正态分布特征描述

概览与结论
- 日新增用户分布近似正态（Shapiro-Wilk p=0.12，不拒绝正态性假设），中心在340，波动幅度由标准差28刻画。
- 形状上几乎对称（偏度≈0.05），尾部略微平缓（峰度≈-0.1，相对正态略“平顶”），极端值较正态略少见。
- 9/18的420属高值但仍在3σ范围内，更可能是活动驱动的外生波动，而非基础分布的常态。

正态分布的关键特征（结合当前数据）
- 中心位置：均值≈340（在正态分布下，均值≈中位数≈众数）。
- 离散程度：标准差≈28（方差=784），决定日波动的典型幅度。
- 形状与尾部：近似钟形、对称；偏度≈0表示左右对称；峰度≈0表示与正态相当的尾部厚度。当前偏度≈0.05、峰度≈-0.1，说明轻微对称、尾部稍“薄”。
- 经验法则（68–95–99.7）：在正态假设下
  - 约68%的天数在[312, 368]（±1σ）；
  - 约95%在[284, 396]（±2σ）；
  - 约99.7%在[256, 424]（±3σ）。
- 典型百分位（近似）：第25/75百分位约为[321, 359]（340 ± 0.674σ）。

异常点说明
- 9/18=420的Z分数≈(420−340)/28≈2.86，单侧尾概率约0.21%。在无活动条件下属于低概率事件，但仍在3σ区间内；考虑到活动影响，更应归类为可解释的事件型异常。

业务含义与建议
- 基线波动：大多数天的新增将集中在340±28的范围，超出396（+2σ）的高值应重点关注是否有营销、渠道或产品事件。
- 监测与告警：
  - 建议使用Z分数或均值±3σ作为稳定告警线；对活动窗口采用单独基线或加入事件变量，避免误报。
  - 补充稳健指标（如中位数、IQR）用于周报对比，减少单日极值对趋势解读的影响。
- 统计推断：样本量n=1200使均值估计稳定，均值的95%置信区间约[338.4, 341.6]；可基于正态假设开展预测与容量规划。

结论与要点
- 次日留存率在当前样本下可视为近似正态（正态性检验p=0.20），可安全使用基于正态分布的区间估计与显著性检验。
- 总体估计：均值42%，标准差3.5%，n=800。标准误约0.124个百分点，95%置信区间为[41.76%，42.24%]，估计精度高。
- A/B 对比：A=41.8%，B=43.1%，差异1.3个百分点；若两组样本量相等且合并标准差为3.6%，差异的标准误约0.255个百分点，Z≈5.11，p<10^-6，差异在统计上显著；差异的95%置信区间约为[0.8，1.8]个百分点。标准化效应量d≈0.36（小到中等）。
- 样本量与检验力（基于σ=3.6%，差异δ=1.3个百分点）：达到80%/90%检验力所需的每组样本量约为120/162；若当前为均分800（每组≈400），检验力>99%。请确认各组实际样本量以最终定论。

正态分布的关键特征（结合本指标解读）
- 由两个参数完全刻画：均值μ与标准差σ。对称、钟形，均值=中位数=众数。对次日留存这种比例型指标，当样本量足够大且p不极端（此处p≈0.42），样本均值在中心极限定理下近似正态。
- 标准化与区间：任意观测可用Z=(X−μ)/σ转为标准正态。对样本均值，需使用标准误SE=σ/√n。当前SE≈0.124个百分点，意味着均值估计的随机波动很小。
- 经验法则（68-95-99.7）：X~N(μ,σ^2)时，约68%/95%/99.7%的值落在μ±1σ/±2σ/±3σ内。以μ=42%，σ=3.5%计，μ±3σ≈[31.5%，52.5%]，仍在0–100%边界内，正态近似合理。
- 线性与可加性：独立正态的线性组合仍为正态。对比A/B的均值差（在大样本下）也近似正态，便于用Z检验/置信区间评估差异。
- 推断友好性：正态假设允许使用z/t检验、置信区间、效应量与功效分析，便于运营评估中的快速决策。

A/B 评估要点与计算
- 假设与方法：检验H0: μA=μB。已知各组方差近似相等（合并σ≈3.6%），可用两独立样本z/t检验。差异的标准误SE(Δ)=σp√(1/nA+1/nB)。
- 在均分nA=nB=400的示例下：SE(Δ)≈0.036×√(1/400+1/400)≈0.255个百分点；Z=1.3/0.255≈5.11，p<10^-6；95% CI≈[0.8，1.8]个百分点；Cohen’s d=0.013/0.036≈0.36。
- 样本量指引（两侧α=0.05）：为检测δ=1.3个百分点，σ=3.6%，
  - 80%检验力：每组约120
  - 90%检验力：每组约162
- 注：如nA≠nB或方差不齐，应改用Welch检验；如存在多轮查看或多指标并行，对显著性需做校正或使用序贯方法。

注意事项
- 指标边界与分布：留存率∈[0,1]。在当前p与n下，用正态近似评估均值与均值差是恰当的；若p接近0或1或样本很小，应改用二项/比例检验或广义线性模型（如logit）。
- 稳健性：关注异常值与方差同质性；必要时进行Levene/Brown-Forsythe检验。
- 行动建议：确认A/B各组样本量并复算SE与p值；报告差异的置信区间与效应量；结合业务阈值评估1.3个百分点的实际价值，并规划后续发布或扩量验证。

Objective
Describe the characteristics of a normal distribution and apply them to your marketing metrics: conversion rate (CR) and average order value (AOV), with validity checks and actionable implications.

Key characteristics of a normal distribution
- Shape and symmetry: Bell-shaped, perfectly symmetric around the mean; mean = median = mode.
- Defined by two parameters: Mean (μ) sets the center; standard deviation (σ) sets the spread. Variance is σ².
- Empirical rule (68–95–99.7): 
  - About 68% of values lie within ±1σ of μ.
  - About 95% within ±2σ.
  - About 99.7% within ±3σ.
- Z-scores: Standardizing (z = (x − μ) / σ) allows probability calculations and anomaly detection.
- Additivity/approximation: Sums/averages of many independent effects tend toward normality (central limit theorem), making normal a practical model for aggregated marketing metrics.
- Tails: The tails are thin; extreme values are rare but possible and quantifiable.

Application to your metrics
Normality checks
- CR normality test p = 0.09; AOV p = 0.15. Both > 0.05 → fail to reject normality; “approximately normal” is reasonable for these data.
- Note: The provided CR variance (0.0016) is inconsistent with σ = 0.4% if σ is in proportion units (0.004), which implies variance 0.000016. I’ll use σ for interval and probability estimates.

Conversion rate (μ = 3.2%, σ = 0.4%, n = 3000)
- One-sigma range (≈68% of observations): 2.8% to 3.6%.
- Two-sigma range (≈95%): 2.4% to 4.0%.
- Three-sigma range (≈99.7%): 2.0% to 4.4%.
- Example probabilities:
  - P(CR > 3.9%): z = (3.9 − 3.2) / 0.4 = 1.75 → ≈ 4.0%.
  - P(CR < 2.5%): same z magnitude → ≈ 4.0%.
- Precision of the mean (95% CI for μ): SE = 0.4% / √3000 ≈ 0.0073%; CI ≈ 3.2% ± 0.014% → [3.186%, 3.214%]. The mean estimate is very stable.

Average order value (μ = 86, σ = 12, n = 3000)
- One-sigma range (≈68%): 74 to 98.
- Two-sigma range (≈95%): 62 to 110.
- Three-sigma range (≈99.7%): 50 to 122.
- Precision of the mean (95% CI for μ): SE = 12 / √3000 ≈ 0.219; CI ≈ 86 ± 0.429 → [85.571, 86.429].

Anomaly assessment: Thursday AOV = 125
- z = (125 − 86) / 12 = 3.25.
- One-sided tail probability ≈ 0.0006; two-sided ≈ 0.0012. This exceeds the 3σ upper bound (~122), making it a statistically rare event under normality.
- Business implications: Treat as an outlier likely driven by a discrete factor (promotion, campaign targeting, mix shift, data error). Investigate and document; consider robust handling (e.g., winsorize or analyze with/without the point).

Practical notes for marketing analysis
- CR is bounded between 0 and 1 and often arises from binomial processes; at μ = 3.2% with modest σ and large n, the normal approximation is acceptable, but beta/binomial modeling can be considered for finer inference.
- Normal modeling supports:
  - Setting alert thresholds (e.g., 2σ bands for monitoring).
  - Quantifying the rarity of spikes/drops.
  - Building confidence intervals for KPIs to separate signal from noise.

Conclusions
- Both CR and AOV are approximately normal in your sample, enabling reliable use of z-scores, sigma bands, and confidence intervals.
- Expected operational ranges:
  - CR: 2.4%–4.0% (95% band); mean tightly estimated at ~3.2%.
  - AOV: 62–110 (95% band).
- The Thursday AOV = 125 is a statistically rare outlier (>3σ); investigate cause and treat carefully in reporting and optimization.