Varians terkumpul Kalkulator

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What Is Pooled Variance?

Pooled variance combines the variances of two or more independent groups into a single estimate, on the assumption that the groups come from populations with the same variance. Each group's variance is weighted by its degrees of freedom (its size minus one), so larger groups count for more. It is the variance used in the classic two-sample t-test and the "mean square within" in one-way ANOVA.

Formula

Each group's sample variance is weighted by its degrees of freedom:

Pooled Variance (Two Groups)

sp² = [(n₁−1)s₁² + (n₂−1)s₂²] / (n₁ + n₂ − 2)

sp²Pooled variance
n₁, n₂Sizes of group 1 and group 2
s₁², s₂²Sample variances of group 1 and group 2

Pooled Variance (k Groups)

sp² = Σ[(nᵢ−1)sᵢ²] / Σ(nᵢ−1)

kNumber of groups
nᵢSize of group i
sᵢ²Sample variance of group i

How to Calculate Pooled Variance Step by Step

Find each group's sample variance.Calculate s² for every group separately, dividing by n − 1.
Note each group's size.Record n for each group.
Weight each variance.Multiply each s² by its group's n − 1 and add the results.
Add the degrees of freedom.Add up all the (n − 1) values.
Divide.sp² = weighted total / total degrees of freedom.

Pooled Variance Examples

1Comparing Two Teaching Methods

Data:Method A (n=12, s²=18) | Method B (n=15, s²=22)

Solution:sp² = [(11)(18) + (14)(22)] / (11 + 14) = (198 + 308) / 25 = 506 / 25.

Answer:Pooled variance = 20.24

2Drug Trial: Two Dosage Groups

Data:Low dose (n=20, s²=9.5) | High dose (n=25, s²=11.2)

Solution:sp² = [(19)(9.5) + (24)(11.2)] / (19 + 24) = (180.5 + 268.8) / 43 = 449.3 / 43.

Answer:Pooled variance ≈ 10.45

3Three Manufacturing Shifts

Data:Morning (n=30, s²=4.2) | Afternoon (n=28, s²=5.1) | Night (n=25, s²=4.8)

Solution:sp² = [(29)(4.2) + (27)(5.1) + (24)(4.8)] / (29 + 27 + 24) = (121.8 + 137.7 + 115.2) / 80 = 374.7 / 80.

Answer:Pooled variance ≈ 4.68

4Two Class Sections

Data:Section 1 (n=18, s²=30) | Section 2 (n=22, s²=25)

Solution:sp² = [(17)(30) + (21)(25)] / (17 + 21) = (510 + 525) / 38 = 1035 / 38.

Answer:Pooled variance ≈ 27.24

How to Use This Calculator for Pooled Variance

The calculator above works on one list of numbers at a time. To pool several groups:

Set the calculator to Sample.
Enter the first group's raw values and note its sample variance (s²) and its count (n).
Clear the input, enter the next group and note its s² and n. Repeat for every group.
Put the figures into the formula above: multiply each s² by its n − 1, add them, and divide by the total of the (n − 1) values.

Why Pool Variances?

A pooled estimate uses all the data, so it is more stable than any single group's variance.
It is the variance the equal-variance (Student's) two-sample t-test is built on.
It generalises directly to three or more groups as the within-group mean square in ANOVA.

Where Pooled Variance Is Used

Two-sample t-test

The equal-variance t-test divides the difference in means by a standard error built from sp².

ANOVA

The mean square within groups (MSW) is the pooled variance of all the groups.

Effect sizes

Cohen's d divides the difference in means by the pooled standard deviation √sp².

Clinical trials

Comparing treatment and control groups when their variances are similar.

Education research

Comparing outcomes between classes taught different ways.

Assumptions and Limitations of Pooled Variance

Pooling is only appropriate when its assumptions hold:

Equal population variances

If the groups' variances clearly differ, the pooled figure describes none of them well. Levene's test is a common check.

Independent groups

If the same people appear in more than one group (before-and-after measurements, for example), use a paired analysis instead.

Unequal group sizes

A much larger group dominates the pooled estimate. Unequal sizes combined with unequal variances are the case where the pooled t-test is least reliable.

Pooled is not the same as combined

Pooled variance ignores differences between group means. The variance of all the data lumped together includes them and is larger whenever the means differ.

The alternative

Welch's t-test does not assume equal variances and does not pool. Many statisticians use it by default.

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Soalan Lazim: Varians terkumpul Kalkulator

Semua yang anda perlu tahu tentang varians.

Apakah varians terkumpul?

Varians terkumpul ialah purata wajaran varians daripada dua atau lebih sampel bebas. Ia mengandaikan bahawa populasi mempunyai varians yang sama (homoskedastisitas). Formula untuk dua kumpulan ialah: sp² = [(n₁−1)s₁² + (n₂−1)s₂²] / (n₁ + n₂ − 2). Ia memberikan lebih berat kepada sampel yang lebih besar dan digunakan sebagai anggaran varians dalam ujian-t terkumpul.

Bilakah anda menggunakan varians terkumpul?

Varians terkumpul digunakan apabila: 1. Melakukan ujian-t dua sampel (dengan mengandaikan varians yang sama). 2. Menjalankan ANOVA untuk membandingkan cara merentas berbilang kumpulan. 3. Menggabungkan hasil daripada berbilang kajian (analisis meta). 4. Menganggarkan varians biasa apabila kumpulan dipercayai datang daripada populasi dengan sebaran yang sama. Sentiasa sahkan andaian varians yang sama dahulu menggunakan ujian Levene atau ujian F.

Bagaimana untuk mengira varians terkumpul langkah demi langkah?

Untuk dua kumpulan: 1. Kira varians setiap kumpulan (s₁² dan s₂²). 2. Perhatikan saiz sampel (n₁ dan n₂). 3. Kira: sp² = [(n₁−1)·s₁² + (n₂−1)·s₂²] / (n₁ + n₂ − 2). Contoh: Kumpulan 1 (n=10, s²=8), Kumpulan 2 (n=15, s²=12): sp² = [(9)(8) + (14)(12)] / 23 = (72 + 168) / 23 = 10.43.

Apakah perbezaan antara varians terkumpul dan varians gabungan?

Varians terkumpul menganggap variasi populasi dan berat yang sama mengikut darjah kebebasan — digunakan dalam ujian hipotesis. Varian gabungan ialah istilah yang lebih luas untuk menggabungkan set data yang mungkin menyumbang kepada perbezaan dalam kedua-dua min dan varians antara kumpulan. Untuk menggabungkan dua set data menjadi satu, anda juga perlu mengambil kira perbezaan antara min kumpulan, bukan hanya varians dalam kumpulan.

Apakah andaian yang diperlukan varians terkumpul?

Varians terkumpul menganggap kehomogenan varians (homokedastisitas) — bahawa semua kumpulan berasal daripada populasi dengan varians yang sama. Jika andaian ini dilanggar, anggaran terkumpul tidak boleh dipercayai. Uji andaian ini dengan ujian Levene, ujian Bartlett atau ujian F untuk kesamaan varians. Jika varians tidak sama, gunakan ujian-t Welch dan bukannya ujian-t terkumpul.

Bolehkah saya mengumpulkan varians daripada lebih daripada dua kumpulan?

Ya. Formula umum untuk kumpulan k ialah: sp² = Σ[(nᵢ−1)·sᵢ²] / Σ(nᵢ−1). Ini ialah purata wajaran di mana varians setiap kumpulan ditimbang mengikut darjah kebebasannya (nᵢ−1). Inilah yang digunakan ANOVA sebagai Mean Square Within (MSW) — anggaran terkumpul varians merentas semua kumpulan.