Write the normal distribution as a mathematical function. If you have not memorized this long and complex function, consult a statistics textbook. Use the variables “m” and “s” to represent the mean and standard deviation of the distribution, respectively. Let “n” represent the number of data points.
Take the natural log of the distribution. For example, the "(2pi)^(-n/2)/s^n" piece of the function should become "log[(2pi)^(-n/2)/s^n]."
Simplify the new function according to the properties of the natural log function, bringing the exponents outside and converting multiplication and division to addition and subtraction, respectively. For example, the "log[(2pi)^(-n/2)/s^n]" piece of the function should simplify to "–(1/2)n*log(2pi)-n*log(s)."
Take the derivative of the simplified function with respect to "m." If done correctly, the result will be much less complex: "sigma(xi – m)/s^2," where “sigma” represents the summation function, and “xi” represents the “ith” data point.
Set the derivative equal to zero. You will have the equation "sigma(xi – m)/s^2 = 0."
Multiply both sides of the equation by "s^2" and simplify. The result is "sigma(xi – m) = 0."
Simplify the function "sigma(xi – m)." Since "m" does not rely on how many data points are in the data set, the result is "sigma(xi – m) = sigma(xi) * nm."
Solve for "m." Basic algebra gives the solution "m = sigma(xi)/n."