### 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. ΣP(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 goal for score 8 and above 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 92% of students score 7 and above. The goal should be to make every student score 7 or more, and the difficulty of questions should be realistic to achieve.

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. 