Dans un ensemble de données, la moyenne \( \mu \) est de 50 et l'écart type \( \sigma \) est de 10. Si chaque point de données est augmenté de 5 puis multiplié par 2, quelle est la nouvelle moyenne et l'écart type ?

["Transform Data: How Adding 5 and Multiplying by 2 Affects Mean and Standard Deviation", "In statistics, understanding how transformations impact key measures like the mean and standard deviation is essential for accurate data analysis. This article explores what happens to the average and spread of a dataset when each value is first increased by 5 and then multiplied by 2.", "---", "### Starting Point: The Original Data", "We are given:", "- Original mean (( \mu )) = 50\n- Original standard deviation (( \sigma )) = 10", "The mean represents the central tendency of the data, while the standard deviation measures how spread out the values are around the mean.", "---", "### Applying the Transformations", "Every data point ( x ) in the original dataset undergoes two operations:", "1. Add 5: ( x \ o x + 5 )\n2. Multiply by 2: ( x + 5 \ o 2(x + 5) = 2x + 10 )", "This transformation can be written as:\n[\ny = 2x + 10\n]", "---", "### New Mean After Transformation", "The mean transforms predictably under linear operations. For a linear transformation ( y = ax + b ):", "[\n\mu_y = a\mu_x + b\n]", "Here, ( a = 2 ), ( b = 10 ), and ( \mu_x = 50 ):\n[\n\mu_y = 2 \cdot 50 + 10 = 100 + 10 = 110\n]", "New mean = 110", "---", "### New Standard Deviation After Transformation", "For a linear transformation ( y = ax + b ), the standard deviation scales only by the factor ( |a| ), since adding a constant (( b )) does not affect spread:", "[\n\sigma_y = |a| \cdot \sigma_x\n]", "Here, ( a = 2 ), ( \sigma_x = 10 ):\n[\n\sigma_y = 2 \cdot 10 = 20\n]", "New standard deviation = 20", "---", "### Conclusion", "When each data point in the original set is transformed by adding 5 and then multiplying by 2:", "- The mean increases from 50 to 110\n- The standard deviation increases from 10 to 20", "These changes reflect how mathematical transformations scale central tendency and dispersion, helping analysts accurately interpret transformed datasets.", "---", "Key Takeaways:\n✅ Mean transforms as ( \mu_y = 2\mu_x + 5 )\n✅ Standard deviation transforms as ( \sigma_y = 2\sigma_x )\n✅ Adding a constant shifts the mean but not the spread\n✅ Multiplying by a factor scales both the mean and standard deviation", "Understanding these principles ensures reliable statistical inference after data manipulation — a cornerstone of effective data science and research.", "---", "Keywords: mean transformation, standard deviation change, data normalization, linear transformation statistics, statistical analysis,データ変換, mean and standard deviation"]