Function to determine number of significant figures (e.g. in answer)

[FloMiLe]

Description of proposed feature

I think it would be helpful to have a function that returns the number of significant figures of some given number, e.g. an answer. This should probably be done after converting to a string in order to catch trailing zeros.

As significant figures convey meaning in physics, I would like to attribute points for rounding numeric answers to a meaningful number of significant figures.

How can the new feature be used?

Say the function was called numsigfig, then it should be used as:

numsigfig(1234) => 4
numsigfig(1.234) => 4
numsigfig(0200) => 3
numsigfig(0.200) => 3

Additional comments

I currently use the following code to determine the number of significant figures nSigFigAns in an answer:

origAns = _0;

expAns = floor( log10(abs(origAns)));
mantAns = origAns / ( 10**( expAns ));
nSigFigAns = 1;
for (i:[0:6]) {
    shiftedNum = round( 10**(i) * abs(mantAns) , 6 );
    divTest = round( shiftedNum - floor(shiftedNum) , 6);
    nSigFigAns = ( divTest > 1e-5 ) ? nSigFigAns+1 : nSigFigAns ;
}

I don't think, it would be doable using Formula's current string functions.

[FloMiLe]

I would like to suggest the following function to be added to the source code:

/**
 * Calculates the number of significant figures in a numerical string.
 * Handles standard notation and scientific notation (e.g., 1.23E4).
 * Preserves trailing zeros after a decimal point.
 *
 * @param mixed $x The number as a string or number type. Will be cast to string.
 * @return int The number of significant figures. Returns 1 for zero by convention.
 */
function get_sigfig($x): int
{
    // Cast the input to a string to preserve original formatting (especially trailing zeros).
    $num_str = (string) $x;

    // 1. Handle Scientific Notation
    // Look for 'e' or 'E'. The exponent part does NOT affect the number of sig figs.
    $parts = explode('e', str_ireplace('E', 'e', $num_str)); // Use str_ireplace for case-insensitivity
    $mantissa_str = $parts[0];
    // $exponent_str = isset($parts[1]) ? $parts[1] : null; // Exponent is ignored for sig figs

    // 2. Pre-process the Mantissa String
    // Remove leading/trailing whitespace.
    $mantissa_str = trim($mantissa_str);

    // Remove leading plus/minus signs, as they are not significant figures.
    if (substr($mantissa_str, 0, 1) === '+' || substr($mantissa_str, 0, 1) === '-') {
        $mantissa_str = substr($mantissa_str, 1);
    }

    // Check if the mantissa contains a decimal point. This affects trailing zero significance.
    $decimal_found = strpos($mantissa_str, '.') !== false;

    // Create a string containing only the digits for easier processing.
    $digits_only = str_replace('.', '', $mantissa_str);

    // 3. Handle the case where the number is zero (or effectively zero like 0.000)
    // If the digits-only string is all zeros after trimming, it's zero.
    if (trim($digits_only, '0') === '') {
        return 1; // By convention, zero has 1 significant figure.
    }

    // 4. Find the first and last non-zero digits in the digits-only string
    // strspn counts the length of the initial segment that consists of characters from '0'.
    $first_non_zero_index = strspn($digits_only, '0');

    // Find the index of the last non-zero digit by searching backwards.
    $last_non_zero_index = strlen($digits_only) - 1;
    while ($last_non_zero_index > $first_non_zero_index && $digits_only[$last_non_zero_index] === '0') {
        $last_non_zero_index--;
    }

    // 5. Calculate the number of significant figures
    if ($decimal_found) {
        // If a decimal point is present, all digits from the first non-zero digit
        // to the very end of the digits-only string are significant.
        // This includes trailing zeros after the last non-zero digit.
        $sig_figs = strlen($digits_only) - $first_non_zero_index;
    } else {
        // If no decimal point is present, only the digits from the first non-zero
        // digit up to the last non-zero digit are considered significant.
        $sig_figs = $last_non_zero_index - $first_non_zero_index + 1;
    }

    return $sig_figs;
}

It works in all of the following test cases:

echo "15.440E3: " . get_sigfig("15.440E3") . " = 5 ?<br>";
echo "15.440: " . get_sigfig("15.440") . " = 5 ?<br>";
echo "15440: " . get_sigfig("15440") . " = 4 ?<br>";
echo "15440.0: " . get_sigfig("15440.0") . " = 6 ?<br>";
echo "0.00123: " . get_sigfig("0.00123") . " = 3 ? (leading zeros are not significant)<br>";
echo "0.00120: " . get_sigfig("0.00120") . " = 3 ? (trailing zero after decimal is significant)<br>";
echo "101: " . get_sigfig("101") . " = 3 ? (zero between non-zeros is significant)<br>";
echo "100: " . get_sigfig("100") . " = 1 ? (trailing zeros without decimal not significant)<br>";
echo "100.: " . get_sigfig("100.") . " = 3 ? (trailing zeros with decimal are significant)<br>";
echo "1.00: " . get_sigfig("1.00") . " = 3 ?<br>";
echo "1.0E1: " . get_sigfig("1.0E1") . " = 2 ? (from 1.0)<br>";
echo "10: " . get_sigfig("10") . " = 1 ?<br>";
echo "0: " . get_sigfig("0") . " = 1 ? (convention)<br>";
echo "0.0: " . get_sigfig("0.0") . " = 1 ? (convention)<br>";
echo "-5.2: " . get_sigfig("-5.2") . " = 2 ? (sign ignored)<br>";
echo "+123: " . get_sigfig("+123") . " = 3 ? (sign ignored)<br>";
echo "12300000000: " . get_sigfig("12300000000") . " = 3 ?<br>";

[PhilippImhof]

Thanks @FloMiLe

I will take care of that, and I hope to be able to do it sooner rather than later. I will keep you posted.

[FloMiLe]

Additionally, I would like to suggest to somehow determine and store the number of significant figures for an answer before unit conversions are applied if that is possible. With the example question attached below, determining the number of significant figures of a student's answer given in years fails because of the unit conversion to seconds applied beforehand.

questions-Zerfallsgesetz (Halbwertszeit, Aktivität, Zeit)-20250626-2137.xml.zip

[PhilippImhof]

For most things, there is no "before" and "after" unit conversion. If the units are compatible, a conversion factor is calculated and the student's answer is multiplied by that factor before the evaluation of the grading criterion.

I can imagine (but I haven't thought that through yet) adding one more special variable (like we currently have _a or _r) that would hold the student's answer pre-conversion and could then be used in an advanced grading criterion.

[FloMiLe]

I very much look forward to these features being added.

Upon review of the function get_sigfig, I noticed that it might make sense to allow for an optional flag to also count trailing zeros in whole numbers (i.e. without a decimal point) as that is often the convention in German textbooks. So here would a possible implementation:

/**
 * Option flag: Count trailing zeros in whole numbers as significant.
 * E.g., for "1200", if this flag is set, it will return 4 sig figs.
 * If not set, it will return 2 sig figs (default behavior).
 */
const SIGFIG_OPTION_COUNT_TRAILING_ZEROS_NO_DECIMAL = 1;

/**
 * Calculates the number of significant figures in a numerical string.
 * Handles standard notation and scientific notation (e.g., 1.23E4).
 * Preserves trailing zeros after a decimal point.
 *
 * @param mixed $x The number as a string or number type. Will be cast to string.
 * @param int $options A bitmask of options. Use SIGFIG_OPTION_ constants.
 *                     Defaults to 0 (no special options).
 * @return int The number of significant figures. Returns 1 for zero by convention.
 */
function get_sigfig($x, int $options = 0): int
{
    // Cast the input to a string to preserve original formatting (especially trailing zeros).
    $num_str = (string) $x;

    // 1. Handle Scientific Notation
    // Look for 'e' or 'E'. The exponent part does NOT affect the number of sig figs.
    $parts = explode('e', str_ireplace('E', 'e', $num_str)); // Use str_ireplace for case-insensitivity
    $mantissa_str = $parts[0];
    // $exponent_str = isset($parts[1]) ? $parts[1] : null; // Exponent is ignored for sig figs

    // 2. Pre-process the Mantissa String
    // Remove leading/trailing whitespace.
    $mantissa_str = trim($mantissa_str);

    // Remove leading plus/minus signs, as they are not significant figures.
    if (substr($mantissa_str, 0, 1) === '+' || substr($mantissa_str, 0, 1) === '-') {
        $mantissa_str = substr($mantissa_str, 1);
    }

    // Check if the mantissa contains a decimal point. This affects trailing zero significance.
    $decimal_found = strpos($mantissa_str, '.') !== false;

    // Create a string containing only the digits for easier processing.
    $digits_only = str_replace('.', '', $mantissa_str);

    // 3. Handle the case where the number is zero (or effectively zero like 0.000)
    // If the digits-only string is all zeros after trimming, it's zero.
    if (trim($digits_only, '0') === '') {
        return 1; // By convention, zero has 1 significant figure.
    }

    // 4. Find the first non-zero digit in the digits-only string
    // strspn counts the length of the initial segment that consists of characters from '0'.
    $first_non_zero_index = strspn($digits_only, '0');

    // 5. Check if the option to count trailing zeros without a decimal is enabled
    $count_trailing_zeros_without_decimal = ($options & SIGFIG_OPTION_COUNT_TRAILING_ZEROS_NO_DECIMAL) !== 0;


    // 6. Calculate the number of significant figures
    if ($decimal_found || $count_trailing_zeros_without_decimal) {
        // If a decimal point is present, OR if the option to count trailing zeros without a decimal is true:
        // All digits from the first non-zero digit to the very end of the digits-only string are significant.
        // This s all trailing zeros.
        $sig_figs = strlen($digits_only) - $first_non_zero_index;
    } else {
        // If no decimal point is present AND we are NOT counting trailing zeros without decimal:
        // Find the index of the last non-zero digit by searching backwards.
        $last_non_zero_index = strlen($digits_only) - 1;
        while ($last_non_zero_index > $first_non_zero_index && $digits_only[$last_non_zero_index] === '0') {
            $last_non_zero_index--;
        }
        // Only the digits from the first non-zero digit up to the last non-zero digit are considered significant.
        $sig_figs = $last_non_zero_index - $first_non_zero_index + 1;
    }

    return $sig_figs;
}

Test cases:

echo "<h2>'English convention'</h2>";
echo "15.440E3: " . get_sigfig("15.440E3") . " = 5 ?<br>";
echo "15.440: " . get_sigfig("15.440") . " = 5 ?<br>";
echo "15440: " . get_sigfig("15440") . " = 4 ?<br>";
echo "15440.0: " . get_sigfig("15440.0") . " = 6 ?<br>";
echo "0.00123: " . get_sigfig("0.00123") . " = 3 ? (leading zeros are not significant)<br>";
echo "0.00120: " . get_sigfig("0.00120") . " = 3 ? (trailing zero after decimal is significant)<br>";
echo "101: " . get_sigfig("101") . " = 3 ? (zero between non-zeros is significant)<br>";
echo "100: " . get_sigfig("100") . " = 1 ? (trailing zeros without decimal not significant)<br>";
echo "100.: " . get_sigfig("100.") . " = 3 ? (trailing zeros with decimal are significant)<br>";
echo "1.00: " . get_sigfig("1.00") . " = 3 ?<br>";
echo "1.0E1: " . get_sigfig("1.0E1") . " = 2 ? (from 1.0)<br>";
echo "0: " . get_sigfig("0") . " = 1 ? (convention)<br>";
echo "0.0: " . get_sigfig("0.0") . " = 1 ? (convention)<br>";
echo "-5.2: " . get_sigfig("-5.2") . " = 2 ? (sign ignored)<br>";
echo "+123: " . get_sigfig("+123") . " = 3 ? (sign ignored)<br>";
echo "12300000000: " . get_sigfig("12300000000") . " = 3 ?<br>";


echo "<h2>'German convention'</h2>";
echo "15.440E3: " . get_sigfig("15.440E3", 1) . " = 5 ?<br>";
echo "15.440: " . get_sigfig("15.440", 1) . " = 5 ?<br>";
echo "15440: " . get_sigfig("15440", 1) . " = 5 ? (trailing zeros are significant)<br>";
echo "15440.0: " . get_sigfig("15440.0", 1) . " = 6 ?<br>";
echo "0.00123: " . get_sigfig("0.00123", 1) . " = 3 ? (leading zeros are not significant)<br>";
echo "0.00120: " . get_sigfig("0.00120", 1) . " = 3 ? (trailing zero is significant)<br>";
echo "101: " . get_sigfig("101", 1) . " = 3 ? (zero between non-zeros is significant)<br>";
echo "100: " . get_sigfig("100", 1) . " = 3 ? (trailing zeros are significant)<br>";
echo "100.: " . get_sigfig("100.", 1) . " = 3 ? (trailing zeros are significant)<br>";
echo "1.00: " . get_sigfig("1.00", 1) . " = 3 ?<br>";
echo "1.0E1: " . get_sigfig("1.0E1", 1) . " = 2 ? (from 1.0)<br>";
echo "0: " . get_sigfig("0", 1) . " = 1 ? (convention)<br>";
echo "0.0: " . get_sigfig("0.0", 1) . " = 1 ? (convention)<br>";
echo "-5.2: " . get_sigfig("-5.2", 1) . " = 2 ? (sign ignored)<br>";
echo "+123: " . get_sigfig("+123", 1) . " = 3 ? (sign ignored)<br>";
echo "12300000000: " . get_sigfig("12300000000", 1) . " = 11 ?<br>";

[FloMiLe]

@PhilippImhof, please don't take this as offensive or bugging. I just noticed that you're already past the 6.2.0 milestone (https://github.com/FormulasQuestion/moodle-qtype_formulas/milestone/10). So I wanted to ask whether you still plan to include these features around determining the number of significant figures in responses. – Thanks!

[PhilippImhof]

No worries at all @FloMiLe

Yes, you are right, I have passed to 6.2.0. I try to always follow semantic versioning, so when I push a release (e. g. after a bug fix or after the release of a new Moodle version) that contains new features, I always increase the minor version number. This is how Formulas moved to 6.2.x although I was not yet ready with all things I would have wanted to have in that version. (It's not always easy to anticipate how much I will be able to implement.)

And yes I absolutely plan to include that feature. It will most likely happen in combination with other stuff, e. g. the possibility to have empty model answers and/or other functions to be used in the definition of grading vars or the grading criterion.

[FloMiLe]

That is great to hear, and much appreciated!

I very much look forward to streamline grading and make it more robust. As you can see from my grading algorithm below, I rely quite heavily on the determined number of significant figures of the given answer when comparing it with the correct solution. So it would be nice not to get them wrong, e.g. when there are trailing zeros...

origAns = _0;

# Analyze solution for mantissa, exponent, significant figures
expSol = floor( log10(abs(origSol)) );
mantSol = origSol / ( 10**( expSol ));

# Analyze answer for mantissa, exponent, significant figures
expAns = floor( log10(abs(origAns)));
mantAns = origAns / ( 10**( expAns ));
nSigFigAns = 1;
for (i:[0:6]) {
    shiftedNum = round( 10**(i) * abs(mantAns) , 6 );
    divTest = round( shiftedNum - floor(shiftedNum) , 6);
    nSigFigAns = ( divTest > 1e-5 ) ? nSigFigAns+1 : nSigFigAns ;
}

# Correct exponent
critExp= ( abs( expAns - expSol ) < 0.1 );

# Correct number of significant figures
critSigFig = ( abs( nSigFigAns - nSigFig ) < 0.1 );

# Correct mantissa with respect to significant figures
mantSolSFA = sigfig( mantSol , nSigFigAns );
critMantCorrect = ( abs( mantAns - mantSolSFA ) < 0.5 * 0.1**(nSigFigAns-1) );

# Mantissa with rounding errors with respect to significant figures
critMantRnd1 = ( abs( mantAns - mantSolSFA ) < 1.5 * 0.1**(nSigFigAns-1) );
critMantRnd2 = ( abs( mantAns - mantSolSFA ) < 2.5 * 0.1**(nSigFigAns-1) );

# Calculate grade
critMant = max( critMantCorrect, 0.8*critMantRnd1, 0.5*critMantRnd2 );
grade = ( critMant > 0 ) ? 0.3*critExp + 0.1*critSigFig + 0.6*critMant : 0;


# Feedback
feedback = feedback
+ ( ( critMant < 1) ? "The mantissa is " + ( ( critMant > 0) ? "almost correct." :  "wrong." ) + "<br>" : "" )
+ ( ( critExp < 1) ? "The order of magnitude is off.<br>" : "" )
+ ( ( nSigFigAns - nSigFig > 0.1) 
        ? "The number of significant figures is " + str(nSigFigAns) + " and therefore too high.<br>" : "" )
+ ( ( nSigFig - nSigFigAns > 0.1) 
        ? "The number of significant figures is " + str(nSigFigAns) + " and therefore too low.<br>" : "" )
;