In this article, we\u2019ll break down the main regression metrics in simple, relatable terms. We\u2019ll cover simplified formulas, real-world examples, and when to use (or avoid) each metric. You\u2019ll also learn how to combine metrics for a clearer picture of your model\u2019s performance.<\/p>\n\n\n\n
\n\n\n\n
Regression Metrics: Which One Should You Use?<\/strong><\/h3>\n\n\n\nEach metric measures a specific aspect of your model\u2019s error. The right choice depends on your data type and project goals. For instance, if your dataset has extreme values (outliers), you\u2019ll need a specific metric. Let\u2019s dive in:<\/p>\n\n\n\n
\n\n\n\n1. Mean Squared Error (MSE) \u2013 Heavy Penalty for Large Errors<\/strong><\/h4>\n\n\n\nWhat is it?<\/strong>
MSE calculates the average of the squared errors. Large errors are punished more harshly. For example, a prediction error of 50 million would be squared to 2500 billion (!) in MSE.<\/p>\n\n\n\nSimplified Formula:<\/strong>
<\/span> <\/span>![\"\[MSE = \frac{(Error_1)^2 + (Error_2)^2 + … + (Error_n)^2}{Number\ of\ Samples}\]\"](\"https:\/\/amirhooshang.com\/blog\/wp-content\/ql-cache\/quicklatex.com-f9f93a0159217adf6b84a2719318053a_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\nExample:<\/strong><\/p>\n\n\n\n\n- Model Prediction: 190 million \u2192 Actual Price: 200 million \u2192 Error: 10 million<\/li>\n\n\n\n
- Model Prediction: 250 million \u2192 Actual Price: 300 million \u2192 Error: 50 million<\/li>\n\n\n\n
- Model Prediction: 140 million \u2192 Actual Price: 150 million \u2192 Error: 10 million<\/li>\n<\/ul>\n\n\n\n
<\/span> <\/span>![\"\[MSE = \frac{10^2 + 50^2 + 10^2}{3} = \frac{100 + 2500 + 100}{3} = 900\]\"](\"https:\/\/amirhooshang.com\/blog\/wp-content\/ql-cache\/quicklatex.com-976c2144ce6e654b5a517cfe8335f9e6_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\nPros & Cons:<\/strong><\/p>\n\n\n\n\n- Pros:<\/strong> Prioritizes large errors \u2192 Ideal for critical predictions (e.g., life-saving drug prices).<\/li>\n\n\n\n
- Cons:<\/strong> Sensitive to outliers \u2192 Misleading if your data has extreme values.<\/li>\n\n\n\n
- Cons:<\/strong> Unit is squared (e.g., “Toman\u00b2”) \u2192 Hard to interpret.<\/li>\n<\/ul>\n\n\n\n
Comparison with MAE:<\/strong><\/p>\n\n\n\n\n- Use MAE if your data has many outliers, as it treats all errors equally.<\/li>\n<\/ul>\n\n\n\n
\n\n\n\n2. Mean Absolute Error (MAE) \u2013 Fairly Measures All Errors<\/strong><\/h4>\n\n\n\nWhat is it?<\/strong>
MAE calculates the average of absolute errors without squaring them. It treats small and large errors equally.<\/p>\n\n\n\nSimplified Formula:<\/strong>
<\/span> <\/span>![\"\[MAE = \frac{|Error_1| + |Error_2| + … + |Error_n|}{Number\ of\ Samples}\]\"](\"https:\/\/amirhooshang.com\/blog\/wp-content\/ql-cache\/quicklatex.com-be750a1df4cc9c657e06327c2bea2b01_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\nExample (Same Data as Above):<\/strong>
<\/span> <\/span>![\"\[MAE = \frac{10 + 50 + 10}{3} ≈ 23.33\]\"](\"https:\/\/amirhooshang.com\/blog\/wp-content\/ql-cache\/quicklatex.com-8184aa9cd0728524c528054c4b6b6df5_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\nPros & Cons:<\/strong><\/p>\n\n\n\n\n- Pros:<\/strong> Robust to outliers \u2192 Great for noisy data.<\/li>\n\n\n\n
- Cons:<\/strong> Can\u2019t distinguish between small and large errors \u2192 A 50M error is treated the same as a 10M error.<\/li>\n\n\n\n
- Cons:<\/strong> Less popular in optimization (e.g., Gradient Descent) due to non-differentiable nature.<\/li>\n<\/ul>\n\n\n\n
Comparison with MSE:<\/strong><\/p>\n\n\n\n\n- Use MSE if large errors are critical (e.g., disaster prediction).<\/li>\n\n\n\n
- Use MAE for datasets with outliers or to avoid exaggerating errors.<\/li>\n<\/ul>\n\n\n\n
\n\n\n\n3. Root Mean Squared Error (RMSE) \u2013 MSE in Understandable Units<\/strong><\/h4>\n\n\n\nWhat is it?<\/strong>
RMSE is the square root of MSE. It aligns the error unit with the original data for easier interpretation.<\/p>\n\n\n\nSimplified Formula:<\/strong>
<\/span> <\/span>![\"\[RMSE = \sqrt{MSE}\]\"](\"https:\/\/amirhooshang.com\/blog\/wp-content\/ql-cache\/quicklatex.com-353181378906f00e865e5459ed0ea408_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\nExample (Using MSE Above):<\/strong>
<\/span> <\/span>![\"\[RMSE = \sqrt{900} = 30\]\"](\"https:\/\/amirhooshang.com\/blog\/wp-content\/ql-cache\/quicklatex.com-5b582e5a8f726d0c2b0b9740b777a635_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\nThis means the average prediction error is 30 million.<\/p>\n\n\n\n
Pros & Cons:<\/strong><\/p>\n\n\n\n\n- Pros:<\/strong> Easy to explain \u2192 Non-technical stakeholders get it.<\/li>\n\n\n\n
- Cons:<\/strong> Still sensitive to outliers.<\/li>\n<\/ul>\n\n\n\n
Use Case:<\/strong><\/p>\n\n\n\n\n- Business reports where clarity matters.<\/li>\n\n\n\n
- Problems with normally distributed errors.<\/li>\n<\/ul>\n\n\n\n
\n\n\n\n4. R-Squared (R\u00b2) \u2013 How Much Better Is Your Model Than a Simple Baseline?<\/strong><\/h4>\n\n\n\nWhat is it?<\/strong>
R\u00b2 measures how well your model performs compared to a simple baseline (e.g., predicting the mean). Its value ranges from 0 (worst) to 1 (best).<\/p>\n\n\n\nSimplified Formula:<\/strong>
<\/span> <\/span>![\"\[R^2 = 1 - \frac{Sum\ of\ Squared\ Errors\ (Your\ Model)}{Sum\ of\ Squared\ Errors\ (Baseline\ Model)}\]\"](\"https:\/\/amirhooshang.com\/blog\/wp-content\/ql-cache\/quicklatex.com-260dffe573ab6a2fa0dafa3fc7f09335_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\nExample:<\/strong><\/p>\n\n\n\n\n- Baseline model error: 2500<\/li>\n\n\n\n
- Your model error: 500
<\/span> <\/span>![\"\[R^2 = 1 - \frac{500}{2500} = 0.8\]\"](\"https:\/\/amirhooshang.com\/blog\/wp-content\/ql-cache\/quicklatex.com-08200bf673f2ab1d72b238a27090a74e_l3.png\" "\\"Rendered")
<\/p>
This means your model explains 80% of the data variance.<\/li>\n<\/ul>\n\n\n\nPros & Cons:<\/strong><\/p>\n\n\n\n\n- Pros:<\/strong> Easy to compare models.<\/li>\n\n\n\n
- Cons:<\/strong> High R\u00b2 can be misleading if the model performs poorly on new data.<\/li>\n\n\n\n
- Cons:<\/strong> Doesn\u2019t indicate bias.<\/li>\n<\/ul>\n\n\n\n
Use Case:<\/strong><\/p>\n\n\n\n\n- Initial model evaluation to gauge overall performance.<\/li>\n<\/ul>\n\n\n\n
\n\n\n\n5. Mean Absolute Percentage Error (MAPE) \u2013 Error in Percentage Terms<\/strong><\/h4>\n\n\n\nWhat is it?<\/strong>
MAPE expresses error as a percentage, useful for data with varying scales.<\/p>\n\n\n\nSimplified Formula:<\/strong>
<\/span> <\/span>![\"\[MAPE = \frac{100\%}{n} \times \sum \left| \frac{Error}{Actual\ Value} \right|\]\"](\"https:\/\/amirhooshang.com\/blog\/wp-content\/ql-cache\/quicklatex.com-7322b393705278c57ed587cb8076ecea_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\nExample:<\/strong><\/p>\n\n\n\n\n- Actual Value: 200 million \u2192 Prediction: 180 million \u2192 Error: 20 million
<\/span> <\/span>![\"\[MAPE = \left| \frac{20}{200} \right| \times 100\% = 10\%\]\"](\"https:\/\/amirhooshang.com\/blog\/wp-content\/ql-cache\/quicklatex.com-429766ff170b3a23266956268f0c62bb_l3.png\" "\\"Rendered")
<\/p><\/li>\n<\/ul>\n\n\n\nPros & Cons:<\/strong><\/p>\n\n\n\n\n- Pros:<\/strong> Scale-independent \u2192 Compare errors across datasets.<\/li>\n\n\n\n
- Cons:<\/strong> Fails if actual values are zero (division by zero).<\/li>\n\n\n\n
- Cons:<\/strong> Unreliable for values close to zero.<\/li>\n<\/ul>\n\n\n\n
Use Case:<\/strong><\/p>\n\n\n\n\n- Sales forecasting for products with varying prices (e.g., $10 to $10,000).<\/li>\n<\/ul>\n\n\n\n
\n\n\n\n6. Mean Squared Logarithmic Error (MSLE) \u2013 For Wide-Ranging Data<\/strong><\/h4>\n\n\n\nWhat is it?<\/strong>
MSLE uses logarithms to reduce the impact of large value differences. Ideal for data with exponential growth (e.g., population or sales over decades).<\/p>\n\n\n\nSimplified Formula:<\/strong>
<\/span> <\/span>![\"\[MSLE = \frac{1}{n} \sum (\log(Actual + 1) - \log(Prediction + 1))^2\]\"](\"https:\/\/amirhooshang.com\/blog\/wp-content\/ql-cache\/quicklatex.com-b99af678f3723176e22f8ba03ed1a20b_l3.png\" "\\"Rendered")
<\/p><\/p>\n\n\n\nExample:<\/strong><\/p>\n\n\n\n\n- Actual Value: 1000 \u2192 Prediction: 1200
<\/span> <\/span>![\"\[MSLE ≈ (\log(1001) - \log(1201))^2 ≈ (6.9 - 7.09)^2 ≈ 0.033\]\"](\"https:\/\/amirhooshang.com\/blog\/wp-content\/ql-cache\/quicklatex.com-bd025fc937dd1190f2acdcdf28d57936_l3.png\" "\\"Rendered")
<\/p><\/li>\n<\/ul>\n\n\n\nPros & Cons:<\/strong><\/p>\n\n\n\n\n- Pros:<\/strong> Reduces emphasis on large absolute errors.<\/li>\n\n\n\n
- Cons:<\/strong> Harder to interpret due to logarithms.<\/li>\n<\/ul>\n\n\n\n
Use Case:<\/strong><\/p>\n\n\n\n\n- Predicting startup revenue growth (from $100 to billions).<\/li>\n<\/ul>\n\n\n\n
\n\n\n\nCombining Metrics: Why One Metric Isn\u2019t Enough<\/strong><\/h3>\n\n\n\nRelying on one metric is like painting with one eye closed! Combining metrics gives you a broader perspective.<\/p>\n\n\n\n
Smart Combinations:<\/strong><\/h4>\n\n\n\n\n- MSE + MAE:<\/strong><\/li>\n<\/ol>\n\n\n\n
\n- High MSE but low MAE? \u2192 Your model has huge errors in specific outliers.<\/li>\n\n\n\n
- Example: Predicting luxury home prices \u2192 MSE flags errors in expensive homes, while MAE shows good performance for average homes.<\/li>\n<\/ul>\n\n\n\n
\n- R\u00b2 + RMSE:<\/strong><\/li>\n<\/ol>\n\n\n\n
\n- R\u00b2 shows overall performance; RMSE gives real-world error magnitude.<\/li>\n\n\n\n
- Example: R\u00b2=0.9 is great, but RMSE=50M might be unacceptable for business.<\/li>\n<\/ul>\n\n\n\n
\n- MAPE + MSLE:<\/strong><\/li>\n<\/ol>\n\n\n\n
\n- Use for datasets with mixed scales (e.g., $10 apps vs. $50M laptops).<\/li>\n<\/ul>\n\n\n\n
Practical Example:<\/strong><\/h4>\n\n\n\nIf you build a model to predict building electricity consumption:<\/p>\n\n\n\n
\n- Using only MSE might hide poor performance in industrial buildings (high consumption).<\/li>\n\n\n\n
- Combining MSE (absolute error) with MAPE (% error) gives a complete picture.<\/li>\n<\/ul>\n\n\n\n
\n\n\n\nConclusion: How to Choose the Right Metric?<\/strong><\/h3>\n\n\n\n\n- Define Your Goal:<\/strong><\/li>\n<\/ol>\n\n\n\n
\n- Care about % error? \u2192 MAPE.<\/li>\n\n\n\n
- Need absolute error? \u2192 RMSE.<\/li>\n<\/ul>\n\n\n\n
\n- Understand Your Data:<\/strong><\/li>\n<\/ol>\n\n\n\n
\n- Many outliers? \u2192 MAE or MAPE.<\/li>\n\n\n\n
- Wide value ranges? \u2192 MSLE.<\/li>\n<\/ul>\n\n\n\n
\n- Combine Metrics:<\/strong><\/li>\n<\/ol>\n\n\n\n
\n- Always use 2-3 metrics to avoid blind spots.<\/li>\n<\/ul>\n\n\n\n
Remember, there\u2019s no universal “best” metric. The right one depends on your problem, data, and business needs!<\/p>\n\n\n\n
<\/p>\n","protected":false},"excerpt":{"rendered":"
Regression in machine learning is used to predict continuous values like house prices, tomorrow\u2019s temperature, or someone\u2019s income. But once you\u2019ve built your model, how do you know how accurate…<\/p>\n","protected":false},"author":1,"featured_media":91,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[61,62],"tags":[80,83,82,51,81],"class_list":["post-75","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-machine-learning","category-model-evaluation","tag-evaluation-metrics","tag-metric-combination","tag-metric-selection","tag-prediction-error","tag-regression"],"_links":{"self":[{"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/posts\/75","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/comments?post=75"}],"version-history":[{"count":15,"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/posts\/75\/revisions"}],"predecessor-version":[{"id":90,"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/posts\/75\/revisions\/90"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/media\/91"}],"wp:attachment":[{"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/media?parent=75"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/categories?post=75"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/tags?post=75"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
What is it?<\/strong> Simplified Formula:<\/strong> <\/span> <\/span> Example:<\/strong><\/p>\n\n\n\n <\/span> <\/span> Pros & Cons:<\/strong><\/p>\n\n\n\n Comparison with MAE:<\/strong><\/p>\n\n\n\n What is it?<\/strong> Simplified Formula:<\/strong> <\/span> <\/span> Example (Same Data as Above):<\/strong> <\/span> <\/span> Pros & Cons:<\/strong><\/p>\n\n\n\n Comparison with MSE:<\/strong><\/p>\n\n\n\n What is it?<\/strong> Simplified Formula:<\/strong> <\/span> <\/span> Example (Using MSE Above):<\/strong> <\/span> <\/span> This means the average prediction error is 30 million.<\/p>\n\n\n\n Pros & Cons:<\/strong><\/p>\n\n\n\n Use Case:<\/strong><\/p>\n\n\n\n What is it?<\/strong> Simplified Formula:<\/strong> <\/span> <\/span> Example:<\/strong><\/p>\n\n\n\n <\/span> <\/span> Pros & Cons:<\/strong><\/p>\n\n\n\n Use Case:<\/strong><\/p>\n\n\n\n What is it?<\/strong> Simplified Formula:<\/strong> <\/span> <\/span> Example:<\/strong><\/p>\n\n\n\n <\/span> <\/span> Pros & Cons:<\/strong><\/p>\n\n\n\n Use Case:<\/strong><\/p>\n\n\n\n What is it?<\/strong> Simplified Formula:<\/strong> <\/span> <\/span> Example:<\/strong><\/p>\n\n\n\n <\/span> <\/span> Pros & Cons:<\/strong><\/p>\n\n\n\n Use Case:<\/strong><\/p>\n\n\n\n Relying on one metric is like painting with one eye closed! Combining metrics gives you a broader perspective.<\/p>\n\n\n\n If you build a model to predict building electricity consumption:<\/p>\n\n\n\n Remember, there\u2019s no universal “best” metric. The right one depends on your problem, data, and business needs!<\/p>\n\n\n\n <\/p>\n","protected":false},"excerpt":{"rendered":" Regression in machine learning is used to predict continuous values like house prices, tomorrow\u2019s temperature, or someone\u2019s income. But once you\u2019ve built your model, how do you know how accurate…<\/p>\n","protected":false},"author":1,"featured_media":91,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[61,62],"tags":[80,83,82,51,81],"class_list":["post-75","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-machine-learning","category-model-evaluation","tag-evaluation-metrics","tag-metric-combination","tag-metric-selection","tag-prediction-error","tag-regression"],"_links":{"self":[{"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/posts\/75","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/comments?post=75"}],"version-history":[{"count":15,"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/posts\/75\/revisions"}],"predecessor-version":[{"id":90,"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/posts\/75\/revisions\/90"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/media\/91"}],"wp:attachment":[{"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/media?parent=75"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/categories?post=75"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/amirhooshang.com\/blog\/wp-json\/wp\/v2\/tags?post=75"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
MSE calculates the average of the squared errors. Large errors are punished more harshly. For example, a prediction error of 50 million would be squared to 2500 billion (!) in MSE.<\/p>\n\n\n\n<\/p><\/p>\n\n\n\n\n
<\/p><\/p>\n\n\n\n\n
\n
\n\n\n\n2. Mean Absolute Error (MAE) \u2013 Fairly Measures All Errors<\/strong><\/h4>\n\n\n\n
MAE calculates the average of absolute errors without squaring them. It treats small and large errors equally.<\/p>\n\n\n\n<\/p><\/p>\n\n\n\n<\/p><\/p>\n\n\n\n\n
\n
\n\n\n\n3. Root Mean Squared Error (RMSE) \u2013 MSE in Understandable Units<\/strong><\/h4>\n\n\n\n
RMSE is the square root of MSE. It aligns the error unit with the original data for easier interpretation.<\/p>\n\n\n\n<\/p><\/p>\n\n\n\n<\/p><\/p>\n\n\n\n\n
\n
\n\n\n\n4. R-Squared (R\u00b2) \u2013 How Much Better Is Your Model Than a Simple Baseline?<\/strong><\/h4>\n\n\n\n
R\u00b2 measures how well your model performs compared to a simple baseline (e.g., predicting the mean). Its value ranges from 0 (worst) to 1 (best).<\/p>\n\n\n\n<\/p><\/p>\n\n\n\n\n
<\/p>
This means your model explains 80% of the data variance.<\/li>\n<\/ul>\n\n\n\n\n
\n
\n\n\n\n5. Mean Absolute Percentage Error (MAPE) \u2013 Error in Percentage Terms<\/strong><\/h4>\n\n\n\n
MAPE expresses error as a percentage, useful for data with varying scales.<\/p>\n\n\n\n<\/p><\/p>\n\n\n\n\n
<\/p><\/li>\n<\/ul>\n\n\n\n\n
\n
\n\n\n\n6. Mean Squared Logarithmic Error (MSLE) \u2013 For Wide-Ranging Data<\/strong><\/h4>\n\n\n\n
MSLE uses logarithms to reduce the impact of large value differences. Ideal for data with exponential growth (e.g., population or sales over decades).<\/p>\n\n\n\n<\/p><\/p>\n\n\n\n\n
<\/p><\/li>\n<\/ul>\n\n\n\n\n
\n
\n\n\n\nCombining Metrics: Why One Metric Isn\u2019t Enough<\/strong><\/h3>\n\n\n\n
Smart Combinations:<\/strong><\/h4>\n\n\n\n
\n
\n
\n
\n
\n
\n
Practical Example:<\/strong><\/h4>\n\n\n\n
\n
\n\n\n\nConclusion: How to Choose the Right Metric?<\/strong><\/h3>\n\n\n\n
\n
\n
\n
\n
\n
\n