# Standard Deviation

#### [Expected Values](http://mathworld.wolfram.com/ExpectationValue.html)

The expected value of a discrete random variable, $$X$$, is more or less another way of referring to the mean ($$\mu$$). We can also refer to this as the *mathematical expectation* (or just the *expectation*) of $$X$$. 

#### [Variance](http://mathworld.wolfram.com/Variance.html) $$\sigma^2$$

This is the average magnitude of fluctuations of $$X$$ from its expected value, $$\mu$$. You can also think of it as the expectation of a random variable's squared deviation from its mean. Given a data set, $$X$$, of size :

$$\sigma^2 = \frac{\sum\_{i=1}^n (x\_i - \mu)^2}{n}$$

where $$x\_i$$ is the $$i^{th}$$ element of the data set and $$\mu$$ is the *mean* of all the elements. 

#### [Standard Deviation](http://mathworld.wolfram.com/StandardDeviation.html) $$\sigma$$

The standard deviation quantifies the amount of variation in a set of data values. Given a data set, $$X$$, of size :

$$\sigma = \sqrt{\frac{\sum\_{i=1}^n (x\_i - \mu)^2}{n}}$$

where $$x\_i$$ is the $$i^{th}$$ element of the data set and $$\mu$$ is the *mean* of all the elements. 

```
import math

num = int(input())
values = list(map(int, input().split()))

mean = sum(values)/num

result = 0
for i in values:
    result += (i - mean)**2

print('%.1f' % math.sqrt(result/num))
```

## Interquartile Range

```
num = 5
elements = [10, 40, 30, 50, 20]
freq = [1, 2, 3, 4, 5]

values = []
for k, v in enumerate(elements):
    values.extend([v] * freq[k])
values.sort()

lower = values[:len(values)//2]
upper = values[len(values)//2:]

num = len(values)-1
if num % 2 == 0:
    lower = values[:len(values)//2]
    upper = values[len(values)//2:]
else:
    lower = values[:len(values)//2]
    upper = values[len(values)//2:]
    lower.append(upper[0])

if len(lower) % 2 == 0:
    v1 = lower[int(len(lower)/2)-1]
    v2 = lower[int(len(lower)/2)]
    q1 = (v1+v2)/2
    v1 = upper[int(len(upper)/2)-1]
    v2 = upper[int(len(upper)/2)]
    q3 = (v1+v2)/2
else:
    q1 = lower[int(len(lower)/2)]
    q3 = upper[int(len(upper)/2)]

print(float(q3 - q1))
```


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