Step by Step solution about How to resolve the algorithm Fivenum step by step in the Julia programming language

The provided Julia code defines two functions: mediansorted and fivenum. Here's how these functions work:

1. mediansorted Function:

   • It takes three parameters: an abstract vector x, an index i (representing the starting index), and an index l (representing the ending index).
   • It calculates the length of the sub-array as len = l - i + 1.
   • It checks if len > zero(len); if not, it throws an ArgumentError indicating that the array slice cannot be empty.
   • It computes the middle index mid = i + len ÷ 2.
   • Depending on whether the length len is odd or even, it calculates the median as follows:
     • If len is odd, it uses x[mid] as the median.
     • If len is even, it computes the median as (x[mid-1] + x[mid]) / 2.

2. fivenum Function:

   • It takes one parameter: x, an abstract vector of type T where T is constrained to be a floating-point type (AbstractFloat).
   • It creates an empty vector r of type Vector{T} with a size of 5. This vector will store the five-number summary statistics.
   • It sorts the input vector x in ascending order and stores the sorted vector in xs.
   • It calculates the middle index mid of the sorted vector xs as length(xs) ÷ 2.
   • It calculates lowerend, which is set to mid if the length of xs is odd and mid - 1 if it is even.
   • It computes the five-number summary statistics:
     • r[1] is set to the minimum value of xs (xs[1]).
     • r[2] is set to the median of the lower half of xs (mediansorted(xs, 1, lowerend)).
     • r[3] is set to the median of the entire sorted vector xs (mediansorted(xs, 1, endof(xs))).
     • r[4] is set to the median of the upper half of xs (mediansorted(xs, mid, endof(xs))).
     • r[end] is set to the maximum value of xs (xs[end]).
   • Finally, it returns the vector r containing the five-number summary: minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum.

In the provided examples, the code calculates and prints the five-number summary for three different input vectors:

1. [15.0, 6.0, 42.0, 41.0, 7.0, 36.0, 49.0, 40.0, 39.0, 47.0, 43.0]
2. [36.0, 40.0, 7.0, 39.0, 41.0, 15.0]
3. A vector of 20 random floating-point numbers

For each input vector, the code prints the original vector and its corresponding five-number summary.

function mediansorted(x::AbstractVector{T}, i::Integer, l::Integer)::T where T
    len = l - i + 1
    len > zero(len) || throw(ArgumentError("Array slice cannot be empty."))
    mid = i + len ÷ 2
    return isodd(len) ? x[mid] : (x[mid-1] + x[mid]) / 2
end

function fivenum(x::AbstractVector{T}) where T<:AbstractFloat
    r = Vector{T}(5)
    xs = sort(x)
    mid::Int = length(xs) ÷ 2
    lowerend::Int = isodd(length(xs)) ? mid : mid - 1
    r[1] = xs[1]
    r[2] = mediansorted(xs, 1, lowerend)
    r[3] = mediansorted(xs, 1, endof(xs))
    r[4] = mediansorted(xs, mid, endof(xs))
    r[end] = xs[end]
    return r
end

for v in ([15.0, 6.0, 42.0, 41.0, 7.0, 36.0, 49.0, 40.0, 39.0, 47.0, 43.0],
          [36.0, 40.0, 7.0, 39.0, 41.0, 15.0],
          [0.14082834,  0.09748790,  1.73131507,  0.87636009, -1.95059594,  0.73438555,
          -0.03035726,  1.46675970, -0.74621349, -0.72588772,  0.63905160,  0.61501527,
          -0.98983780, -1.00447874, -0.62759469,  0.66206163,  1.04312009, -0.10305385,
           0.75775634,  0.32566578])
    println("# ", v, "\n -> ", fivenum(v))
end