PertengahanAplikasi·9 min

Mengesan Pencilan dengan Sisihan Piawai

Pelajari cara mengenal pasti pencilan dalam data anda menggunakan sisihan piawai. Kuasai peraturan 3-sigma, kaedah IQR, dan fahami bila pencilan patut dibuang.

Apakah Pencilan?

Pencilan ialah titik data yang berbeza secara ketara daripada pemerhatian lain. Ia boleh disebabkan oleh ralat pengukuran, kesilapan kemasukan data, atau mungkin mewakili kes luar biasa yang tulen yang patut disiasat.

Titik oren pada (10, 50) ialah pencilan

Peraturan 3-Sigma

Untuk data bertaburan normal, titik melebihi 3 sisihan piawai daripada min dianggap sebagai pencilan. Ia berlaku kurang daripada 0.3% daripada masa secara kebetulan.

Pencilan jika

x < μ - 3σ OR x > μ + 3σ

Contoh

Jika markah ujian mempunyai μ = 75 dan σ = 10: - Had bawah: 75 - 30 = 45 - Had atas: 75 + 30 = 105 - Sebarang markah di bawah 45 atau di atas 105 adalah pencilan

Kaedah Skor-Z

Kira skor-z untuk setiap titik data. Jika |z| > 3 (atau kadangkala 2.5), ia adalah pencilan.

Skor-Z

z = (x - μ) / σ

Pilihan Ambang

- |z| > 3: Konservatif (menangkap lebih sedikit pencilan) - |z| > 2.5: Sederhana - |z| > 2: Liberal (menangkap lebih banyak pencilan)

Kaedah IQR (Alternatif)

Kaedah Julat Antara Kuartil (IQR) lebih teguh terhadap pencilan kerana ia tidak menggunakan min atau sisihan piawai.

Langkah 1

Cari Q1 (persentil ke-25) dan Q3 (persentil ke-75)

Langkah 2

Kira IQR = Q3 - Q1

Langkah 3

Pagar bawah = Q1 - 1.5 × IQR

Langkah 4

Pagar atas = Q3 + 1.5 × IQR

Langkah 5

Titik di luar pagar adalah pencilan

Mengendalikan Pencilan

Jangan Padam Secara Automatik!

Pencilan tidak sentiasa merupakan ralat. Sebelum membuangnya, siasat: - Adakah ia ralat kemasukan data atau pengukuran? - Adakah ia nilai ekstrem yang tulen? - Adakah ia mewakili kes tepi yang penting?

Bila Perlu Dibuang

- Ralat kemasukan data yang disahkan - Kerosakan peralatan pengukuran - Di luar julat nilai yang mungkin

Bila Perlu Dikekalkan

- Mewakili kebolehubahan sebenar - Penting untuk analisis anda - Membuangnya akan menyebabkan berat sebelah dalam keputusan

How to Read This Article

A statistics tutorial is a practical interpretation guide, not just a formula dump. It refers to the assumptions, notation, and reporting language that analysts need when they explain a result to a teacher, manager, client, or reviewer. The article body covers the specific topic, while the sections below create a common interpretation frame that readers can reuse across related metrics.

Reading goal	What to focus on	Common mistake
Definition	What the metric is and what quantity it summarizes	Treating the formula as self-explanatory
Formula choice	Sample versus population assumptions and notation	Using n when n-1 is required or vice versa
Interpretation	Whether the result indicates concentration, spread, or risk	Calling a large value good or bad without context

Frequently Asked Questions

How should I interpret a high standard deviation?

A high standard deviation means the observations are spread farther from the mean on average. Whether that spread is acceptable depends on the context: wide dispersion might signal risk in finance, instability in manufacturing, or genuine natural variation in scientific data.

Why do some articles mention n while others mention n-1?

The denominator reflects the difference between population and sample formulas. Population variance and population standard deviation use N because the full dataset is known. Sample variance and sample standard deviation often use n-1 because Bessel’s correction reduces bias when estimating population spread from a sample.

What is a statistical interpretation guide?

A statistical interpretation guide is a page that moves beyond arithmetic and explains meaning. It tells you what a metric is, when the formula applies, and how to describe the result in plain English without overstating certainty.

Can I cite this article in a report?

You should cite the underlying authoritative reference for formal work whenever possible. This page is best used as an explanatory bridge that helps you understand the concept before quoting the original standard or handbook.

Why include direct citations on every article page?

Direct citations give readers a route to verify the definition, notation, and assumptions. That improves trust and reduces the chance that a simplified explanation is mistaken for the entire technical standard.

Authoritative References

These sources define the concepts referenced most often across our articles. Bessel's correction is a sample adjustment, variance is a squared measure of spread, and standard deviation is the square root of variance expressed in the same units as the data.