### Probability distribution of marks should not be normal.

**What type of variable is the mark, discrete or continuous?**

Marks is a discrete random variable that has a finite number of values or a countable number of values.

A continuous random variable has infinitely many values, and those values can be associated with measurements on a continuous scale in such a way that there are no gaps or interruptions.

**Requirements for a Probability Distribution**

1. Î£P(x) = 1 where x assumes all possible values of marks

2. 0 ≤ P(x) ≤ 1 for every individual value of x

For example, 2000 students gave exams with full marks of 10, the probability distribution of marks to have a normal like curve will have following frequency distribution given in the table.

Marks x | Frequency f |
Probability P(X=x) |
---|---|---|

0 | 4 | 0.002 |

1 | 23 | 0.0115 |

2 | 99 | 0.0495 |

3 | 227 | 0.1135 |

4 | 399 | 0.1995 |

5 | 497 | 0.2485 |

6 | 390 | 0.195 |

7 | 251 | 0.1255 |

8 | 84 | 0.042 |

9 | 22 | 0.011 |

10 | 4 | 0.002 |

import matplotlib.pyplot as plt import random import numpy as np from collections import Counter, OrderedDict fig, ax = plt.subplots(1, 1) od = OrderedDict([(0.0, 4), (1.0, 23), (2.0, 99), (3.0, 227), (4.0, 399), (5.0, 497), (6.0, 390), (7.0, 251), (8.0, 84), (9.0, 22), (10.0, 4)]) print(od) val = sum(od.values()) probability = [] for prob in od.values(): probability.append(prob/val) print(probability) print(sum(probability)) ax.plot(list(od.keys()), probability, 'bo', ms=8, label='probability distribution') ax.vlines(list(od.keys()), 0, probability , colors='b', lw=5, alpha=0.5) plt.show()

But frequency distribution like this is

**very difficult to achieve**and based on many different factors, such as question difficulty, learning levels of students.

**Do we even require such a curve?**

About 37% of students will score below marks 5. So the

*.*

**performance of many students is too low**Only about 5% will score 8 and above. The

**for score 8 and above**

*goal**.*

**becomes too unrealistic**So we don't want a bell curve in education.

*If we are getting a bell curve, then our education system is very wrongly designed and implemented.***What type of graph should our education system have?**

This is perhaps a better distribution, which our education system should have, and should achieve where about

**. The goal should be to make every student score 7 or more, and the difficulty of questions should be realistic to achieve.**

*92% of students score 7 and above*Marks x | Frequency f |
Probability P(X=x) |
---|---|---|

0 | 0 | 0 |

1 | 15 | 0.007503751875937969 |

2 | 23 | 0.01150575287643822 |

3 | 18 | 0.009004502251125562 |

4 | 16 | 0.0080040020010005 |

5 | 19 | 0.009504752376188095 |

6 | 27 | 0.013506753376688344 |

7 | 575 | 0.2876438219109555 |

8 | 621 | 0.3106553276638319 |

9 | 559 | 0.27963981990995496 |

10 | 126 | 0.06303151575787894 |

import matplotlib.pyplot as plt import random import numpy as np from collections import Counter, OrderedDict fig, ax = plt.subplots(1, 1) od = OrderedDict([(1, 15), (2, 23), (3, 18), (4, 16), (5, 19), (6, 27), (7, 575), (8, 621), (9, 559), (10, 126)]) print(od) val = sum(od.values()) probability = [] for prob in od.values(): probability.append(prob/val) print(probability) print(sum(probability)) ax.plot(list(od.keys()), probability, 'bo', ms=8, label='probability distribution') ax.vlines(list(od.keys()), 0, probability , colors='b', lw=5, alpha=0.5) plt.show()

**An ideal marks distribution with total of 100 marks.**