Z Score Calculator

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z score calculator
Calculate Z-score (z)Calculate Observed Value (x)Calculate Population Mean (μ)Calculate Standard Deviation (σ)
Answer:
function updateLabels(){var mode=document.getElementById('calc_mode').value;var l1=document.getElementById('label1');var l2=document.getElementById('label2');var l3=document.getElementById('label3′);if(mode=='z'){l1.innerHTML='Observed Value (x):';l2.innerHTML='Population Mean (μ):';l3.innerHTML='Standard Deviation (σ):';}else if(mode=='x'){l1.innerHTML='Z-score (z):';l2.innerHTML='Population Mean (μ):';l3.innerHTML='Standard Deviation (σ):';}else if(mode=='m'){l1.innerHTML='Observed Value (x):';l2.innerHTML='Z-score (z):';l3.innerHTML='Standard Deviation (σ):';}else if(mode=='s'){l1.innerHTML='Observed Value (x):';l2.innerHTML='Population Mean (μ):';l3.innerHTML='Z-score (z):';}}function getPValue(z){z=Math.abs(z);var b1=0.319381530;var b2=-0.356563782;var b3=1.781477937;var b4=-1.821255978;var b5=1.330274429;var p=0.2316419;var c=0.39894228;if(z>=6.0){return 1.0;}var t=1.0/(1.0+p*z);var cp=1.0-c*Math.exp(-z*z/2.0)*(t*(b1+t*(b2+t*(b3+t*(b4+t*b5)))));return cp;}function calculateZ(){var mode=document.getElementById('calc_mode').value;var v1=parseFloat(document.getElementById('input1').value);var v2=parseFloat(document.getElementById('input2').value);var v3=parseFloat(document.getElementById('input3').value);var ansDiv=document.getElementById('answer');var resDiv=document.getElementById('final_res');var stepDiv=document.getElementById('step_box');var showSteps=document.getElementById('show_steps').checked;if(isNaN(v1)||isNaN(v2)||isNaN(v3)){alert('Please enter valid numeric values.');return;}var result=0;var steps=";if(mode=='z'){result=(v1-v2)/v3;steps='z = (x – μ) / σ
z = ('+v1+' – '+v2+') / '+v3+'
z = '+(v1-v2).toFixed(4)+' / '+v3+'
z = '+result.toFixed(4);var p=getPValue(result);var p_lt=result<0? (1-p):p;var p_gt=1-p_lt;resDiv.innerHTML='z = '+result.toFixed(4);steps+='

Probabilities:
P(X < x) = '+p_lt.toFixed(5)+'
P(X > x) = '+p_gt.toFixed(5);}else if(mode=='x'){result=(v1*v3)+v2;steps='x = (z * σ) + μ
x = ('+v1+' * '+v3+') + '+v2+'
x = '+(v1*v3).toFixed(4)+' + '+v2+'
x = '+result.toFixed(4);resDiv.innerHTML='x = '+result.toFixed(4);}else if(mode=='m'){result=v1-(v2*v3);steps='μ = x – (z * σ)
μ = '+v1+' – ('+v2+' * '+v3+')
μ = '+v1+' – '+(v2*v3).toFixed(4)+'
μ = '+result.toFixed(4);resDiv.innerHTML='μ = '+result.toFixed(4);}else if(mode=='s'){result=(v1-v2)/v3;steps='σ = (x – μ) / z
σ = ('+v1+' – '+v2+') / '+v3+'
σ = '+(v1-v2).toFixed(4)+' / '+v3+'
σ = '+result.toFixed(4);resDiv.innerHTML='σ = '+result.toFixed(4);}ansDiv.style.display='block';if(showSteps){stepDiv.innerHTML=steps;stepDiv.style.display='block';}else{stepDiv.style.display='none';}}function resetCalc(){document.getElementById('answer').style.display='none';document.getElementById('step_box').style.display='none';}

Calculator Use

The z score calculator is a professional tool used to calculate the "standard score," which represents how many standard deviations a data point is from the mean of a population. This calculator is essential for statistics students, researchers, and data analysts who need to normalize data or compare scores from different distributions.

By using this tool, you can solve for any of the four variables in the z-score formula: the Z-score itself, the raw observed value (x), the population mean (μ), or the standard deviation (σ).

Observed Value (x)
The raw score or specific data point you are analyzing.
Population Mean (μ)
The average value of the entire dataset or population.
Standard Deviation (σ)
A measure of the amount of variation or dispersion in the set of values.
Z-score (z)
The final calculated standard score indicating the distance from the mean.

How It Works

The calculation is based on the standard normal distribution. When you convert a value to a z-score, you are effectively "standardizing" it so that the mean becomes 0 and the standard deviation becomes 1. The fundamental formula used by our z score calculator is:

z = (x – μ) / σ

  • x is the raw score to be transformed.
  • μ (mu) is the mean of the population.
  • σ (sigma) is the standard deviation of the population.

A positive z-score indicates the value is above the mean, while a negative z-score indicates it is below the mean. A z-score of zero means the value is exactly the same as the mean.

Calculation Example

Example: Suppose you took a math test and scored 85. The class mean was 75, and the standard deviation was 5. How many standard deviations away from the mean is your score?

Step-by-step solution:

  1. Identify Observed Value (x) = 85
  2. Identify Mean (μ) = 75
  3. Identify Standard Deviation (σ) = 5
  4. Apply the formula: z = (85 – 75) / 5
  5. Subtract: 10 / 5
  6. Result: z = 2.0

In this case, your score is 2 standard deviations above the average. Using a standard normal table, this z-score would tell us you scored better than approximately 97.7% of the class.

Common Questions

What is a "good" z-score?

In most standardized contexts, a "good" z-score depends on whether you want to be above or below the mean. For tests or income, a positive z-score (e.g., +1.5 or +2.0) is usually better. For things like golf scores or debt, a negative z-score is often preferred.

How do you interpret a Z-score of 0?

A z-score of 0 means the data point is exactly equal to the mean. It represents the 50th percentile in a perfectly normal distribution.

Is the Z-score the same as a P-value?

No. The z-score tells you the distance from the mean in units of standard deviation. The p-value tells you the probability of observing a value at least as extreme as your score, given the null hypothesis is true. You use the z-score to look up the p-value in a standard normal table.

What if the Standard Deviation is zero?

If the standard deviation is zero, all values in the dataset are identical to the mean. In this case, the z-score is mathematically undefined because you cannot divide by zero.

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