Calculate Probability
Z-Score Converter
Convert between raw values and Z-scores using your current mean and standard deviation.
Empirical Rule Ranges
Automatically computed from your mean and standard deviation above.
Standard Normal Z-Table
P(Z ≤ z) values. Click any cell to highlight. Rows = tenths place, Columns = hundredths place.
Formulas & Theory
The normal distribution (Gaussian distribution) is a continuous probability distribution symmetric about its mean. It is parameterised by the mean \(\mu\) and standard deviation \(\sigma\).
1. Probability Density Function (PDF)
The PDF gives the height of the curve at point \(x\). It is not a probability itself; probabilities are areas under the curve.
2. Cumulative Distribution Function (CDF)
3. Z-Score (Standardisation)
Transforms any normal distribution to the standard normal \(N(0,1)\), enabling use of standard Z-tables.
4. Inverse CDF (Quantile Function)
5. Key Properties
Mean = Median = Mode = \(\mu\). Total area under the PDF = 1. The curve is asymptotic; it approaches but never touches the x-axis. The inflection points occur at \(\mu \pm \sigma\).
6. Empirical Rule
For any normal distribution: \(\approx 68.27\%\) of values lie within \(\pm 1\sigma\); \(\approx 95.45\%\) within \(\pm 2\sigma\); \(\approx 99.73\%\) within \(\pm 3\sigma\). This is derived from exact CDF values, not rounded estimates.
Worked Examples (\(\mu=50,\;\sigma=10\))
Example 1 — PDF: Density at \(x=60\).
Example 2 — CDF: \(P(X\le65)\).
Example 3 — Upper tail: \(P(X\ge70)\).
Example 4 — Between: \(P(40\le X\le60)\).
Exactly the 68% of the empirical rule since \(40=\mu-\sigma\) and \(60=\mu+\sigma\).
Example 5 — Inverse CDF: Find \(x\) such that \(P(X\le x)=0.95\).
Frequently Asked Questions
1. What is the normal distribution?
The normal distribution is a continuous probability distribution that forms a symmetric bell-shaped curve. It is fully defined by its mean (centre) and standard deviation (spread). Most natural phenomena such as heights, measurement errors and test scores approximate this distribution.
2. What is the difference between PDF and CDF?
The PDF (Probability Density Function) gives the relative likelihood at a single point; it is the height of the bell curve. The CDF (Cumulative Distribution Function) gives the total probability that X is less than or equal to a value; it is the area under the PDF curve from negative infinity to that point.
3. What is a Z-score and why is it useful?
A Z-score measures how many standard deviations a value is above or below the mean. It standardises values from any normal distribution into the standard normal N(0,1), enabling comparison across different distributions and use of a single probability table.
4. What does P(X = x) equal for a continuous distribution?
Exactly zero. Continuous distributions assign zero probability to individual points. The PDF value at x represents density, not probability. Probabilities are only meaningful over intervals (areas under the curve).
5. What is the Empirical Rule (68-95-99.7)?
Approximately 68.27% of data falls within ±1σ, 95.45% within ±2σ, and 99.73% within ±3σ of the mean. These are exact values from the standard normal CDF, providing a quick mental model of distribution spread.
6. How does the standard deviation affect the bell curve shape?
A smaller σ produces a taller, narrower curve (data tightly clustered). A larger σ produces a shorter, wider curve (data more spread out). The total area always equals 1 regardless of shape.
7. What is the inverse CDF (quantile function)?
The inverse CDF finds the value x at which the cumulative probability equals a specified p. For example, the 95th percentile is the x such that P(X ≤ x) = 0.95. It is used to find critical values and confidence interval boundaries.
8. Can the normal distribution take negative values?
Yes. The normal distribution extends from −∞ to +∞ in theory. For variables that cannot be negative (like heights or incomes), other distributions like log-normal may be more appropriate.
9. What is the Central Limit Theorem?
The CLT states that the sampling distribution of the mean of any independent, identically distributed random variables approaches a normal distribution as sample size grows, regardless of the original distribution. This underpins most inferential statistics.
10. How do I calculate P(x1 ≤ X ≤ x2)?
Compute P(X ≤ x2) - P(X ≤ x1) using the CDF at both values. This equals the area under the bell curve between x1 and x2, representing the probability of a value falling in that interval.
11. What is the standard normal distribution?
The standard normal distribution N(0,1) has mean = 0 and standard deviation = 1. Every normal distribution can be transformed to it using Z = (x − μ)/σ, allowing use of a single universal probability table (the Z-table).
12. Are normal distribution calculations exact?
The CDF has no simple closed form and is calculated using numerical approximations of the error function (erf). Modern implementations achieve accuracy to many decimal places, well beyond practical requirements.
13. What are real-world applications of the normal distribution?
Quality control (product dimensions), finance (daily returns approximation), medicine (blood pressure, cholesterol), engineering (tolerances and measurement errors), education (standardised test scores), and reliability analysis all use the normal distribution.
14. What is skewness and how does it relate to normal distributions?
Skewness measures asymmetry. A perfect normal distribution has zero skewness. Positive skew means a longer right tail; negative skew means a longer left tail. Many real datasets are approximately but not perfectly normal.
15. What is kurtosis in normal distributions?
Kurtosis measures tail heaviness. The normal distribution has excess kurtosis = 0 (mesokurtic). Positive excess kurtosis (leptokurtic) means heavier tails with more extreme values; negative (platykurtic) means lighter tails.