Computing Two-Sample Degrees of Freedom With the Welch Formula

Why Two-Sample t Procedures Need Degrees of Freedom

In “Standard Error for a Difference in Means,” we calculated the standard error of \(\bar{x}_1-\bar{x}_2\) using both groups’ sample standard deviations and sample sizes. A two-sample t procedure uses that standard error to compare the observed difference with a hypothesized population difference or to build an interval for \(\mu_1-\mu_2\). It also needs degrees of freedom, or \(df\), to specify which t distribution to use.

For a one-sample t procedure, the degrees of freedom are \(n-1\), as explained in “Why the t Test Uses n Minus 1 Degrees of Freedom.” With two independent samples, each group contributes information about the uncertainty in the difference. The resulting degrees of freedom depend on both groups’ sample sizes and standard deviations. They are not generally just \(n_1+n_2-2\).

Definition: In a two-sample t procedure, the degrees of freedom determine the particular t distribution used to find a p-value or a critical value. A common calculation is the Welch-Satterthwaite formula, which uses both samples’ sizes and standard deviations.

Welch degrees of freedom can be a non-integer. That is not an error: the formula estimates the appropriate degrees of freedom for the t distribution when the population standard deviations are not assumed equal. A calculator’s two-sample t procedure can report this value. Another accepted approach in many AP Statistics settings is the conservative choice \(\min(n_1-1,n_2-1)\), the smaller of the two one-sample degrees of freedom.

Follow the method required by the problem, course, or calculator directions. Do not calculate both values and then choose the one that gives the result you prefer. Once the degrees of freedom are set, use that same choice consistently for the p-value in a test or the critical value in an interval.

Welch’s Formula

Let \(n_1\) and \(n_2\) be the sample sizes, and let \(s_1\) and \(s_2\) be the sample standard deviations. The standard error from the previous tutorial is \(SE_{\bar{x}_1-\bar{x}_2}=\sqrt{s_1^2/n_1+s_2^2/n_2}\). Welch’s formula uses the two variance contributions in that standard error:

df_{\text{Welch}} = \frac{\left(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}\right)^2} {\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1-1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2-1}}

Each variance contribution, \(s_i^2/n_i\), is squared in the denominator and divided by that group’s \(n_i-1\). The numerator squares the sum of the two contributions. Keep each standard deviation and sample size matched to its own group.

This formula is often easiest to use through a calculator’s two-sample t function. In Stats mode, enter each group’s \(\bar{x}\), \(s\), and \(n\); choose the alternative hypothesis that matches the question; and select Pooled: No, if the calculator offers that option. The output’s \(df\) is the Welch value. “Pooled: No” does not combine the standard deviations; it uses the separate-variance two-sample procedure.

Formula: For Welch degrees of freedom, use the two groups’ separate variance contributions in the formula above. For the conservative approach, use \(df=\min(n_1-1,n_2-1)\). Do not substitute \(n_1+n_2-2\) unless a procedure specifically assumes equal population variances.

The Conservative Minimum Approach

The conservative approach avoids the longer calculation. Find each group’s one-sample degrees of freedom, \(n_1-1\) and \(n_2-1\), and use the smaller. For example, if \(n_1=14\) and \(n_2=22\), the conservative choice is \(\min(13,21)=13\).

“Conservative” refers to how the choice generally affects inference: a smaller \(df\) gives a t distribution with heavier tails. For a fixed confidence level, it generally produces a larger critical value and a wider interval; for a fixed observed t statistic, it generally produces a larger p-value for a two-sided test or for a one-sided test when the statistic points in the direction of the alternative. Welch’s formula often uses more of the information in both groups and can give a larger \(df\), but it need not always do so. Neither method changes the observed difference in sample means or its standard error.

If a calculator reports a fractional Welch \(df\), use it as reported when calculator-based inference is requested. If a t table does not have that exact row, follow the table’s directions; a common conservative practice is to use the next lower listed integer. If a problem directs you to use the conservative minimum, use that instead of the calculator’s Welch value.

Worked Examples

Worked Example: Testing a Difference in Component Strength

An invented quality-control example compares the breaking strengths, in newtons, of components made by two production processes. Independent random samples give \(\bar{x}_1=52.4\), \(s_1=4\), and \(n_1=10\) for process 1, and \(\bar{x}_2=47.0\), \(s_2=12\), and \(n_2=30\) for process 2. Test whether the population mean strengths differ at \(\alpha=0.05\), using Welch degrees of freedom.

State.
Let \(\mu_1\) and \(\mu_2\) be the true mean breaking strengths of components made by processes 1 and 2, respectively. Test \(H_0:\mu_1-\mu_2=0\) against \(H_a:\mu_1-\mu_2\ne0\), with \(\alpha=0.05\).

Plan and check conditions.
Use a two-sample t test for independent means. The samples are described as independent random samples. If each sample is less than 10% of its respective process’s output, the 10% condition is met. For these relatively small samples, suppose plots of the strengths show no pronounced skewness or outliers. These checks support using a two-sample t procedure.

Do.
Calculate the standard error and Welch degrees of freedom from the sample summaries, then find the t statistic and two-sided p-value.

Conclude.
Compare the p-value with \(\alpha\), state the decision, and describe the evidence about the population mean strengths in context.

First, calculate the standard error:

SE_{\bar{x}_1-\bar{x}_2} = \sqrt{\frac{4^2}{10}+\frac{12^2}{30}} = \sqrt{1.6+4.8} = \sqrt{6.4} \approx 2.5298\text{ newtons}

Now substitute the variance contributions into Welch’s formula:

df_{\text{Welch}} = \frac{(1.6+4.8)^2}{\frac{1.6^2}{9}+\frac{4.8^2}{29}} = \frac{40.96}{0.284444+0.794483} \approx 37.964

The statistic is \(t=(52.4-47.0)/2.5298\approx2.135\). Using \(df=37.964\), the two-sided p-value is approximately \(0.03932\), rounded to five decimal places. Since \(0.03932<0.05\), reject \(H_0\). There is convincing evidence that the population mean breaking strengths differ between the two production processes.

For comparison, the conservative degrees of freedom are \(\min(10-1,30-1)=9\). Using the same t statistic but \(df=9\) gives a two-sided p-value of approximately \(0.06157\). That value is above 0.05, so the conservative approach would lead to failing to reject \(H_0\) at this significance level. This example shows why it matters to use the method specified and report which degrees of freedom were used. Do not switch methods after seeing which conclusion you prefer.

Worked Example: Finding Degrees of Freedom for a Study-Time Comparison

In an invented comparison of independent student samples, group 1 has \(\bar{x}_1=81\) minutes of study time, \(s_1=6\) minutes, and \(n_1=16\). Group 2 has \(\bar{x}_2=76\) minutes, \(s_2=10\) minutes, and \(n_2=25\). Find the Welch and conservative degrees of freedom.

The variance contributions are \(6^2/16=36/16=2.25\) and \(10^2/25=100/25=4\). Thus the standard error is \(\sqrt{2.25+4}=\sqrt{6.25}=2.5\) minutes. Welch’s calculation is:

df_{\text{Welch}} = \frac{(2.25+4)^2}{\frac{2.25^2}{15}+\frac{4^2}{24}} = \frac{39.0625}{0.3375+0.666667} \approx 38.900

As a check, the denominator is about \(1.004167\), and \(39.0625/1.004167\approx38.900\). The conservative choice is \(\min(15,24)=15\). The sample means are needed for the observed difference, but they do not enter either degrees-of-freedom calculation.

Worked Example: A Smaller Sample with Greater Variability

An invented environmental comparison measures a quantitative outcome in two independent samples. Group 1 has \(\bar{x}_1=14.8\), \(s_1=3\), and \(n_1=12\); group 2 has \(\bar{x}_2=12.1\), \(s_2=8\), and \(n_2=20\). Find both choices of degrees of freedom and identify the one to use if the instructions require the conservative approach.

The group contributions to the standard error are \(3^2/12=0.75\) and \(8^2/20=3.2\). Welch’s formula gives:

df_{\text{Welch}} = \frac{(0.75+3.2)^2}{\frac{0.75^2}{11}+\frac{3.2^2}{19}} = \frac{15.6025}{0.051136+0.538947} \approx 26.441

The conservative value is \(\min(11,19)=11\). The standard error is \(\sqrt{3.95}\approx1.9875\), in the outcome’s units, and the observed difference in sample means is \(14.8-12.1=2.7\). Those quantities do not determine which degrees-of-freedom convention to use: that comes from the calculator setting or instructions. If the directions specify the conservative method, use \(df=11\); if they specify the calculator’s separate-variance method, use the Welch value of about 26.441.

Using Degrees of Freedom in Tests and Intervals

For a two-sample t test, the test statistic compares the observed difference in sample means with the null difference, measured in standard errors. The degrees of freedom then determine the t distribution used to calculate the p-value. For a two-sample t interval, the degrees of freedom determine the t critical value for the chosen confidence level. In both settings, the difference in means and standard error are calculated separately from the degrees of freedom.

When using a calculator, check that the output matches the requested method. In a two-sample test or interval using separate sample standard deviations, a “Pooled: No” setting gives the Welch degrees of freedom. The calculator may also display the test statistic and p-value, or the interval endpoints. When doing part of the work by hand, use the same Welch or conservative \(df\) for the appropriate t probability or critical value.

Key takeaway: For two-sample t procedures about independent population means, use the calculator’s Welch degrees of freedom or the required conservative value \(\min(n_1-1,n_2-1)\). Keep the method consistent throughout a test or interval, and never choose it based on which result is more favorable.

Common Mistakes and AP Exam Tips

Using \(n_1+n_2-2\) automatically: That is not the general degrees-of-freedom rule for the separate-variance two-sample t procedure. Use the instructed Welch or conservative method.
Confusing the two approaches: Welch’s formula uses both variance contributions and can give a fractional value. The conservative method takes the smaller of \(n_1-1\) and \(n_2-1\).
Mixing up a group’s values: Keep \(s_1\) with \(n_1\), and \(s_2\) with \(n_2\), in every term of Welch’s formula.
Rounding too early: Keep full calculator precision for the variance contributions, degrees of freedom, and t statistic where possible. Round the reported answer only at the end.
Assuming the means determine \(df\): The sample means affect the observed difference and t statistic, but Welch’s formula uses the standard deviations and sample sizes.
Changing methods to change the conclusion: State and follow the method specified before interpreting a p-value. A complete response names the degrees-of-freedom approach and uses it consistently.

On an AP response, show the formula or identify the calculator’s Welch output, state the degrees of freedom, and use that value in the test or interval calculation. If using the conservative method, show the minimum explicitly. For a test conclusion, compare the p-value to \(\alpha\), say “reject” or “fail to reject,” and describe the evidence in context; do not say that a null hypothesis has been proved.

Check Your Understanding

Show the key substitution for each calculation, and state which degrees-of-freedom method you are using.

For independent samples with \(n_1=8\), \(s_1=5\), \(n_2=18\), and \(s_2=9\), find the conservative degrees of freedom.
Why can Welch degrees of freedom be fractional, while the conservative minimum is an integer?
A calculator reports Welch \(df=22.7\). What does that value determine in a two-sample t test, and what does it determine in a two-sample t interval?
Do the sample means appear in Welch’s formula? Explain which sample summaries do appear.
Why should you not calculate both methods and then select the one that gives the desired test conclusion?