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Normal Distribution কি

যখন আমরা কোন রিয়েল লাইফ ডাটা কালেকশনকে রিপ্রেসেন্ট/ডিস্ট্রিবিউট (স্প্রেড আউট) করি তখন সেটার চেহারা বিভিন্ন রকম হতে পারে। যেমন নিচের ডাটাসেটের হিস্টোগ্রাম শো করলে দেখা যাচ্ছে বাম দিকে লম্বা বার বেশি,

x = np.array([20, 30, 30, 30, 20, 20, 50, 20, 20, 30, 40, 40, 40,  50, 60, 60, 90])

plt.hist(x, 15)
plt.show()

![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAW4AAAD8CAYAAABXe05zAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz AAALEgAACxIB0t1+/AAADLVJREFUeJzt3GGMZXV9xvHv012qgkQERkJZpkOjwRBSQCeIgRpdq11d gkljE0hrfGE7bzSFxsSsadqEdzRprH3RNNmIbdMqtlVoza4iVDHWpoXuwmJ3WahUtwpBV9pStE1s wV9f3LM6nc7s3IW5M+dHv5/kZu459z8nT07OPHPu/55zU1VIkvr4sa0OIEk6NRa3JDVjcUtSMxa3 JDVjcUtSMxa3JDVjcUtSMxa3JDVjcUtSM9tnsdFzzz23FhYWZrFpSXpBOnjw4JNVNTfN2JkU98LC AgcOHJjFpiXpBSnJP0871qkSSWrG4pakZixuSWrG4pakZixuSWpmqqtKkhwDvgs8CzxTVYuzDCVJ WtupXA74pqp6cmZJJElTcapEkpqZtrgLuCvJwSRLswwkSTq5aadKrqmqx5O8Arg7ycNV9aXlA4ZC XwKYn59/zoEW9ux/zr+7mmO37N7Q7UnSVpvqjLuqHh9+HgfuAK5cZczeqlqsqsW5ualut5ckPQfr FneSM5KceeI58Fbg8KyDSZJWN81UyXnAHUlOjP94Vd0501SSpDWtW9xV9TXgsk3IIkmagpcDSlIz FrckNWNxS1IzFrckNWNxS1IzFrckNWNxS1IzFrckNWNxS1IzFrckNWNxS1IzFrckNWNxS1IzFrck NWNxS1IzFrckNWNxS1IzFrckNWNxS1IzFrckNWNxS1IzFrckNWNxS1IzFrckNWNxS1IzFrckNWNx S1IzFrckNWNxS1IzFrckNWNxS1IzFrckNTN1cSfZluSBJPtmGUiSdHKncsZ9I3B0VkEkSdOZqriT 7AB2Ax+ZbRxJ0nqmPeP+MPAB4AczzCJJmsL29QYkuRY4XlUHk7zxJOOWgCWA+fn5DQs4Ngt79m/4 No/dsnvDtynphWuaM+6rgeuSHAM+AexM8icrB1XV3qparKrFubm5DY4pSTph3eKuqg9W1Y6qWgCu B75QVb8082SSpFV5HbckNbPuHPdyVfVF4IszSSJJmopn3JLUjMUtSc1Y3JLUjMUtSc1Y3JLUjMUt Sc1Y3JLUjMUtSc1Y3JLUjMUtSc1Y3JLUjMUtSc1Y3JLUjMUtSc1Y3JLUjMUtSc1Y3JLUjMUtSc1Y 3JLUjMUtSc1Y3JLUjMUtSc1Y3JLUjMUtSc1Y3JLUjMUtSc1Y3JLUjMUtSc1Y3JLUjMUtSc1Y3JLU jMUtSc1Y3JLUzLrFneTFSe5L8mCSI0lu3oxgkqTVbZ9izPeBnVX1vSSnAV9O8tmq+rsZZ5MkrWLd 4q6qAr43LJ42PGqWoSRJa5tqjjvJtiSHgOPA3VV172xjSZLWMs1UCVX1LHB5krOAO5JcWlWHl49J sgQsAczPz294UE1vYc/+Dd3esVt2b+j2JD0/p3RVSVU9BdwD7Frltb1VtVhVi3NzcxuVT5K0wjRX lcwNZ9okeQnwFuDhWQeTJK1umqmS84E/SrKNSdH/WVXtm20sSdJaprmq5CvAFZuQRZI0Be+clKRm LG5JasbilqRmLG5JasbilqRmLG5JasbilqRmLG5JasbilqRmLG5JasbilqRmLG5JasbilqRmLG5J asbilqRmLG5JasbilqRmLG5JasbilqRmLG5JasbilqRmLG5JasbilqRmLG5JasbilqRmLG5Jasbi lqRmLG5JasbilqRmLG5JasbilqRmLG5Jambd4k5yYZJ7kjyU5EiSGzcjmCRpddunGPMM8P6quj/J mcDBJHdX1UMzziZJWsW6Z9xV9URV3T88/y5wFLhg1sEkSas7pTnuJAvAFcC9swgjSVrfNFMlACR5 KfAp4KaqenqV15eAJYD5+fkNC6gXpoU9+zd0e8du2b2h2xt7Pv3/NtUZd5LTmJT2x6rq9tXGVNXe qlqsqsW5ubmNzChJWmaaq0oC3AocraoPzT6SJOlkpjnjvhp4F7AzyaHh8fYZ55IkrWHdOe6q+jKQ TcgiSZqCd05KUjMWtyQ1Y3FLUjMWtyQ1Y3FLUjMWtyQ1Y3FLUjMWtyQ1Y3FLUjMWtyQ1Y3FLUjMW tyQ1Y3FLUjMWtyQ1Y3FLUjMWtyQ1Y3FLUjMWtyQ1Y3FLUjMWtyQ1Y3FLUjMWtyQ1Y3FLUjMWtyQ1 Y3FLUjMWtyQ1Y3FLUjMWtyQ1Y3FLUjMWtyQ1Y3FLUjMWtyQ1Y3FLUjPrFneSjyY5nuTwZgSSJJ3c NGfcfwjsmnEOSdKU1i3uqvoS8K+bkEWSNIXtG7WhJEvAEsD8/PxGbVaSZmZhz/4N3d6xW3Zv6PbW smEfTlbV3qparKrFubm5jdqsJGkFryqRpGYsbklqZprLAW8D/ha4OMljSd4z+1iSpLWs++FkVd2w GUEkSdNxqkSSmrG4JakZi1uSmrG4JakZi1uSmrG4JakZi1uSmrG4JakZi1uSmrG4JakZi1uSmrG4 JakZi1uSmrG4JakZi1uSmrG4JakZi1uSmrG4JakZi1uSmrG4JakZi1uSmrG4JakZi1uSmrG4JakZ i1uSmrG4JakZi1uSmrG4JakZi1uSmrG4JakZi1uSmrG4JamZqYo7ya4kjyR5NMmeWYeSJK1t3eJO sg34PeBtwCXADUkumXUwSdLqpjnjvhJ4tKq+VlX/BXwCeMdsY0mS1jJNcV8AfHPZ8mPDOknSFkhV nXxA8k5gV1X98rD8LuB1VfW+FeOWgKVh8WLgkeeY6Vzgyef4u5utU1bolbdTVuiVt1NW6JX3+WT9 yaqam2bg9inGPA5cuGx5x7Duf6mqvcDeqeKdRJIDVbX4fLezGTplhV55O2WFXnk7ZYVeeTcr6zRT JX8PvCrJRUl+HLge+PRsY0mS1rLuGXdVPZPkfcDngG3AR6vqyMyTSZJWNc1UCVX1GeAzM85ywvOe btlEnbJCr7ydskKvvJ2yQq+8m5J13Q8nJUnj4i3vktTMlhV3kguT3JPkoSRHktw4rD87yd1Jvjr8 fPlWZVwuyYuT3JfkwSHvzcP6i5LcO3wdwJ8OH+COQpJtSR5Ism9YHnPWY0n+IcmhJAeGdWM9Fs5K 8skkDyc5muT1I8568bBPTzyeTnLTiPP+2vD3dTjJbcPf3SiP2yQ3DjmPJLlpWLcp+3Urz7ifAd5f VZcAVwHvHW6l3wN8vqpeBXx+WB6D7wM7q+oy4HJgV5KrgN8CfqeqXgn8G/CeLcy40o3A0WXLY84K 8KaqunzZ5VRjPRZ+F7izql4NXMZkH48ya1U9MuzTy4HXAv8J3MEI8ya5APhVYLGqLmVyMcT1jPC4 TXIp8CtM7iy/DLg2ySvZrP1aVaN4AH8JvIXJjTvnD+vOBx7Z6myrZD0duB94HZOL7bcP618PfG6r 8w1ZdgwHzk5gH5CxZh3yHAPOXbFudMcC8DLg6wyfD4056yrZ3wr8zVjz8qO7tM9mcuHEPuDnxnjc Ar8A3Lps+TeAD2zWfh3FHHeSBeAK4F7gvKp6YnjpW8B5WxTr/ximHg4Bx4G7gX8CnqqqZ4YhY/o6 gA8zOZB+MCyfw3izAhRwV5KDw124MM5j4SLgO8AfDNNQH0lyBuPMutL1wG3D89HlrarHgd8GvgE8 Afw7cJBxHreHgZ9Jck6S04G3M7lRcVP265YXd5KXAp8Cbqqqp5e/VpN/W6O57KWqnq3JW84dTN4i vXqLI60qybXA8ao6uNVZTsE1VfUaJt9C+d4kb1j+4oiOhe3Aa4Dfr6orgP9gxdvhEWX9oWFe+Drg z1e+Npa8w3zwO5j8c/wJ4Axg15aGWkNVHWUyhXMXcCdwCHh2xZiZ7dctLe4kpzEp7Y9V1e3D6m8n OX94/XwmZ7ejUlVPAfcwedt2VpIT18Ov+nUAW+Bq4Lokx5h8m+NOJvOyY8wK/PBsi6o6zmQO9krG eSw8BjxWVfcOy59kUuRjzLrc24D7q+rbw/IY8/4s8PWq+k5V/TdwO5NjeZTHbVXdWlWvrao3MJl7 /0c2ab9u5VUlAW4FjlbVh5a99Gng3cPzdzOZ+95ySeaSnDU8fwmT+fijTAr8ncOwUeStqg9W1Y6q WmDy9vgLVfWLjDArQJIzkpx54jmTudjDjPBYqKpvAd9McvGw6s3AQ4ww6wo38KNpEhhn3m8AVyU5 feiHE/t2rMftK4af88DPAx9ns/brFk7uX8PkbcRXmLzNOMRknugcJh+qfRX4K+Dsrcq4Iu9PAw8M eQ8Dvzms/yngPuBRJm9DX7TVWVfkfiOwb8xZh1wPDo8jwK8P68d6LFwOHBiOhb8AXj7WrEPeM4B/ AV62bN0o8wI3Aw8Pf2N/DLxoxMftXzP5x/Ig8ObN3K/eOSlJzWz5h5OSpFNjcUtSMxa3JDVjcUtS Mxa3JDVjcUtSMxa3JDVjcUtSM/8Dikg1VWc8lD4AAAAASUVORK5CYII= )

অথবা কিছু ডাটার হিস্টোগ্রাম বার গুলো হতে পারে নিচের মত অগোছালো,

x = np.array([40, 30, 80, 30, 20, 80, 50, 20, 20, 60, 40, 40, 40,  50, 60, 60, 90, 80, 70])

plt.hist(x, 15)
plt.show()

![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAXcAAAD8CAYAAACMwORRAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz AAALEgAACxIB0t1+/AAAEd1JREFUeJzt3X+MZXV9xvH347Liz4iyU93u7jg2EI0SAZ0gRmso1HYV AknFBNOqGO0mRiI0JgZsioG/NGnUWo1mI7ZIrWLR2hXwB1WM2sTVWVyQZaWuSmUJygICUhW7+ukf 96DT21numZk7O3e+vl/JzZ4f3zn3yc3ZZ86ce869qSokSW151GoHkCSNn+UuSQ2y3CWpQZa7JDXI cpekBlnuktQgy12SGmS5S1KDLHdJatARq/XEGzZsqJmZmdV6eklak3bt2nV3VU2NGrdq5T4zM8Pc 3NxqPb0krUlJ/qvPOE/LSFKDLHdJapDlLkkNstwlqUGWuyQ1qHe5J1mX5FtJrl5g3ZFJrkyyL8nO JDPjDClJWpzFHLmfD+w9xLrXAz+pqmOAdwPvXG4wSdLS9Sr3JJuB04EPHWLIWcDl3fRVwGlJsvx4 kqSl6Hvk/h7grcCvD7F+E3A7QFUdBO4Hjl52OknSkoy8QzXJGcBdVbUrySnLebIk24BtANPT08vZ lMZg5sJrxrq9295x+li3J2np+hy5vwg4M8ltwMeBU5P809CYO4AtAEmOAJ4E3DO8oaraXlWzVTU7 NTXyoxEkSUs0styr6qKq2lxVM8A5wJeq6i+Ghu0AXttNn92NqbEmlST1tuQPDktyKTBXVTuAy4Ar kuwD7mXwS0CStEoWVe5V9WXgy930xfOW/wJ45TiDSZKWzjtUJalBlrskNchyl6QGWe6S1CDLXZIa ZLlLUoMsd0lqkOUuSQ2y3CWpQZa7JDXIcpekBlnuktQgy12SGmS5S1KDLHdJapDlLkkNstwlqUEj 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কিন্তু অনেক সময় বাস্তবের কিছু ডাটাকে (কিছু ছাত্রের উচ্চতার মান, তাদের পরীক্ষায় প্রাপ্ত নম্বর, একটি মেশিন দারা তৈরি কোন প্রোডাক্টের সাইজ, জনগণের আয় ইত্যাদি) ডিস্ট্রিবিউট করলে নিচের মত চেহারা পাওয়া যায়,

sizes = np.array([9, 8, 8, 9, 9, 1, 4, 5, 6, 7, 8, 9, 10, 10, 11, 11, 11, 11, 11, 13, 12, 12, 12, 12, 13, 13, 14, 14, 14, 15, 15, 15, 16, 17, 18, 20])

plt.hist(sizes)
plt.show()

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যেটা অনেকটা বেল (ঘণ্টি) কার্ভের মত অর্থাৎ মাঝখানের বার গুলো লম্বা এবং তার দুপাশের বার গুলো ক্রমান্বয়ে ছোট। এরকম কোন ডাটার ডিস্ট্রিবিউশনকে বলা হয় নরমালি ডিস্ট্রিবিউটেড।

সব ডাটা এমনি এমনি এমন চেহারা নাও পেতে পারে। সেক্ষেত্রে ডাটা গুলোর গড় বা মধ্যক বের করে সেটাকে মাঝখানে রেখে ওই মধ্যম মানের চেয়ে ছোট ও বড় মান গুলোকে যথাক্রমে বাম পাশে এবং ডানপাশে রেখে একটি ডিস্ট্রিবিউশন তৈরি করাকে নরমাল ডিস্ট্রিবিউশন বলা হয়। ডাটাকে এভাবে ডিস্ট্রিবিউট করলে পরবর্তীতে অনেক রকম হিসাব, পর্যবেক্ষণ বা সম্ভাব্যতা বের করা সহজ হয়ে যায়।

নরমালি ডিস্ট্রিবিউটেড কোন ডাটাসেটের mean, median এবং mode মোটামুটি একই হয়। নিচে প্রমাণ করে দেখা যেতে পারে,

np.mean(sizes)
11.194444444444445
np.median(sizes)
11.0
stats.mode(sizes)
ModeResult(mode=array([11]), count=array([5]))