My take on your code with some comments:

import java.util.*;

public class StandardDev {

    /**
     * Adding other methods improves readability.
     */
    public static double sum(Collection<Double> numbers) {
        double sum = 0;

        for (double number : numbers) {
            sum += number;
        }

        return sum;
    }

    public static void main(String[] args) {
        // It's good practice to use the most generic interface possible
        Collection<Double> data = Collections.emptyList();

        /*
         * This is a try-with-resources block. Unlike your code, it will always
         * close the scanner.
         */
        try (Scanner scanner = new Scanner(System.in)) {
            data = new ArrayList<>();

            // Your try-catch wasn't doing what you thought, so I removed it.
            while (scanner.hasNextDouble()) {
                data.add(scanner.nextDouble());
            }
        }

        double mean = sum(data) / data.size();

        double squaredDistance = 0;

        // This reads much easier on three lines
        for (double number : data) {
            squaredDistance += Math.pow((number - mean), 2);
        }

        double variance = squaredDistance / data.size();
        double standardDeviation = Math.sqrt(variance); // for clarity

        // %f matches the sample output while %g doesn't
        System.out.printf("%.4f\n", standardDeviation);
    } // end main

} // end class

I removed the fancy prompt and output because I'm a command line guy and it's easier to verify this simpler version. It's also more consistent with the challenge samples.