build in extra uncertainty from estimating the standard deviation with few data points.
For a sample mean, degrees of freedom are usually n - 1. The interval is the sample mean plus or minus a critical t value times the standard error.
The NIST table gives these critical values. For a 95% two-sided interval, the relevant column is 0.975.
The critical value is 2.776 with 4 degrees of freedom, 2.262 with 9, 2.045 with 29, and 1.96 for the normal curve. The smaller the sample, the wider the interval.
NIST also explains how to use the table in a hypothesis test. For a two-sided test, find the column for 1 - alpha/2 and reject the null hypothesis if the absolute value of the test statistic exceeds the table value.
For one-sided tests, use the column for 1 - alpha. In finance, the t-distribution appears in regression output and in tests of average returns.
Because return data often have fat tails, the t shape can also be a more realistic model than the normal. Always check the sample size and independence before trusting the result.
In practice
Real-world examples.
Example
A fictional analyst has 10 monthly returns with a mean of 4.0 and a standard deviation of 3. The standard error is 3 / sqrt(10) = 0.949, and the t value with 9 degrees of freedom is 2.262. The 95% interval is 4.0 +/- 2.146, or 1.85 to 6.15.
Example
A fictional shop tests delivery times for 5 orders with standard deviation 2. The standard error is 2 / sqrt(5) = 0.894. With 4 degrees of freedom the margin is 2.776 x 0.894 = 2.48, wider than the 1.75 a normal value of 1.96 would give.
Example
A fictional team has 30 observations, so 29 degrees of freedom. The t value of 2.045 is only about 4.3% above the normal value of 1.96. At this size, the choice between the two matters little.
Formula
Calculation
Confidence interval = sample mean +/- t x (s / sqrt(n)), where t uses n - 1 degrees of freedom.
Worked example 1. With n = 10, mean 4.0 and s = 3, the margin is 2.262 x 0.949 = 2.146. The interval is 1.85 to 6.15, a width of 4.29.
Worked example 2. With n = 30, mean 4.0 and s = 3, the standard error is 3 / sqrt(30) = 0.548 and the critical t value with 29 degrees of freedom is 2.045. The margin is 2.045 x 0.548 = 1.12, so the interval is 2.88 to 5.12, a width of 2.24.
Same mean and same standard deviation, but triple the data roughly halves the width, because both the standard error and the critical value fall.Case study
Seen in the real world.
This case study is fictional and illustrative. Ravi tests whether a fund's average monthly excess return is above zero using 10 months of data. The mean is 4.0 and the standard deviation is 3. The t statistic is 4.0 / 0.949 = 4.2, which is well above the table value of 2.262 for a two-sided 5% test with 9 degrees of freedom. He rejects zero.
He then notes that 10 months is a short record and returns may have fat tails. He repeats the work with 60 months and keeps the t-distribution, since the standard deviation is still estimated. He reports both intervals and does not claim the fund will repeat the result. Ravi's note to the investment committee states the sample size, the degrees of freedom, the critical value and the interval, so readers can judge how much weight a short record deserves. The numbers are invented to show the method, and no fund result is implied.
Watch out
Common mistakes.
- Using the normal curve for a small sample, when the t-distribution gives a wider and more honest interval.
- Using n instead of n - 1 for degrees of freedom in a one-sample mean test.
- Assuming the t-distribution fixes biased or dependent data, when it only adjusts for estimating the standard deviation.
Questions
People also ask.
What is the t-distribution?
It is a bell-shaped distribution with heavier tails than the normal. It is used for means when the population standard deviation is unknown. Its shape depends on degrees of freedom.
When does it match the normal curve?
NIST says it approaches the normal distribution as degrees of freedom grow, and the match is quite good above 30. For smaller samples, the difference is large enough to matter.
What is the 95% critical value?
For a two-sided 95% interval it is 2.776 with 4 degrees of freedom, 2.262 with 9 and 2.045 with 29, from the NIST table. The normal value is 1.96.
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