The technique used to check quantitatively whether the given data distribution is close to Gaussian distribution is

Gaussian distribution:

• All Gaussian distributions can be standardized to the reference Gaussian distribution, which is called the standard Gaussian distribution. Standardization in general is accomplished by subtracting the center of the distribution from a given element in the distribution and dividing the result by the standard deviation of the distribution.
• The distribution of a standardized Gaussian distribution—that is, a Gaussian distribution that has its elements standardized in this form—has its center at zero and has a variance of unity.
• Gaussian distribution (also known as the normal distribution) is a bell-shaped curve,
• If a distribution is normal, then the values of the mean, median, and mode are the same. However, the value of the mean, median, and mode may be different if the distribution is skewed (not Gaussian distribution). 
• The standard deviation of the mean used to check the given data distribution is close to Gaussian distribution.

The probability density function of a zero-mean Gaussian variable is as shown:

The probability distribution function of a Gaussian Random Variable is defined as;

\(f\left( x \right) = \frac{1}{{\sqrt {2\pi {\sigma ^2}} }}\;{e^{\frac{{{{\left( {x - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}\)

Given distribution has zero mean i.e. μ = 0, so the above distribution can be written as:

\(f\left( x \right) = \frac{1}{{\sqrt {2\pi {\sigma ^2}} }}\;{e^{\frac{{{x^2}}}{{2{\sigma ^2}}}}}\)