Why my calculated value is far different from actual value?

Question

I using CharacterController and not using a Rigidbody so I writing a script for downward velocity which is velocity=-9.81*(time) and I declare my initial velocity as 12. Supposing from this calculation, the max height it can reach from the equation is roughly 6.7891 unit on y axis. By using the formulae distance travel=initial velocity*(time)-0.5g*(time)^2,differentiate this w.r.t time, getting time=u/g=12/9.81=1.223, then max distance travel=12*(1.223)-0.5*(-9.81)*(1.223)^2=7.339 but from the coordinate below, can clearly seen the max distance travel before falling down is 13.2803unit--0.5499unit=13.8302unit? This is far exceed calculated value, wonder what went wrong?

Initial coordinate of y on ground=-0.5499

approximate final coordinate of y=13.2803

using UnityEngine;

public class NewPlayerMovement : MonoBehaviour
{
    public CharacterController Controller;
    //speed of the character
    public float Speed = 12f;
    // earth gravitational value
    public float Gravity = -9.81f;
    Vector3 velocity;
    //hit point of character
    float hp = 90;

    bool Grounded = true;

    public void OnControllerColliderHit(ControllerColliderHit hit)
    {
        Grounded = true;
        hp -= velocity.magnitude;
        velocity.y = 0;
        Debug.Log(hp);
    }

    public void OnControllerColliderExit(ControllerColliderHit hit)
    {
        Grounded = false;
    }

    public void Checkstatus()
    {
        if(Grounded==false)
        {
            velocity.y += 0.5f * Gravity * Time.deltaTime;
        }
    }

    void Update()
    {
        float x = Input.GetAxis("Horizontal");
        float z = Input.GetAxis("Vertical");
        Vector3 move = transform.right * x + transform.forward * z;

        Controller.Move(move * Speed * Time.deltaTime);

        if (Input.GetKeyDown(KeyCode.Space) && velocity.y == 0)
        {
            velocity.y = 12;
            Grounded = false;
        }

       //check if it grounded
        Checkstatus();
        Controller.Move(velocity * Time.deltaTime);
    }
}

Answer 1

Calculating the new velocity should be

velocity.y += Gravity * Time.deltaTime;

without the 0.5f *. This is because the velocity remains the same during the whole time step unlike in the continuous case where we have an inclined velocity curve approximated with infinitesimal small triangles calculated as baseline * height / 2. This is where the 0.5 comes from when we integrate.