How do I convert probability into z-score

Question

Javascript>

If you are in the data science industry, you would be bothered if you don't have normal distribution table. I came across the article in Stackoverflow that converts z-score to probability in JavaScript. What I really want to know is the reverse calculation of this function.

/**
 * @param {number} z - Number of standard deviations from the mean.
 */
function GetZPercent(z) {
   // If z is greater than 6.5 standard deviations from the mean
   // the number of significant digits will be outside of a reasonable 
   // range.
   if (z < -6.5)
     return 0.0;

   if (z > 6.5)
     return 1.0;

   var factK    = 1;
   var sum      = 0;
   var term     = 1;
   var k        = 0;
   var loopStop = Math.exp(-23);
   
   while (Math.abs(term) > loopStop) {
     term = 0.3989422804 * Math.pow(-1, k) * Math.pow(z, k) / (2 * k + 1) /
            Math.pow(2, k) * Math.pow(z, k + 1) / factK;
     sum += term;
     k++;
     factK *= k;
   }

   sum += 0.5;

   return sum;
 }

I have a sense of how to convert z-score into the probability. But, I have no idea how to calculate the z-score(Standard deviation) from corresponding probability in javascript. For example, If I put in 0.95 (or 95%), I can expect to get 2.25 standard deviation. Above code gives me 95%, if I enter 2.25.

Answer 1

Here is a function that does an opposite calculation (probability to z-score). This snippet allows you to input the probability and the the corresponding z-score is displayed:

function percentile_z(p) {
    if (p < 0.5) return -percentile_z(1-p);

    if (p > 0.92) {
        if (p == 1) return Infinity;
        let r = Math.sqrt(-Math.log(1-p));
        return (((2.3212128*r+4.8501413)*r-2.2979648)*r-2.7871893)/
               ((1.6370678*r+3.5438892)*r+1);
    }
    p -= 0.5;
    let r = p*p;
    return p*(((-25.4410605*r+41.3911977)*r-18.6150006)*r+2.5066282)/
             ((((3.1308291*r-21.0622410)*r+23.0833674)*r-8.4735109)*r+1);
}

// I/O handling
function calc() {
    var p = +document.getElementById("prob").value;
    var z = percentile_z(p);
    document.getElementById("z").textContent = z.toFixed(4);
}
calc();

input { width: 5em }

Probability (between 0 and 1):
<input type="number" id="prob" step="0.0001" min="0" max="1" value="0.9500" oninput="calc()"><p>

Z Score: <span id="z"></span>

For a probability of 0.95 it returns a z-score of 1.6449. See also this table as reference.

---

_{Derived from easycalculation.com}

Answer 2

I found that this code also works. Use critz(p) to convert probability to z-score. For example we can expect 1.65 from critz(0.95) as 95% corresponds to 1.65 standard deviation in z-score.

/*  The following JavaScript functions for calculating normal and
    chi-square probabilities and critical values were adapted by
    John Walker from C implementations
    written by Gary Perlman of Wang Institute, Tyngsboro, MA
    01879.  Both the original C code and this JavaScript edition
    are in the public domain.  */

/*  POZ  --  probability of normal z value

    Adapted from a polynomial approximation in:
            Ibbetson D, Algorithm 209
            Collected Algorithms of the CACM 1963 p. 616
    Note:
            This routine has six digit accuracy, so it is only useful for absolute
            z values <= 6.  For z values > to 6.0, poz() returns 0.0.
*/
    var Z_MAX = 6;
function poz(z) {

    var y, x, w;

    if (z == 0.0) {
        x = 0.0;
    } else {
        y = 0.5 * Math.abs(z);
        if (y > (Z_MAX * 0.5)) {
            x = 1.0;
        } else if (y < 1.0) {
            w = y * y;
            x = ((((((((0.000124818987 * w
                     - 0.001075204047) * w + 0.005198775019) * w
                     - 0.019198292004) * w + 0.059054035642) * w
                     - 0.151968751364) * w + 0.319152932694) * w
                     - 0.531923007300) * w + 0.797884560593) * y * 2.0;
        } else {
            y -= 2.0;
            x = (((((((((((((-0.000045255659 * y
                           + 0.000152529290) * y - 0.000019538132) * y
                           - 0.000676904986) * y + 0.001390604284) * y
                           - 0.000794620820) * y - 0.002034254874) * y
                           + 0.006549791214) * y - 0.010557625006) * y
                           + 0.011630447319) * y - 0.009279453341) * y
                           + 0.005353579108) * y - 0.002141268741) * y
                           + 0.000535310849) * y + 0.999936657524;
        }
    }
    return z > 0.0 ? ((x + 1.0) * 0.5) : ((1.0 - x) * 0.5);
}


/*  CRITZ  --  Compute critical normal z value to
               produce given p.  We just do a bisection
               search for a value within CHI_EPSILON,
               relying on the monotonicity of pochisq().  */

function critz(p) {
    var Z_EPSILON = 0.000001;     /* Accuracy of z approximation */
    var minz = -Z_MAX;
    var maxz = Z_MAX;
    var zval = 0.0;
    var pval;
    if( p < 0.0 ) p = 0.0;
    if( p > 1.0 ) p = 1.0;

    while ((maxz - minz) > Z_EPSILON) {
        pval = poz(zval);
        if (pval > p) {
            maxz = zval;
        } else {
            minz = zval;
        }
        zval = (maxz + minz) * 0.5;
    }
    return(zval);
}