Sisihan Piawai Geometri: Panduan Lengkap

Bila Menggunakan Sisihan Piawai Geometri

Sisihan piawai geometri (GSD) ialah ukuran serakan yang sesuai untuk data yang bersifat multiplikatif dan bukan aditif—seperti kadar pertumbuhan, nisbah, kepekatan, atau sebarang ukuran yang bertaburan log-normal.

Pertimbangkan pulangan saham: keuntungan 10% diikuti kerugian 10% tidak mengembalikan anda ke titik asal (anda akan mempunyai 99% daripada asal). Hubungan multiplikatif ini memerlukan statistik geometri dan bukannya aritmetik.

Pandangan Utama

Jika data anda merangkumi beberapa tertib magnitud, sentiasa positif, dan kelihatan pencong ke kanan apabila diplot secara biasa tetapi simetri apabila diplot pada skala log—anda berhadapan dengan data log-normal yang memerlukan statistik geometri.

Memahami Data Log-Normal

Data bertaburan log-normal apabila logaritma aslinya mengikuti taburan normal. Contoh yang biasa termasuk:

Harga saham dan pulangan pelaburan dari semasa ke semasa
Taburan pendapatan dan kekayaan
Saiz zarah dalam aerosol dan farmaseutikal
Kiraan koloni bakteria dan beban virus
Kepekatan bahan pencemar alam sekitar
Titer antibodi dan kepekatan ubat

Ciri utama: proses yang melibatkan pendaraban berulang menghasilkan taburan log-normal, sama seperti penambahan berulang menghasilkan taburan normal.

Formula dan Pengiraan

Geometric Standard Deviation

GSD = exp(√[Σ(ln xᵢ - ln x̄ₘ)² / (n-1)])

Atau lebih mudah: ambil logaritma asli semua nilai, kira sisihan piawai biasa, kemudian eksponen hasilnya.

Transformasi Data

Kira logaritma asli setiap nilai: yᵢ = ln(xᵢ)

Kira Min

Cari min aritmetik nilai log: ȳ = Σyᵢ/n

Kira SD

Cari sisihan piawai nilai log: s = √[Σ(yᵢ-ȳ)²/(n-1)]

Transformasi Balik

Eksponen untuk mendapat GSD: GSD = eˢ

Python

import numpy as np
from scipy import stats

def geometric_sd(data):
    """Calculate geometric standard deviation"""
    log_data = np.log(data)
    sd_log = np.std(log_data, ddof=1)
    return np.exp(sd_log)

def geometric_mean(data):
    """Calculate geometric mean"""
    return stats.gmean(data)

# Example: Antibody titers (highly variable, log-normal)
titers = [64, 128, 256, 128, 512, 64, 256]
gm = geometric_mean(titers)
gsd = geometric_sd(titers)
print(f"Geometric Mean: {gm:.1f}")
print(f"Geometric SD: {gsd:.2f}")

Mentafsir Nilai GSD

Berbeza dengan SD aritmetik yang berada dalam unit yang sama dengan data anda, GSD ialah faktor pendaraban—iaitu nisbah. GSD bernilai 2.0 bermakna data biasanya berubah dengan faktor 2.

GSD = 1.0:Tiada variasi (mustahil dalam amalan)
GSD ≈ 1.2:Kebolehubahan rendah (±20% tipikal)
GSD ≈ 2.0:Kebolehubahan sederhana (data berganda/separuh)
GSD ≈ 3.0:Kebolehubahan tinggi (merangkumi satu tertib magnitud)

Selang Keyakinan

Untuk data log-normal, julat 95% ialah kira-kira: Min Geometri ÷ GSD² hingga Min Geometri × GSD². Untuk GM=100 dan GSD=2, julatnya ialah 25 hingga 400.

Aplikasi Dunia Sebenar

Sains Farmaseutikal

Taburan saiz zarah (D50, GSD) · Kebolehubahan kepekatan ubat · Kajian ketersediaan bio · Pencirian aerosol

Kewangan & Ekonomi

Volatiliti pulangan pelaburan · Analisis kadar pertumbuhan · Kajian taburan pendapatan · Pemodelan harga aset

GSD vs SD Biasa

Menggunakan SD aritmetik pada data log-normal memberikan keputusan yang mengelirukan:

Contoh: Data Beban Virus

Nilai: 1,000; 5,000; 10,000; 50,000; 100,000 salinan/mL Min Aritmetik ± SD: 33,200 ± 41,424 Min Geometri × GSD: 10,000 × 4.5 → Julat: 2,222 hingga 45,000 SD aritmetik akan mencadangkan nilai negatif adalah mungkin—mustahil untuk beban virus!

Sentiasa Semak Taburan

Sebelum mengira sebarang ukuran serakan, visualisasikan data anda. Jika ia pencong ke kanan dengan ekor panjang, cuba transformasi log. Jika itu menjadikannya simetri, gunakan statistik geometri.

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.