Normal Distribution Calculator
Normal Distribution Calculator
Enter a value (x), the distribution's mean, and its standard deviation. This calculates the cumulative probability of landing below x and the complementary probability of landing above x, using the standard normal (bell curve) model.
The normal distribution describes an enormous range of real-world measurements that cluster around an average — heights, test scores, measurement errors, and many natural and manufacturing processes all approximate a bell curve. Internally, the calculator converts your value to a z-score (how many standard deviations x is from the mean) and evaluates the area under the curve up to that point using the cumulative distribution function; since every outcome falls either above or below x, the "below" and "above" probabilities always add up to exactly 1.
- Uses the cumulative distribution function (CDF): converts x to a z-score ((x − mean) ÷ standard deviation) and evaluates the area under the bell curve up to that point.
- Below + above always sum to 1: since every outcome falls either below or above x, the two probabilities are complementary.
- Different from the Z-Score Calculator: this tool goes a step further than converting to a z-score — it converts that z-score all the way into an actual probability/percentage.
What does "probability below x" actually mean?
It's the proportion of the entire distribution that falls at or below the value x — for example, if x is one standard deviation above the mean, roughly 84% of the distribution falls below it.
Does this work for any data, or only "normal" data?
This specifically models data that follows (or approximately follows) a normal/bell-curve distribution — many real-world measurements do (heights, test scores, measurement errors), but skewed or bounded data may not fit this model well.
Normal Distribution Calculator
Enter a value (x), the distribution's mean, and its standard deviation. This calculates the cumulative probability of landing below x and the complementary probability of landing above x, using the standard normal (bell curve) model.
The normal distribution describes an enormous range of real-world measurements that cluster around an average — heights, test scores, measurement errors, and many natural and manufacturing processes all approximate a bell curve. Internally, the calculator converts your value to a z-score (how many standard deviations x is from the mean) and evaluates the area under the curve up to that point using the cumulative distribution function; since every outcome falls either above or below x, the "below" and "above" probabilities always add up to exactly 1.
- Uses the cumulative distribution function (CDF): converts x to a z-score ((x − mean) ÷ standard deviation) and evaluates the area under the bell curve up to that point.
- Below + above always sum to 1: since every outcome falls either below or above x, the two probabilities are complementary.
- Different from the Z-Score Calculator: this tool goes a step further than converting to a z-score — it converts that z-score all the way into an actual probability/percentage.
What does "probability below x" actually mean?
It's the proportion of the entire distribution that falls at or below the value x — for example, if x is one standard deviation above the mean, roughly 84% of the distribution falls below it.
Does this work for any data, or only "normal" data?
This specifically models data that follows (or approximately follows) a normal/bell-curve distribution — many real-world measurements do (heights, test scores, measurement errors), but skewed or bounded data may not fit this model well.