Free · Normal distribution

Standard Scores and Normal Tail Probabilities

Standardize a value and compare approximate left, right and two-sided normal probabilities.

Formula & worked example
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Enter the known values in the units shown. Results update as you type.
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Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

A value of 115 with mean 100 and standard deviation 15 has z score 1.

z_score = (experimental_result - mean) / standard_deviation; left_tailed_p_value = (z_score == 0 ? 0.5 : z_score > 0 ? (1+(((abs(z_score)/2) > 3 ? 1 : (abs(z_score)/2) < 1 ? (((((((((((((((((0.000124818987)*(((abs(z_score)/2))^2)+(-0.001075204047)))*(((abs(z_score)/2))^2)+(0.005198775019)))*(((abs(z_score)/2))^2)+(-0.019198292004)))*(((abs(z_score)/2))^2)+(0.059054035642)))*(((abs(z_score)/2))^2)+(-0.151968751364)))*(((abs(z_score)/2))^2)+(0.319152932694)))*(((abs(z_score)/2))^2)+(-0.5319230073)))*(((abs(z_score)/2))^2)+(0.797884560593)))*((abs(z_score)/2))*2 : ((((((((((((((((((((((((((((-0.000045255659)*(((abs(z_score)/2)-2))+(0.00015252929)))*(((abs(z_score)/2)-2))+(-0.000019538132)))*(((abs(z_score)/2)-2))+(-0.000676904986)))*(((abs(z_score)/2)-2))+(0.001390604284)))*(((abs(z_score)/2)-2))+(-0.00079462082)))*(((abs(z_score)/2)-2))+(-0.002034254874)))*(((abs(z_score)/2)-2))+(0.006549791214)))*(((abs(z_score)/2)-2))+(-0.010557625006)))*(((abs(z_score)/2)-2))+(0.011630447319)))*(((abs(z_score)/2)-2))+(-0.009279453341)))*(((abs(z_score)/2)-2))+(0.005353579108)))*(((abs(z_score)/2)-2))+(-0.002141268741)))*(((abs(z_score)/2)-2))+(0.000535310849)))*(((abs(z_score)/2)-2))+(0.999936657524)))))/2 : (1-(((abs(z_score)/2) > 3 ? 1 : (abs(z_score)/2) < 1 ? (((((((((((((((((0.000124818987)*(((abs(z_score)/2))^2)+(-0.001075204047)))*(((abs(z_score)/2))^2)+(0.005198775019)))*(((abs(z_score)/2))^2)+(-0.019198292004)))*(((abs(z_score)/2))^2)+(0.059054035642)))*(((abs(z_score)/2))^2)+(-0.151968751364)))*(((abs(z_score)/2))^2)+(0.319152932694)))*(((abs(z_score)/2))^2)+(-0.5319230073)))*(((abs(z_score)/2))^2)+(0.797884560593)))*((abs(z_score)/2))*2 : ((((((((((((((((((((((((((((-0.000045255659)*(((abs(z_score)/2)-2))+(0.00015252929)))*(((abs(z_score)/2)-2))+(-0.000019538132)))*(((abs(z_score)/2)-2))+(-0.000676904986)))*(((abs(z_score)/2)-2))+(0.001390604284)))*(((abs(z_score)/2)-2))+(-0.00079462082)))*(((abs(z_score)/2)-2))+(-0.002034254874)))*(((abs(z_score)/2)-2))+(0.006549791214)))*(((abs(z_score)/2)-2))+(-0.010557625006)))*(((abs(z_score)/2)-2))+(0.011630447319)))*(((abs(z_score)/2)-2))+(-0.009279453341)))*(((abs(z_score)/2)-2))+(0.005353579108)))*(((abs(z_score)/2)-2))+(-0.002141268741)))*(((abs(z_score)/2)-2))+(0.000535310849)))*(((abs(z_score)/2)-2))+(0.999936657524)))))/2); right_tailed_p_value = one_way(1 - left_tailed_p_value); two_tailed_p_value = 2*min(left_tailed_p_value, right_tailed_p_value); two_tailed_confidence_level = one_way(1 - two_tailed_p_value)

Worked example

Enter these known values and leave the other values blank.

Mean value (μ): 100
Standard deviation (σ): 15
Experimental result (X): 115

Z-score (z): 1
Left tailed p-value: 0.8413
Right tailed p-value: 0.15866
Two tailed p-value: 0.3173
Two tailed confidence level: 0.6827

Assumptions and limitations

A z score is (value−mean)/standard deviation, with strictly positive standard deviation. Tail probabilities assume a normal distribution and retain the captured polynomial approximation, clamped outside ±6; they are not an exact numerical integral.
All original editable controls remain available. Clear one unknown and provide sufficient independent inputs to solve a supported inverse; outputs marked read-only cannot be entered.

Common questions

Are the probabilities exact?

A z score is (value−mean)/standard deviation, with strictly positive standard deviation. Tail probabilities assume a normal distribution and retain the captured polynomial approximation, clamped outside ±6; they are not an exact numerical integral.

Can I solve for the mean or standard deviation?

Clear the target and provide the other values. Standard deviation must be positive. A zero score with a value equal to the mean does not determine the standard deviation.

References

Replace external quantized normal-tail images with an original accessible description and current score/tail result references; preserve every captured image variant condition. All numeric controls and computed tail values are retained. Translate the complete captured pLeft helper into its signed piecewise Horner polynomials using decimal arithmetic. Retain all score inverses and four tail outputs, including the captured clamp beyond six standard deviations. JavaScript binary rounding is removed; the captured approximation is disclosed rather than asserted as an exact normal CDF.

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