Normal Distribution Calculator

by
Normal Distribution Calculator
P(X < x) – Area to the LeftP(X > x) – Area to the RightP(x1 < X < x2) – Area BetweenP(X x2) – Area Outside
Result:
Please enter your values and click calculate.
function updateInputs(){var type=document.getElementById('calc_type').value;var row2=document.getElementById('row_x2');var label1=document.getElementById('label_x1′);if(type==='between'||type==='outside'){row2.style.display='table-row';label1.innerHTML='Lower Value (x1):';}else{row2.style.display='none';label1.innerHTML='Value (x):';}}function resetForm(){setTimeout(function(){updateInputs();document.getElementById('answer').innerHTML='Please enter your values and click calculate.';},10);}function getZProb(z){var t=1/(1+0.2316419*Math.abs(z));var d=0.3989423*Math.exp(-z*z/2);var p=d*t*(0.3193815+t*(-0.3565638+t*(1.781478+t*(-1.821256+t*1.330274))));if(z>0)p=1-p;return p;}function calculateResult(){var mu=parseFloat(document.getElementById('mean').value);var sigma=parseFloat(document.getElementById('sd').value);var x1=parseFloat(document.getElementById('x1').value);var x2=parseFloat(document.getElementById('x2').value);var type=document.getElementById('calc_type').value;var showSteps=document.getElementById('steps').checked;if(isNaN(mu)||isNaN(sigma)||isNaN(x1)){alert('Please enter valid numeric values.');return;}if((type==='between'||type==='outside')&&isNaN(x2)){alert('Please enter both x1 and x2.');return;}if(sigma<=0){alert('Standard deviation must be greater than 0.');return;}var z1=(x1-mu)/sigma;var p1=getZProb(z1);var resultText="";var finalProb=0;if(type==='left'){finalProb=p1;resultText="P(X < "+x1+") = "+finalProb.toFixed(5)+"";if(showSteps)resultText+="
Z-score = ("+x1+" – "+mu+") / "+sigma+" = "+z1.toFixed(4)+"";}else if(type==='right'){finalProb=1-p1;resultText="P(X > "+x1+") = "+finalProb.toFixed(5)+"";if(showSteps)resultText+="
Z-score = ("+x1+" – "+mu+") / "+sigma+" = "+z1.toFixed(4)+"
P = 1 – Φ(Z)";}else if(type==='between'){var z2=(x2-mu)/sigma;var p2=getZProb(z2);finalProb=Math.abs(p2-p1);resultText="P("+Math.min(x1,x2)+" < X < "+Math.max(x1,x2)+") = "+finalProb.toFixed(5)+"";if(showSteps)resultText+="
Z1 = "+z1.toFixed(4)+", Z2 = "+z2.toFixed(4)+"
P = |Φ(Z2) – Φ(Z1)|";}else if(type==='outside'){var z2=(x2-mu)/sigma;var p2=getZProb(z2);finalProb=1-Math.abs(p2-p1);resultText="P(X "+Math.max(x1,x2)+") = "+finalProb.toFixed(5)+"";if(showSteps)resultText+="
Z1 = "+z1.toFixed(4)+", Z2 = "+z2.toFixed(4)+"
P = 1 – (Φ(Z_upper) – Φ(Z_lower))";}document.getElementById('answer').innerHTML="
"+resultText+"
Percentage: "+(finalProb*100).toFixed(2)+"%
";}

Using the Normal Distribution Calculator

This normal distribution calculator is a powerful tool for statisticians, students, and researchers. It allows you to calculate the area under the normal curve (probability) for any given set of parameters. Whether you are dealing with standardized test scores, biological measurements, or financial modeling, understanding where a data point falls on the "bell curve" is essential.

To get started, follow these simple steps:

Mean (μ)
Enter the average value of your dataset. In a standard normal distribution, this value is 0.
Standard Deviation (σ)
Enter the measure of variability. This must be a positive number. In a standard normal distribution, this is 1.
Calculation Type
Choose whether you want the probability to the left of a value, to the right, between two values, or outside of two values.

How the Normal Distribution Works

The normal distribution is a continuous probability distribution that is symmetrical on both sides of the mean. This means the right side of the center is a mirror image of the left side. Most of the data values cluster around the central peak (the mean), while the probabilities for values further away from the mean taper off equally in both directions.

To find probabilities, we use the Z-score formula to "standardize" any normal distribution into a Standard Normal Distribution (where mean = 0 and SD = 1). The formula is:

Z = (x – μ) / σ

  • x: The value you are evaluating.
  • μ (Mu): The population mean.
  • σ (Sigma): The population standard deviation.
  • Z: The number of standard deviations the value x is from the mean.

The Empirical Rule (68-95-99.7)

One of the most useful concepts associated with the normal distribution is the Empirical Rule. This rule states that for a normal distribution:

  • Approximately 68.27% of the data falls within one standard deviation (μ ± 1σ).
  • Approximately 95.45% of the data falls within two standard deviations (μ ± 2σ).
  • Approximately 99.73% of the data falls within three standard deviations (μ ± 3σ).

Normal Distribution Calculation Example

Scenario: Suppose the IQ scores of a population are normally distributed with a Mean (μ) of 100 and a Standard Deviation (σ) of 15. What is the probability that a randomly selected person has an IQ greater than 130?

Step-by-Step Solution:

  1. Identify the values: μ = 100, σ = 15, x = 130.
  2. Calculate the Z-score: Z = (130 – 100) / 15 = 2.0.
  3. Use a Z-table or this normal distribution calculator to find the area to the left of Z = 2.0, which is approximately 0.97725.
  4. Since we want the probability of being greater than 130, we calculate the area to the right: P = 1 – 0.97725.
  5. Result: P(X > 130) = 0.02275 or 2.275%.

Common Questions

What is a Standard Normal Distribution?

A Standard Normal Distribution is a special case of the normal distribution where the mean is exactly 0 and the standard deviation is exactly 1. Any normal distribution can be converted to this form using the Z-score formula.

When should I use a T-distribution instead?

You typically use a Normal Distribution (Z-test) when the population standard deviation is known and the sample size is large (usually n > 30). If the population standard deviation is unknown or the sample size is small, a T-distribution is more appropriate.

Can the standard deviation be negative?

No. Standard deviation is a measure of spread and is calculated as the square root of variance. It must always be zero or a positive value. Our calculator will return an error if a negative value is entered.

Leave a Comment