A Corrected Welch Satterthwaite Equation. And: What You Always Wanted to Know About Kish's Effective Sample but Were Afraid to Ask

A Corrected W elc h–Satterth w aite Equation. And: What Y ou Alw a ys W an ted to Kno w Ab out Kish’s Effectiv e Sample Size but W ere Afraid to Ask. Matthias v on Davier F ebruary 26, 2026 Abstract This article presents a corrected version of the Satterth waite (1941, 1946) appro ximation for the degrees of freedom of a weigh ted sum of in- dep enden t v ariance comp onen ts. The original formula is kno wn to yield biased estimates when comp onen t degrees of freedom are small. The cor- rection, derived from exact moment matching, adjusts for the bias by in- corp orating a factor that accoun ts for the estimation of fourth moments. W e sho w that Kish’s (1965) effective sample size formula emerges as a spe- cial case when all v ariance comp onents are equal and component degrees of freedom are ignored. Simulation studies demonstrate that the corrected estimator closely matches the exp ected degrees of freedom even for small comp onen t sizes, while the original Satterthw aite estimator exhibits sub- stan tial down w ard bias. A dditional applications are discussed, including jac kknife v ariance estimation, m ultiple imputation total v ariance, and the W elch test for unequal v ariances. 1 In tro duction The estimation of effectiv e degrees of freedom for complex v ariance estimators is a recurring problem in statistics, particularly when combining indep enden t v ariance comp onen ts with different degrees of freedom. Satterth waite (1941, 1 1946) provided a widely used approximation that expresses the degrees of free- dom of a weigh ted sum of v ariance comp onen ts as a function of the comp onen t v ariances and their degrees of freedom. This appro ximation underpins many sta- tistical pro cedures, including the W elch test for t wo-sample comparisons with unequal v ariances, and v ariance estimation in complex survey designs. Ho w- ev er, the Satterth waite formula relies on large-sample approximations and can b e severely biased when the comp onen t degrees of freedom are small or the n umber of comp onen ts is limited (Johnson & Rust, 1992). In parallel, Kish (1965) introduced the concept of effective sample size for w eighted samples, which adjusts the nominal sample size to account for unequal w eighting. Kish’s formula, deriv ed under the assumption of equal v ariances across observ ations, has b ecome a standard to ol in survey sampling. Although dev elop ed in different contexts, both Satterth waite’s and Kish’s estimators share a common structure and can b e link ed through a simplified Satterthw aite ex- pression when comp onen t v ariances are homogeneous. This article has three main ob jectiv es. First, we derive a corrected Satterth waite degrees-of-freedom estimator that mitigates the bias for small comp onen t de- grees of freedom. The correction follows from a careful handling of the exp ected fourth momen ts of the comp onen t v ariance estimates, leading to an impro ved form ula that reduces to the original Satterthw aite expression as comp onen t de- grees of freedom increase. Second, we show that Kish’s effective sample size is a sp ecial case of the Satterthw aite estimator when all v ariance comp onen ts are equal and the comp onen t degrees of freedom are effectively ignored. Third, w e presen t simulation results comparing the original Satterthw aite estimator, the corrected version, and Kish’s estimator under v arious conditions, demonstrating the bias reduction achiev ed b y the correction. Finally , we illustrate applications of the corrected formula in jackknife v ariance estimation, m ultiple imputation total v ariance, and the W elc h test. The article is organized as follo ws. Section 2 reviews Satterth waite’s original deriv ation and presents the corrected form ula. Section 3 discusses Kish’s ef- fectiv e sample size and its relationship to Satterthw aite’s estimator. Section 4 pro vides sim ulation evidence comparing the three estimators. Section 5 gives examples of the corrected estimator beyond effectiv e sample size, including jack- knifing, m ultiple imputation, and the W elc h test. Section 6 concludes with a 2 summary of recommendations. 2 Satterth w aite’s (1941, 1946) Appro ximate d.f. Satterth waite (1941, 1946) examines the appro ximate degrees of freedom for complex v ariance estimates. That is, we hav e weigh ts w 1 , . . . , w k and v ariance comp onen ts S 2 1 , . . . S 2 k , each with degrees of freedom ν k , so that ν k S 2 k σ 2 k ∼ χ 2 ν k that is, E  ν k S 2 k σ 2 k  = ν k and V ar  ν k S 2 k σ 2 k  = 2 ν k . The approximate d.f. are developed for a complex v ariance estimate S 2 = K X k =1 w k S 2 k whic h is a linear com bination of indep enden t v ariance comp onen ts. Satterth- w aite (1941, 1946) argues that with indep enden t X k and reasonably large d.f. for each of the v ariance comp onen ts S 2 k , one may assume that S 2 ∼ χ 2 ν ∗ and that d.f .  S 2  = ν ∗ ≈  P i w i S 2 i  2 P i w 2 i S 4 i ν k . 2.1 Pro of of Satterthw aite’s F orm ula and its Impro v emen t The form ula provided by Satterthw aite (1941, 1946) is widely used, but it is kno wn that it pro duces biased estimated (e.g. Johnson & Rust, 1992) for small comp onen t degrees of freedom ν k . In this section, w e derive the exp ected v alue the approximation is based on and provide a tighter approx imation suitable for small ν k and K , while it conv erges to the original for large K and large ν k . 3 This section starts with reviewing some of the properties of χ 2 distributed v ari- ables that are needed to derive the main result. 2.1.1 Some Properties of χ 2 Distributed Random V ariables Note that for any χ 2 distributed v ariable with ν degrees of freedom we hav e ν 2 V ar  S 2  σ 4 = 2 ν → ν = 2  σ 2  2 V ar ( S 2 ) and ν 2 V ar  S 2  σ 4 = 2 ν → V ar  S 2  = 2  σ 2  2 ν . Also, we hav e V ar  S 2  + E  S 2  2 = E  S 4  = 2 σ 4 ν + σ 4 = σ 4  2 ν + 1  = σ 4  2 + ν ν  and hence σ 4 = E  S 4  ν S ν S + 2 . V ar K X k =1 w k S 2 k ! = K X k =1 w 2 k V ar  S 2 k  = K X k =1 w 2 k E  S 4 k  − K X k =1 w 2 k E 2  S 2 k  and further 2 σ 4 ν ∗ = K X k =1 w 2 k 2 σ 4 k ν k . 2.1.2 Deriv ation of the Satterth w aite F ormula Since the S 2 k are indep enden t w e hav e V ar  S 2  = P k w 2 k V ar  S 2 k  and by using the equiv alence ν = 2 [ σ 2 ] 2 V ar ( S 2 ) ↔ V ar  S 2  = 2 [ σ 2 ] 2 ν for S 2 and S 2 k w e obtained 2 σ 4 ν ∗ = K X k =1 w 2 k 2 σ 4 k ν k 4 whic h, by solving for ν ∗ , pro vides ν ∗ = σ 4 P K k =1 w 2 k σ 4 k ν k Then, an estimate of ν ∗ is obtained by using the equiv alence σ 4 = E  S 4  ν ν + 2 ≈ S 4 b oth for σ 4 k =  σ 2 k  2 =  E  S 2 k  2 and σ 4 . This assumes that the comp onen ts degrees of free dom are sufficiently large so that ν ν +2 ≈ 1 and that S 4 ≈ E  S 4  whic h also is implied by ν → ∞ . Then, plugging in S 4 for σ 4 , as well as S 2 k for σ 2 k yields ν ∗ ≈  P k w k S 2 k  2 P k w 2 k [ S 2 k ] 2 ν k . Satterth waite p oin ts out this approximation may not work w ell for small K, ν k . 2.1.3 Some Equiv alencies Here, it is imp ortant to note that v ∗ satt =  P k w k σ 2 k  2 P k w 2 k [ σ 2 k ] 2 ν k =  P k cw k σ 2 k  2 P k [ cw k σ 2 k ] 2 ν k for any c > 0 . Cho osing c = K " X k w k σ 2 k # − 1 and setting w σ 2 = 1 K P k w k σ 2 k yields ν ∗ satt = h P w k σ 2 k wσ 2 i 2 P k 1 ν k h w k σ 2 k wσ 2 i 2 = K 2 P k 1 ν k h w k σ 2 k wσ 2 i 2 5 whic h is K times the harmonic mean of the q k = ν k h wσ 2 w k σ 2 k i 2 ν ∗ satt = K · H ( q k ) = K · H   ν k " w σ 2 w k σ 2 k # 2   . This section sho ws that the neither the Satterth waite estimator, nor its improv ed v ersion, dep end on the a verage v alue w σ 2 of the weigh ted v ariances w i σ 2 k . 2.1.4 Impro vemen t for Small ν k and Small K While Satterthw aite assumes that K , ν k >> 1 and hence, for all instances, his w ork implicitly uses ν ν +2 ≈ 1 , and then replaces σ 2 k b y S 2 k directly , without the factor for estimation of ν ∗ satt , v on Davier (2025a,b) p oin ts out that for small K, ν k a correction is needed. Recall that we ha ve sho wn ab ov e that ˆ σ 4 k ≈ S 4 k ν k ν k + 2 and σ 4 ≈ S 4 ν ∗ ν ∗ + 2 whic h provide the necessary correction. It follows that ν ∗ ≈ ν ∗ ν ∗ +2  P k w k S 2 k  2 P k w 2 k [ S 2 k ] 2 ν k +2 ↔ ν ∗ ≈  P k w k S 2 k  2 P k w 2 k [ S 2 k ] 2 ν k +2 − 2 . 2.2 Some Results on the Impro ved Effectiv e D.F. The follo wing tables show results for sums of K simple v ariance estimates S 2 k with ν k S 2 k σ 2 k χ 2 distributed. The ideal case where ∀ k , j : σ 2 k = σ 2 j , ν k = ν j , w k = w j w as used to b e able to compare the original Satterthw aite formula to the improv ed estimator, and against the expected v alue E ( ν ∗ ) = K ν that follows in this case since each component S 2 k has the same distribution. 6 T able 1: Simulation results for K = 2 and K = 4 K d f ¯ ν unc SD unc ¯ ν corr SD corr K × ¯ ν 2 1 1.410 0.346 2.229 1.039 2 2 2 3.135 0.640 4.270 1.281 4 2 4 6.837 1.044 8.256 1.566 8 2 8 14.577 1.494 16.222 1.868 16 2 16 30.348 1.938 32.141 2.181 32 2 32 62.199 2.282 64.086 2.425 64 4 1 2.198 0.637 4.595 1.912 4 4 2 5.320 1.202 8.640 2.403 8 4 4 12.400 1.974 16.600 2.961 16 4 8 27.567 2.852 32.459 3.565 32 4 16 58.908 3.567 64.272 4.013 64 4 32 122.482 4.113 128.138 4.370 128 T able 2: Simulation results for K = 8 and K = 16 K d f ¯ ν satt SD unc ¯ ν corr SD corr K × ¯ ν 8 1 3.656 1.023 8.969 3.070 8 8 2 9.473 2.014 16.946 4.029 16 8 4 23.171 3.296 32.757 4.944 32 8 8 53.198 4.720 64.498 5.900 64 8 16 115.997 5.736 128.497 6.453 128 8 32 243.096 6.552 256.290 6.962 256 16 1 6.443 1.608 17.330 4.825 16 16 2 17.586 3.180 33.172 6.360 32 16 4 44.642 5.226 64.963 7.838 64 16 8 104.373 7.278 128.466 9.097 128 16 16 229.741 8.776 256.458 9.873 256 16 32 484.036 9.730 512.289 10.338 512 7 T able 3: Simulation results for K = 32 and K = 64 K d f ¯ ν satt SD unc ¯ ν corr SD corr K × ¯ n 32 1 11.938 2.503 33.815 7.509 32 32 2 33.825 4.924 65.650 9.847 64 32 4 87.394 7.914 129.091 11.870 128 32 8 207.089 10.860 256.861 13.575 256 32 16 457.054 13.031 512.186 14.659 512 32 32 965.813 14.191 1024.177 15.078 1024 64 1 22.725 3.757 66.175 11.270 64 64 2 65.904 7.259 129.807 14.517 128 64 4 172.991 11.588 257.486 17.381 256 64 8 412.037 15.757 513.047 19.696 512 64 16 912.294 18.557 1024.331 20.876 1024 64 32 1929.855 20.471 2048.471 21.750 2048 The ob vious conclusion is that the correction of the Satterthw aite effectiv e sam- ple size is necessary even for large component degrees of freedom. The case where K = 64 and ν k = ν = 32 sho ws this clearly . The av erage d f for the corrected estimate is ν ∗ corr = 2048 . 5 ≈ 2028 = K × ν > ν ∗ satt = 1929 . 9 . The other end of the table sho ws for K = 2 and d f = 1 that ν ∗ corr = 2 . 23 ≈ 2 = K × ν > 1 . 41 = ν ∗ satt . 3 Kish’s (1965) Effectiv e Sample Size Estimate T o start with the similarities, b oth Kish’s (1965) form ula for the effectiv e sam- ple size for an estimator of the v ariance of the mean and the Satterth waite (1941) estimator of effective degrees of freedom of a complex v ariance estimate conceptual similarities. The idea b ehind finding the effective sample size for a weigh ted sample mean can be illustrated using the case where some groups do not carry any weigh t, i.e., for a subset of observ ations R we ha ve that ∀ r ∈ R ⊂ { 1 , . . . , N } : w r = 0 . In this case, only the other observ ation coun t, those in the set i ∈ { 1 , . . . , N } \ R for which the weigh ts w i > 0 do not v anish. In this case, it is immediately clear that the sample size going into the calculation of V ar w ( y ) is reduced, as the cases with zero weigh t do contribute to the estimate of the mean. The sample size b ecomes (at most) N R = N − ∥ R ∥ , i.e., it is reduced by the n umber of observ ations that are eliminated by the fact that their weigh ts are zero. The 8 effectiv e sample size is, assuming unit w eights for i / ∈ R N R = N X i =1 w i = ∥{ 1 , . . . , N } \ R ∥ whic h can b e considered the effective sample size for a w eighted sample where some cases ha ve unit weigh ts ( w i = 1 ), and the rest has v anishing ( w r = 0 ) w eights. In more general terms, the w eights w i ma y b e real v alued and p ositiv e, so that their relative v ariance r elv ar ( w ) = 1 K K X k =1  w k w − 1  2 with w = 1 K P k w k do es not v anish. Consider a random v ariable Y with finite mean µ and finite v ariance σ 2 . Let y i ∈ R , i = 1 , . . . , N denote the observ ed v alues in a sample of size K and assume the sample requires w eighting using the weigh ts w k . W e are studying a w eighted estimator of the sample mean using weigh ts w k ∈ R + , that is y w = M w ( y ) = P K k =1 w k y k P K k =1 w k = 1 N K X k =1 w k w y k . Similarly , we can estimate the v ariance of Y V ar w ( y ) = P K k =1 w k ( y k − y k ) 2 N w = P K k =1 w k y 2 k N w − " P K k =1 w k y k N w # 2 = y 2 w − ( y w ) 2 . The w eigh ts w k can be c hosen to reflect unequal sampling probability . F or example, if the probability of sampling observ ation i is giv en by P i > 0 , the w eights could b e chosen as w i = P − 1 i to counter unequal probabilities of b eing included in the sample. Another scenario is that the y i are statistics based on smaller samples tak en from certain well-defined groups (sampling units, sub-p opulations, etc.). In that case, y k = f k ( y k 1 , . . . , y kN k ) are (potentially also w eighted) aggregates 9 based on the within group observ ations y k 1 , . . . , y kN k . In this case, eac h of the groups may receive a different weigh t to account for and correct discrepancies b et w een exp ected p opulation prop ortion vs. sample sizes for each group. T wo things are worth noting here: 1. The v ariance estimate V ar w ( y ) under non-uniform weigh ts may not be the same as the v ariance calculated under simple random sampling V ar 1 ( y ) (where every observ ation is ’worth the same’, as all w i = 1 ) due to the w eights being used. 2. The quality of the v ariance estimator V ar w ( y ) , may differ from the qual- it y of the unw eigh ted estimate V ar 1 ( y ) . The term quality is chosen to b e delib erately v ague here. One well kno wn issue with weigh ting is that it impro ve one quality (reduce bias) at the cost of another (increase uncer- tain ty). The well cited general formula for the effective sample size is related to this cost of weigh ting. The expression to calculate the effective sample size was first pro vided by Kish (1965) and then giv en in simplified form by Kish (1992). It can b e deriv ed by examining the v ariance of the weigh ted mean y w : V ar ( y w ) = V ar P K k =1 w k y k P K k =1 w k ! = P K k =1 w 2 k V ar ( y k )  P K k =1 w k  2 , whic h follo ws when assuming that the y k are indep enden t. When we additionally assume that ∀ k : V ar ( y k ) = σ 2 , we obtain V ar ( y w ) = P K k =1 w 2 k σ 2  P K k =1 w k  2 = σ 2 P K k =1 w 2 k  P K k =1 w k  2 = σ 2 N w 2 w 2 . Using the same notation as ab o ve, Kish’s (1965) estimate of effectiv e sample size is given by n ef f ≈  P K k =1 w k  2 P K k =1 w 2 k → V ar ( y w ) = σ 2 n ef f . 10 A cross the literature, there is another expression that is often giv en for the effectiv e sample size. This expression is based on the relv ariance, which stands for relative v ariance, of the weigh ts. It is the v ariance of the weigh ts divided b y the av erage of the weigh ts. T hat is D ef f =  1 + cv 2 ( w )  where 1 + cv 2 ( w ) = 1 + V ar  w w  = 1 + 1 N N X i =1  w i w − 1  2 since M ( w ) = w . Then we ha ve 1 + cv 2 ( w ) = 1 N w 2 K X k =1 w 2 k = N P K k =1 w 2 k  P K k =1 w k  2 = N n ef f = D ef f . 4 Satterth w aite’s F ormula implies Kish’s n ef f F or a sample X 1 , ..., X K with weigh ts w 1 , . . . , w K of K observ ations, consider ν ∗ ≈  P K k =1 w k S 2 k  2 P K k =1 w 2 k S 4 k ν k whic h is the w ell kno wn Satterth waite estimator of the d.f. for the v ariance estimate S 2 = K X k =1 w k S 2 k . The Kish (1965) approximation of the effective sample size can obtained by assuming that all v ariance comp onen ts are iden tical, i.e., ∀ i, j : S 2 i = S 2 j = S 2 0 . In that case, we obtain ν ∗ ≈  P K k =1 w k S 2 0  2 P K k =1 w 2 k [ S 2 0 ] 2 =  S 2 0  2  P K k =1 w k  2 [ S 2 0 ] 2 P K k =1 w 2 k = n ef f . In this case, no small K , ν k is needed as the assumption made b y Kish eliminates the need to insert estimates of E  S 4  . The result turns out to b e a simplified 11 Satterth waite-t yp e estimator of degrees of freedom if all comp onen t v ariances are known to b e the same, and only the weigh ts are unequal. 4.1 Comparing Kish, Satterth w aite, and the Corrected V ersion The next table shows a comparison of Kish ’s effectiv e sample size and Sat- terth waite’s degrees of freedom (divided b y comp onen t av erage ν ) for random w eights w k ∼ N (1 , 0 . 3) . The last three columns show ratios that indicate how close the estimates are to the n umber of comp onen ts K , in the case of Kish, or ho w close these are to K ν , the exp ected v alue in the unw eighted case for equal comp onen t ν k . T able 4: Results for different weigh ts (random w k ∼ N (1 , 0 . 3) ). Columns: mean Kish, mean Satterthw aite (uncorrected), mean corrected Satterth waite, total sample size K ν , and ratios relative to K or K ν . K ν M(Kish) M(Satt) M(Corr) K ν Kish/ K Satt/ K ν Corr/ K ν 16 1 14.74 6.16 16.49 16 0.92 0.39 1.03 16 5 14.74 55.36 75.50 80 0.92 0.69 0.94 16 50 14.75 712.31 738.81 800 0.92 0.89 0.92 16 500 14.75 7351.23 7378.63 8000 0.92 0.92 0.92 32 1 29.43 11.28 31.83 32 0.92 0.35 0.99 32 5 29.43 107.84 148.98 160 0.92 0.67 0.93 32 50 29.42 1417.19 1471.87 1600 0.92 0.89 0.92 32 500 29.43 14659.25 14715.88 16000 0.92 0.92 0.92 64 1 58.78 21.27 61.81 64 0.92 0.33 0.97 64 5 58.78 212.92 296.08 320 0.92 0.67 0.93 64 50 58.81 2830.18 2941.39 3200 0.92 0.88 0.92 64 500 58.79 29280.13 29395.25 32000 0.92 0.92 0.92 T able 5: Results for equal weigh ts (all w k = 1 ). Columns: mean Kish, mean Satterth waite (uncorrected), mean corrected Satterthw aite, total sample size K ν , and ratios relativ e to K or K ν . K ν M(Kish) M(Satt) M(Corr) K ν Kish/ K Satt/ K ν Corr/ K ν 16 1 16.00 6.45 17.36 16 1.00 0.40 1.08 16 5 16.00 59.31 81.03 80 1.00 0.74 1.01 16 50 16.00 771.06 799.90 800 1.00 0.96 1.00 16 500 16.00 7970.16 8000.04 8000 1.00 1.00 1.00 32 1 32.00 11.93 33.78 32 1.00 0.37 1.06 32 5 32.00 116.40 160.96 160 1.00 0.73 1.01 32 50 32.00 1540.29 1599.91 1600 1.00 0.96 1.00 32 500 32.00 15938.14 15999.89 16000 1.00 1.00 1.00 64 1 64.00 22.75 66.24 64 1.00 0.36 1.04 64 5 64.00 230.86 321.20 320 1.00 0.72 1.00 64 50 64.00 3078.86 3200.01 3200 1.00 0.96 1.00 64 500 64.00 31874.41 31999.90 32000 1.00 1.00 1.00 12 It can b e seen that with increasing ν → ∞ the Satterthw aite and the Kish estimates conv erge. This is exp ected as Kish’s estimator is obtained by ignoring the comp onent-wise degrees of freedom ν k . Therefore, Kish’s approximation should only b e applied when it is known that all components estimate the same v ariance, and in addition, it is kno wn that the effect of estimating the comp onent v ariances can b e ignored. 5 Examples Bey ond Effectiv e Sample Size 5.1 Jac kknifing and Balanced Rep eated Replications (BRR) In the case of single comp onen t d.f. ν k = 1 and large(-ish) K representing the (non-trivial) jackknife zones we hav e no weigh ts to account for and obtain ν ∗ ≈ 3  P K k =1 [ T · − T k ] 2  2 [ T · − T k ] 4 − 2 since ν k + 2 = 3 for all k . Here, the T k are the pseudo-v alues calculated under the jackknife sc heme of dropping units, and T · = P k T k . In the case of the BRR the T k are the balanced replicated estimates (of the half samples) obtained by adjusting the w eights of the pairs and recalculating the statistics for each zone K . 5.2 T otal V ariance under Multiple Imputations The total v ariance of an estimator when imputations is a prime example of a w eighted v ariance estimate. The total v ariance as defined b y Rubin (1987) is giv en by V ar ( total ) = V ar ( sampling ) + M + 1 M V ar ( imputation ) where the sampling v ariance may b e estimated according to some resampling sc heme (Johnson & Rust, 1992) and the imputation v ariance is the sample standard deviation across M imputation based calculations of the same statistic. In this case, the follo wing weigh ts are used: 13 ( w 1 , w 2 ) =  1 , M + 1 M  and the corrected equation can b e directly applied. 5.3 A Corrrected W elc h T est d.f . The mo dified W elc h (1947) test uses the d.f. ν 1 = N 1 − 1 and ν 2 = N 2 − 1 of eac h of the v ariances of tw o samples to p ool the v ariance. The W elc h-Satterth waite form ula used in this case can b e adjusted using the corrected Satterth waite form ula derived ab o v e. Then we ha ve ν ∗ ≈  1 N 1 S 2 1 + 1 N 2 S 2 2  2 S 4 1 N 2 1 ( v 1 +2) + S 4 1 N 2 2 ( v 2 +2) − 2 where S 2 k are the sample v ariances and weigh ts w k = 1 N k in this case, for the d.f. ν ∗ of the sample size weigh ted p o oled v ariance used in the W elch test. 6 Conclusion The approaches presented here were developed under differen t constraints. The Kish (1965, 1992) effective sample size puts the strongest constrain ts on the estimator, as it has b een developed as to ol for a simple approximation when only the weigh ts are known and all comp onen ts of a v ariance estimate can b e assumed to hav e the same exp ectation, and eac h comp onent is a well estimated v ariance with mo derate to large degrees of freedom. The Satterthw aite (1941, 1946) estimate of the effectiv e degrees of freedom was dev elop ed for weigh ted sums of v ariance comp onen ts with v arying degrees of freedom. This is a m uch more general approach than the simple appro xiamtion pro vided by Kish (1965, 1992), but lacks the abilit y to generate useful estimates when the comp onen t degrees of freedom, ν k , are small. The c orrected Satterthw aite effective degrees of freedom is based on v on Davier’s 14 (2025a/b) research and produces estimates that are muc h closer to the exp ected v alue, when simulating data using the the ideal case, K ν , also for small compo- nen t ν k . It is calculated as ν ∗ ≈  P k w k S 2 k  2 P k w 2 k [ S 2 k ] 2 ν k +2 − 2 and with growing comp onen t degrees of freedom ν k and comp onents K it ap- proac hes the original Satterthw aite expression, while for small ν k it a voids the sev ere bias found with the original form ula. References Johnson, E. G., & Rust, K. F. (1992). Population Inferences and V ariance Estimation for NAEP Data. Journal of Educational Statistics, 17(2), 175–190. doi:10.2307/1165168 Kish, L. (1965). Surv ey samplin g. John Wiley & Sons. Kish, L. (1992). W eighting for unequal Pi. Journal of Official Statistics, 8(2), 183–200. Lipsitz, S., P arzen, M., & Zhao, L. P . (2002). A Degrees-Of-F reedom approxima- tion in Multiple imputation. 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The generalization of ‘Studen t’s’ problem when sev- eral different p opulation v ariances are in v olved. Biometrik a, 34(1/2), 28-35. doi:10.2307/2332510 16