Accessing a two-dimensional array using a dropdown

These kinds of things should always be resolved on the client . The function of the server is to give you the data that resides there, but once with the data, the dynamic changes must be worked on in the client.

What is the reason for this? That if you reflect the dynamic changes on the server, every time there is a change you will have to return to the server to reflect those changes without any need , and going to the server has a very high cost.

Imagine it like this, so that it is understood: the data is on the server (in Australia) and you are in Paris (the client). Every time the selectchange in the client (in Paris) you pay for a Paris-Australia plane ticket to go to Australia to compare data that you already have on hand in Paris. Isn't that absurd? If you have the data in Paris, don't go to Australia to check the data you already have! You're going to end up broke!

Another thing is that some of these data changes, having a different value on the server... then you should go back to the server to look for the new data. But this is not the case here, $arrayResultadosit has some fixed data that, once received in the client, you can work with.

This code, contrary to what you may think, does not burden anyone, but fairly distributes responsibilities. Imagine an application with millions of users, all those millions of users generating Rest of World-Australia air traffic and calculating in Australia everything that should be calculated in the rest of the world. Anyway... a disaster.

Apart from that, the code works much faster being distributed. The Australia server could end up going down with so much traffic.

Look at the proposed solution, working the logic from the client. Here, the variable mDatawould represent the data that would come in $arrayResultados, when the server provides you with that data it will enter rest, and it will be up to each client to work. This is how modern apps work.

/*
  mData sería tu array, que podrías mandar al cliente
  con un simple  echo json_enconde($elArray);
  Aquí lo declaramos a mano para fines de prueba
*/
let mData = `{"Libre":{"No Presentados":12, "Suspenso":10, "Aprobado":6, "Notable":3, "Sobresaliente":1},
"On-line":{"Suspenso":26, "Aprobado":25, "Notable":22, "Sobresaliente":27},
"Presencial":{"Suspenso":30, "Aprobado":25, "Notable":37, "Sobresaliente":28, "Matrícula de Honor":9}
}`;

/*Creamos un JSON con los datos recibidos del servidor*/
let mJson = JSON.parse(mData);

/*Referencia a los elementos*/
let mSelect = document.querySelector('#mSelect');
let mTable = document.querySelector('#mTable');
let tBody = document.querySelector("#mTable tbody");

/*Llenamos de forma dinámica los option*/
for (var key in mJson) {
  const option = document.createElement('option');
  option.textContent = key;
  mSelect.appendChild(option);
}

/*Escuchamos los cambios del select*/
mSelect.addEventListener('change', function(event) {
  let mKey = event.target.value;
  /*Aquí, cuando sea todas recogemos y sumamos los datos*/
  if (mKey === "Todas") {
    var mObj = {};
    for (i in mJson) {
      for (key in mJson[i]) {
        if (mObj.hasOwnProperty(key)) {
          mObj[key] += mJson[i][key];
        } else {
          mObj[key] = mJson[i][key];
        }
      }
    }
  /*O bien construimos la tabla con el option seleccionado*/  
  } else {
    var mObj = mJson[mKey];
  }
  fillTable(mObj);

});

/*Esta función es para llenar la tabla de forma dinámica*/
function fillTable(mObj) {
  /*Limpiamos el tbody de la tabla*/
  document.querySelectorAll("#mTable tbody tr").forEach(function(e) {
    e.remove()
  });
  
  /*Recorremos el objeto para llenar la tabla*/
  for (var key in mObj) {
    let tr = document.createElement("tr");
    let tdA = document.createElement("td");
    let tdB = document.createElement("td");
    tdA.appendChild(document.createTextNode(key));
    tdB.appendChild(document.createTextNode(mObj[key]));
    tr.appendChild(tdA);
    tr.appendChild(tdB);
    tBody.appendChild(tr);
  }
}

<link href="https://cdnjs.cloudflare.com/ajax/libs/bulma/0.9.2/css/bulma.min.css" rel="stylesheet" />

<select id="mSelect">
  <option>-Seleccione-</option>
  <option>Todas</option>
</select>

<table id="mTable" class="table is-bordered is-striped is-narrow is-hoverable">
  <tbody>
  </tbody>
</table>

I update with the complete solution: I have replaced the part of the isset so that when choosing an option it checks if the chosen option is in the array, if it is I add to the array "$listaAlum" all the values ​​of the chosen option.

For the "all" option, I simply go through the array and save the categories in another array and then go through it and accumulate all the students in each category.

    <?php

$arrayResultados =  array 
         ('Libre'=>  array('No Presentados'=>12,'Suspenso'=>10,'Aprobado'=>6,'Notable'=>3,'Sobresaliente'=>1),
          'On-line'=> array('Suspenso'=>26,'Aprobado'=>25,'Notable'=>22,'Sobresaliente'=>27),
          'Presencial'=>array  ('Suspenso'=>30,'Aprobado'=>25,'Notable'=>37,'Sobresaliente'=>28,'Matrícula de Honor'=>9)
          );
?>

<!DOCTYPE html>
<html>
<head>
    <meta charset="utf-8">
</head>
<body>

<form method="POST">
        <select name="desplegable">
        <option value="todas">Todas</option>
        
<?php
    foreach ($arrayResultados as $indice => $valor)
    {
        echo "<option value='".$indice."'>".$indice."</option>";
    }
?>      
</select>
<input type="submit" name="envio" value="Aceptar">
    </form>
<?php
$arrayCalificaciones=array();

        if(isset($_POST["envio"]))
        {
            $listaAlum=[];
            
            foreach ($arrayResultados as $indice => $valor)
            {
                foreach ($valor as $detalles => $numero)
                {                                                           
                    if($_POST["desplegable"]==$indice)
                        {
                            $listaAlum[$detalles] = $numero;
                        }                                           
                }
            }   
            
                        if($_POST["desplegable"]=="todas") //cuando elija la opcion todas
                        {
                            foreach ($arrayResultados as $indice=>$vector)
                            {
                                foreach($vector as $categoria=>$numero)
                                {
                                    if (!array_key_exists($categoria,$listaAlum))
                                    {
                                        $listaAlum[$categoria]=$numero;
                                    }
                                    else
                                    {
                                        $listaAlum[$categoria]=$listaAlum[$categoria]+$numero;
                                    }
                                }
                            }                       
                        }

            echo "<h3>Seleccion: ".$_POST["desplegable"]."</h3>";
            if(sizeof($listaAlum)>0)
            {
                echo "<table border='1'><tr><th>Calificacion</th><th>Alumnos</th></tr>";
                foreach ($listaAlum as $calificacion => $alumnos) 
                {
                    echo "<tr><td>".$calificacion."</td><td>".$alumnos."</td></tr>";
                }
                    echo "</table>";
            }
            else
            { 
                echo "No hay datos"; 
            }
        }
?>  
</body>
</html>