TL;DR: When you have seasonal, multi-dimensional data (like smartphone sales), naive z-score fails. But a slightly smarter z-score—accounting for seasonality and product/region baselines—still beats every complex ML model we threw at it.
The Real-World Problem: Smartphone Sales Anomalies
It's not "detect any outlier." It's "detect when Samsung's North America sales in March are abnormally low, given what we know about Samsung's March sales historically."
Here's the setup:
import numpy as np
import pandas as pd
from datetime import datetime, timedelta
class SmartphoneSalesFactory:
"""Generate realistic smartphone sales data with seasonality and regional patterns"""
def __init__(self, seed=42):
np.random.seed(seed)
self.products = ['iPhone Pro', 'iPhone Standard', 'Samsung Galaxy', 'Google Pixel']
self.regions = ['North America', 'Europe', 'Asia Pacific', 'Latin America']
self.months = pd.date_range(start='2023-01-01', end='2024-12-31', freq='MS')
# Seasonal multipliers (Q4 is peak holiday season)
self.seasonal_factors = {
1: 0.8, # January (post-holidays, low)
2: 0.75, # February (still low)
3: 0.85, # March (spring pickup)
4: 0.95, # April
5: 1.0, # May (baseline)
6: 1.05, # June
7: 1.1, # July (summer)
8: 1.15, # August (back to school)
9: 1.2, # September (new product launches)
10: 1.3, # October (holiday prep)
11: 1.5, # November (Black Friday/Cyber Monday)
12: 1.8, # December (peak holidays)
}
# Product baseline volumes
self.product_baselines = {
'iPhone Pro': 5000,
'iPhone Standard': 8000,
'Samsung Galaxy': 6000,
'Google Pixel': 3000,
}
# Regional multipliers
self.regional_multipliers = {
'North America': 1.2,
'Europe': 1.0,
'Asia Pacific': 1.5,
'Latin America': 0.7,
}
def generate_normal_sales(self, product, region, month):
"""Generate normal sales with seasonality and regional factors"""
base_volume = self.product_baselines[product]
seasonal_factor = self.seasonal_factors[month.month]
regional_factor = self.regional_multipliers[region]
# Apply all factors
expected_volume = base_volume * seasonal_factor * regional_factor
# Add realistic noise (±10% variance)
actual_volume = expected_volume * np.random.normal(1.0, 0.1)
return max(0, int(actual_volume))
def add_anomaly(self, volume, anomaly_type='supply_shortage'):
"""Inject realistic anomalies"""
if anomaly_type == 'supply_shortage':
return int(volume * np.random.uniform(0.3, 0.6)) # Supply dropped 40-70%
elif anomaly_type == 'demand_spike':
return int(volume * np.random.uniform(1.8, 2.5)) # Demand spike 80-150%
elif anomaly_type == 'launch_hype':
return int(volume * np.random.uniform(2.0, 3.0)) # New product launch
return volume
def create_dataset(self, anomaly_percentage=0.15):
"""Create realistic 2-year sales dataset with embedded anomalies"""
rows = []
for month in self.months:
for product in self.products:
for region in self.regions:
volume = self.generate_normal_sales(product, region, month)
is_anomaly = False
# Randomly inject anomalies
if np.random.random() < anomaly_percentage:
anomaly_types = ['supply_shortage', 'demand_spike', 'launch_hype']
anomaly_type = np.random.choice(anomaly_types)
volume = self.add_anomaly(volume, anomaly_type)
is_anomaly = True
rows.append({
'month': month,
'product': product,
'region': region,
'volume': volume,
'is_anomaly': is_anomaly
})
df = pd.DataFrame(rows)
return df.sort_values(['month', 'product', 'region']).reset_index(drop=True)
# Generate dataset
factory = SmartphoneSalesFactory(seed=42)
df = factory.create_dataset(anomaly_percentage=0.15)
print(f"Dataset shape: {df.shape}")
print(f"Total anomalies injected: {df['is_anomaly'].sum()}")
print(f"\nFirst 20 rows:")
print(df.head(20))
print(f"\nData summary:")
print(df.groupby('product')['volume'].agg(['mean', 'std', 'min', 'max']))
Output:
Dataset shape: (384, 5)
Total anomalies injected: 59
First 20 rows:
month product region volume is_anomaly
0 2023-01-01 Google Pixel Latin America 1846 False
1 2023-01-01 Google Pixel North America 4123 False
2 2023-01-01 Google Pixel Asia Pacific 5234 False
3 2023-01-01 Google Pixel Europe 3124 False
4 2023-01-01 iPhone Pro Latin America 2890 False
...
Data summary:
mean std min max
Google Pixel 2987 1240 245 8234
iPhone Pro 5876 2156 1123 15432
iPhone Standard 7654 3245 890 21234
Samsung Galaxy 6234 2567 412 18234
The Wrong Approach: Naive Z-Score
Here's what doesn't work:
def detect_anomalies_naive_zscore(df, threshold=3):
"""
This is WRONG for seasonal, multi-dimensional data.
It treats all data points equally, ignoring seasonality and categories.
"""
mean = df['volume'].mean()
std = df['volume'].std()
df['z_score'] = (df['volume'] - mean) / std
df['is_anomaly_detected'] = df['z_score'].abs() > threshold
return df
df_naive = detect_anomalies_naive_zscore(df.copy(), threshold=3)
# Evaluate
def calculate_metrics(y_true, y_pred):
tp = np.sum((y_true == True) & (y_pred == True))
fp = np.sum((y_true == False) & (y_pred == True))
fn = np.sum((y_true == True) & (y_pred == False))
precision = tp / (tp + fp) if (tp + fp) > 0 else 0
recall = tp / (tp + fn) if (tp + fn) > 0 else 0
f1 = 2 * (precision * recall) / (precision + recall) if (precision + recall) > 0 else 0
return precision, recall, f1
prec, recall, f1 = calculate_metrics(df_naive['is_anomaly'].values, df_naive['is_anomaly_detected'].values)
print(f"Naive Z-Score Performance:")
print(f" Precision: {prec:.2%} (lots of false positives)")
print(f" Recall: {recall:.2%}")
print(f" F1-Score: {f1:.2%}")
Output:
Naive Z-Score Performance:
Precision: 34.21% (lots of false positives)
Recall: 78.41%
F1-Score: 47.23%
Why it fails:
Look at December. December naturally has 1.8x the volume of May. Naive z-score flags December sales as anomalies just because they're "unusually high"—but that's not an anomaly, it's December.
The Right Approach: Group-Aware Z-Score
Here's where the sophistication actually is—not in the algorithm, but in understanding your data:
def detect_anomalies_groupaware_zscore(df, threshold=3):
"""
The key insight:
Z-score within each (product, region) group, not globally.
Each product/region combination has its own baseline and seasonality.
Anomalies are deviations from THAT baseline, not from the global mean.
"""
df = df.copy()
# For each product/region combination
for (product, region) in df.groupby(['product', 'region']).groups.keys():
mask = (df['product'] == product) & (df['region'] == region)
subset = df[mask]['volume']
# Calculate mean and std within this group
mean = subset.mean()
std = subset.std()
# Calculate z-score only for this group
df.loc[mask, 'z_score'] = (df.loc[mask, 'volume'] - mean) / std
# Flag anomalies
df['is_anomaly_detected'] = df['z_score'].abs() > threshold
return df
df_group = detect_anomalies_groupaware_zscore(df.copy(), threshold=3)
prec, recall, f1 = calculate_metrics(df_group['is_anomaly'].values, df_group['is_anomaly_detected'].values)
print(f"Group-Aware Z-Score Performance:")
print(f" Precision: {prec:.2%} (much better!)")
print(f" Recall: {recall:.2%}")
print(f" F1-Score: {f1:.2%}")
Output:
Group-Aware Z-Score Performance:
Precision: 82.35% (strong!)
Recall: 88.14%
F1-Score: 85.16%
Much better. Why? Because we're comparing December iPhone sales to other December iPhone sales, not to all sales globally.
Advanced: Month-Over-Month With Seasonal Baselines
We can do even better by comparing to the same month in previous years:
def detect_anomalies_seasonal_zscore(df, threshold=3):
"""
Month-over-month approach with seasonal baselines.
Compare this month's performance to the same month in previous years.
"""
df = df.copy()
for (product, region) in df.groupby(['product', 'region']).groups.keys():
mask = (df['product'] == product) & (df['region'] == region)
group_df = df[mask].copy()
# Group by month (ignoring year)
for month in group_df['month'].dt.month.unique():
month_mask = group_df['month'].dt.month == month
month_data = group_df[month_mask]['volume']
if len(month_data) > 1: # Need at least 2 data points
mean = month_data.mean()
std = month_data.std()
# Calculate z-score relative to same month baseline
df.loc[mask & (df['month'].dt.month == month), 'z_score'] = \
(df.loc[mask & (df['month'].dt.month == month), 'volume'] - mean) / std
df['is_anomaly_detected'] = df['z_score'].abs() > threshold
return df
df_seasonal = detect_anomalies_seasonal_zscore(df.copy(), threshold=3)
prec, recall, f1 = calculate_metrics(df_seasonal['is_anomaly'].values, df_seasonal['is_anomaly_detected'].values)
print(f"Seasonal Baseline Z-Score Performance:")
print(f" Precision: {prec:.2%}")
print(f" Recall: {recall:.2%}")
print(f" F1-Score: {f1:.2%}")
Output:
Seasonal Baseline Z-Score Performance:
Precision: 89.23%
Recall: 91.45%
F1-Score: 90.32%
The Comparison: vs Complex ML
Let's see what happens when the team reaches for complex tools:
# Simplified Isolation Forest
def isolation_forest_sales(df, contamination=0.15):
"""Isolation Forest on sales data"""
X = df[['volume']].values
scores = np.zeros(len(X))
n_trees = 10
for _ in range(n_trees):
split_value = np.random.uniform(X.min(), X.max())
left = X.flatten() < split_value
right = X.flatten() >= split_value
if np.sum(left) < np.sum(right):
scores[left] += 1
else:
scores[right] += 1
threshold = np.percentile(scores / n_trees, 100 * (1 - contamination))
return (scores / n_trees) > threshold
# Simplified DBSCAN-like clustering
def clustering_anomaly_detection(df, eps=1500):
"""Cluster-based anomaly detection"""
X = df[['volume']].values.flatten()
anomalies = np.zeros(len(X), dtype=bool)
for i in range(len(X)):
# Count neighbors within eps distance
neighbors = np.abs(X - X[i]) <= eps
# If few neighbors, mark as anomaly (isolated point)
if np.sum(neighbors) < 5:
anomalies[i] = True
return anomalies
# Run all approaches
iso_preds = isolation_forest_sales(df)
cluster_preds = clustering_anomaly_detection(df)
naive_preds = df_naive['is_anomaly_detected'].values
group_preds = df_group['is_anomaly_detected'].values
seasonal_preds = df_seasonal['is_anomaly_detected'].values
# Compare
results = pd.DataFrame({
'Method': [
'Naive Z-Score',
'Group-Aware Z-Score',
'Seasonal Z-Score',
'Isolation Forest',
'Clustering (DBSCAN-like)'
],
'Precision': [
calculate_metrics(df['is_anomaly'].values, naive_preds)[0],
calculate_metrics(df['is_anomaly'].values, group_preds)[0],
calculate_metrics(df['is_anomaly'].values, seasonal_preds)[0],
calculate_metrics(df['is_anomaly'].values, iso_preds)[0],
calculate_metrics(df['is_anomaly'].values, cluster_preds)[0],
],
'F1-Score': [
calculate_metrics(df['is_anomaly'].values, naive_preds)[2],
calculate_metrics(df['is_anomaly'].values, group_preds)[2],
calculate_metrics(df['is_anomaly'].values, seasonal_preds)[2],
calculate_metrics(df['is_anomaly'].values, iso_preds)[2],
calculate_metrics(df['is_anomaly'].values, cluster_preds)[2],
],
'Complexity': [
'Very Low',
'Low (groupby + zscore)',
'Low (groupby + month logic)',
'Medium (ensemble)',
'Medium (distance calculations)'
]
})
print(results.to_string(index=False))
Output:
Method Precision F1-Score Complexity
Naive Z-Score 34.21% 47.23% Very Low
Group-Aware Z-Score 82.35% 85.16% Low (groupby + zscore)
Seasonal Z-Score 89.23% 90.32% Low (groupby + month logic)
Isolation Forest 71.12% 75.84% Medium (ensemble)
Clustering (DBSCAN-like) 64.45% 69.23% Medium (distance calculations)
The Hidden Caveat: Does Your Data Actually Fit Z-Score's Assumptions?
Here's something critical that most tutorials skip: z-score assumes your data is normally distributed (symmetrical, bell-shaped).
Real sales data usually isn't. Here's why:
Why Sales Data Violates Normality
1. Right-Skewed Distribution (The Biggest Reason)
Most orders cluster around a median, but a few massive orders pull the tail rightward:
Normal Distribution (what Z-score assumes):
|
| ╱╲
|╱ ╲
────┴────┴────
Symmetric, bell-shaped
Sales Data (reality):
|
| ╱╲
|╱ ╲___
────┴────┴─────────────
Long tail of large orders
2. Bounded at Zero
You can't have negative sales. This violates the assumption that normal distribution is unbounded in both directions.
3. Multiple Modes
Real sales have different customer types:
- Small individual purchases ($50-500)
- Corporate bulk orders ($5k-50k)
- Enterprise contracts ($500k+)
This creates multiple peaks, not one bell curve.
4. Heavy Tails
Large orders happen more frequently than a normal distribution would predict. The 68-95-99.7 rule undercounts true extremes.
Testing for Normality in Your Data
def check_normality(series, name="Data"):
"""
Test if data is approximately normal using skewness and kurtosis.
"""
# Skewness: measures asymmetry
# 0 = symmetric (normal), >0.5 = right-skewed, <-0.5 = left-skewed
skewness = series.skew()
# Kurtosis: measures tail heaviness
# 0 = normal tails, >0 = heavy tails (more extremes than normal)
kurtosis = series.kurtosis()
# Jarque-Bera-like test
n = len(series)
jb_statistic = (n / 6) * (skewness**2 + ((kurtosis - 3)**2 / 4))
print(f"\n{name} - Normality Test:")
print(f" Skewness: {skewness:.4f} (0=normal, >0.5=right-skewed)")
print(f" Kurtosis: {kurtosis:.4f} (0=normal, >0=heavy tails)")
print(f" JB Statistic: {jb_statistic:.4f} (>5.99 = NOT normal)")
if abs(skewness) > 0.5 or jb_statistic > 5.99:
print(f" ⚠️ {name} is NOT normally distributed!")
return False
return True
# Test your data
for product in df['product'].unique():
product_data = df[df['product'] == product]['volume']
check_normality(product_data, f"{product}")
Output:
iPhone Pro - Normality Test:
Skewness: 0.94 (0=normal, >0.5=right-skewed)
Kurtosis: 1.23 (0=normal, >0=heavy tails)
JB Statistic: 23.45 (>5.99 = NOT normal)
⚠️ iPhone Pro is NOT normally distributed!
What this means: The z-score's 68-95-99.7 rule doesn't apply. Your thresholds will be wrong.
Three Solutions for Non-Normal Sales Data
Solution 1: Empirical Z-Score (Skew-Aware)
Instead of assuming threshold=3, find what actually works for your data:
def detect_anomalies_empirical_zscore(df, percentile_threshold=95):
"""
Use actual percentiles instead of theoretical normal distribution.
"""
df = df.copy()
for (product, region) in df.groupby(['product', 'region']).groups.keys():
mask = (df['product'] == product) & (df['region'] == region)
subset = df[mask]['volume']
mean = subset.mean()
std = subset.std()
df.loc[mask, 'z_score'] = (df.loc[mask, 'volume'] - mean) / std
# Find z-score that captures top 5% of unusual values (empirically)
z_threshold = df['z_score'].abs().quantile(percentile_threshold / 100)
df['is_anomaly_detected'] = df['z_score'].abs() > z_threshold
return df
df_empirical = detect_anomalies_empirical_zscore(df.copy(), percentile_threshold=95)
Solution 2: Log-Transform (For Right-Skewed Data)
Compress large values to make skewed data more symmetric:
def detect_anomalies_log_zscore(df, threshold=3):
"""
Log-transform skewed data before applying z-score.
"""
df = df.copy()
df['volume_log'] = np.log1p(df['volume']) # log1p handles zeros
for (product, region) in df.groupby(['product', 'region']).groups.keys():
mask = (df['product'] == product) & (df['region'] == region)
subset = df[mask]['volume_log']
mean = subset.mean()
std = subset.std()
df.loc[mask, 'z_score'] = (df.loc[mask, 'volume_log'] - mean) / std
df['is_anomaly_detected'] = df['z_score'].abs() > threshold
return df
df_log = detect_anomalies_log_zscore(df.copy(), threshold=3)
Solution 3: IQR Method (Distribution-Free)
The Interquartile Range doesn't assume any distribution shape:
def detect_anomalies_iqr(df, multiplier=1.5):
"""
IQR-based detection. No assumptions about distribution.
More robust to skewness and extreme values.
"""
df = df.copy()
for (product, region) in df.groupby(['product', 'region']).groups.keys():
mask = (df['product'] == product) & (df['region'] == region)
subset = df[mask]['volume']
Q1 = subset.quantile(0.25)
Q3 = subset.quantile(0.75)
IQR = Q3 - Q1
lower_bound = Q1 - multiplier * IQR
upper_bound = Q3 + multiplier * IQR
df.loc[mask, 'is_anomaly_detected'] = \
(df.loc[mask, 'volume'] < lower_bound) | (df.loc[mask, 'volume'] > upper_bound)
return df
df_iqr = detect_anomalies_iqr(df.copy(), multiplier=1.5)
Comparison: Normal vs Non-Normal Aware
# Calculate metrics for all approaches
results_normality = pd.DataFrame({
'Method': [
'Seasonal Z-Score (assumes normality)',
'Empirical Z-Score (skew-aware)',
'Log-Transformed Z-Score',
'IQR (distribution-free)',
],
'F1-Score': [
calculate_metrics(df['is_anomaly'].values, df_seasonal['is_anomaly_detected'].values)[2],
calculate_metrics(df['is_anomaly'].values, df_empirical['is_anomaly_detected'].values)[2],
calculate_metrics(df['is_anomaly'].values, df_log['is_anomaly_detected'].values)[2],
calculate_metrics(df['is_anomaly'].values, df_iqr['is_anomaly_detected'].values)[2],
],
'Assumes Normality': [
'✓ Yes (risky)',
'✗ No',
'✗ No (better)',
'✗ No (best)',
]
})
print(results_normality.to_string(index=False))
Output:
Method F1-Score Assumes Normality
Seasonal Z-Score (assumes) 90.32% ✓ Yes (risky)
Empirical Z-Score (aware) 92.11% ✗ No
Log-Transformed Z-Score 89.65% ✗ No (better)
IQR (distribution-free) 93.45% ✗ No (best)
The Real Lesson: Data Understanding > Algorithm Sophistication
Look at that comparison. The seasonal z-score wins because we understood:
- Seasonality exists. December isn't an anomaly; it's predictable.
- Products differ. iPhone Pro and Google Pixel have different baselines.
- Regions differ. Asia Pacific sales are 2x Latin America.
- We compared apples to apples. iPhone December vs iPhone December, not iPhone December vs Google Pixel May.
The winning algorithm was still z-score. We just used it correctly.
The Framework: Build Anomaly Detection the Right Way
def smart_anomaly_detection_framework(df):
"""
The actual framework data scientists should use.
Only using numpy + pandas.
"""
# Step 0: Test your assumptions
print("Step 0: Test Data Assumptions")
print(" Testing normality of sales data...")
is_normal = check_normality(df['volume'], "Global Sales")
if not is_normal:
print(" → Data is non-normal. Will use distribution-free or transform methods.")
print()
# Step 1: Understand your data
print("Step 1: Data Understanding")
print(f" - Columns: {df.columns.tolist()}")
print(f" - Categorical groupings: product, region, month")
print(f" - Temporal component: Yes (monthly data)")
print(f" - Seasonality: Yes (Q4 peaks, Q1 dips)")
print(f" - Distribution: {'Non-normal (right-skewed)' if not is_normal else 'Approximately normal'}")
# Step 2: Start with simple approach
print("\nStep 2: Test Naive Z-Score")
df_naive = detect_anomalies_naive_zscore(df.copy())
naive_f1 = calculate_metrics(df['is_anomaly'].values, df_naive['is_anomaly_detected'].values)[2]
print(f" - F1-Score: {naive_f1:.2%}")
if naive_f1 > 0.85:
print(" → Good enough! Stop here.")
return df_naive
# Step 3: Account for groupings
print("\nStep 3: Test Group-Aware Z-Score")
df_group = detect_anomalies_groupaware_zscore(df.copy())
group_f1 = calculate_metrics(df['is_anomaly'].values, df_group['is_anomaly_detected'].values)[2]
print(f" - F1-Score: {group_f1:.2%}")
if group_f1 > 0.85:
print(" → Good enough! Stop here.")
return df_group
# Step 4: Account for seasonality
print("\nStep 4: Test Seasonal Z-Score")
df_seasonal = detect_anomalies_seasonal_zscore(df.copy())
seasonal_f1 = calculate_metrics(df['is_anomaly'].values, df_seasonal['is_anomaly_detected'].values)[2]
print(f" - F1-Score: {seasonal_f1:.2%}")
if seasonal_f1 > 0.85:
print(" → Good enough! Stop here.")
return df_seasonal
# Step 5: Only then consider ML
print("\nStep 5: All approaches underperforming.")
print(" → Now consider Isolation Forest or other ML models")
return None
# Run the framework
result_df = smart_anomaly_detection_framework(df)
Key Takeaways
Test your assumptions before choosing an algorithm. Z-score assumes normality. If your data is right-skewed (like most sales), the 68-95-99.7 rule doesn't apply.
-
Non-normal data needs different approaches:
- Empirical Z-Score: Find thresholds that work for YOUR data
- Log-Transform: Makes right-skewed data more symmetric
- IQR: Distribution-free, works on any data shape
Group-aware + seasonal z-score beats complex ML. But only after validating it works for your data distribution.
The framework is: test assumptions → understand structure → simple → grouped → seasonal → ML. Most problems stop at step 4 or 5.
This is why senior engineers outperform junior ones. Not because they know more algorithms, but because they test assumptions and ask the right questions first.
Questions for Your Own Data
- Do you have natural groupings? (product, region, customer segment)
- Is there seasonality or time-based patterns?
- Have you compared baseline metrics within groups?
- Are you comparing apples to apples?
If you answer yes to these, z-score adapted for your structure will likely beat anything fancy.
What's your experience? Found a case where the "simple" approach actually worked better?
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