Compute Statistical Measures

Given a list of integers, compute the following statistical measures: mean, median, mode, variance, and standard deviation. Use the formulas below:

• Mean: $$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$$
• Median: the middle value when the list is sorted. If there are an even number of values, then $$median = \frac{x_{\frac{n}{2}} + x_{\frac{n}{2}+1}}{2}$$.
• Mode: the smallest value among those that occur most frequently.
• Variance: $$\sigma^2 = \frac{1}{n}\sum_{i=1}^{n} (x_i - \bar{x})^2$$
• Standard Deviation: $$\sigma = \sqrt{\sigma^2}$$

If the list is empty, then all statistics should be reported as null.

inputFormat

The input is given through stdin:

1. The first line contains a single integer n which represents the number of integers.
2. If n > 0, the second line contains n space-separated integers.

outputFormat

Output a JSON object (printed to stdout) with the following keys: mean, median, mode, variance, and std_deviation. For an empty list (n = 0), all values should be null. Otherwise, output the computed statistical measures. For example:

{
  "mean": 2.5,
  "median": 2.5,
  "mode": 1,
  "variance": 1.25,
  "std_deviation": 1.118033988749895
}

0

{"mean": null, "median": null, "mode": null, "variance": null, "std_deviation": null}