Percentile: 84.13% (this value is higher than about 84.1% of a normal distribution with this mean and standard deviation)
Where This Value Falls on the Distribution
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How This Calculator Works
A z-score measures how many standard deviations a value sits above or below a distribution’s
mean. Enter a value, the distribution’s mean, and its standard deviation to find the z-score and
the corresponding percentile — what share of a normal distribution falls at or below that value.
A z-score of 0 means the value equals the mean exactly. Positive z-scores are above the mean;
negative z-scores are below it. Because roughly 68% of a normal distribution falls within 1
standard deviation of the mean, 95% within 2, and 99.7% within 3, a z-score alone gives an
immediate sense of how unusual a value is.
The Formula
Z-Score=Standard DeviationValue−Mean
The percentile comes from the standard normal distribution’s cumulative distribution function —
the probability that a randomly drawn value from that distribution falls at or below the entered
z-score, computed here via a well-established polynomial approximation of the error function.
Worked Example
An IQ score of 115, where IQ scores have a mean of 100 and a standard deviation of 15:
Z-score: 115−100=15, then 15÷15=1.0.
Percentile: a z-score of exactly 1 corresponds to the 84.13th percentile — this value is
higher than about 84% of the distribution.
Cómo funciona esta calculadora
Un puntaje z mide cuántas desviaciones estándar se encuentra un valor por encima o por debajo
de la media de una distribución. Ingresa un valor, la media de la distribución y su desviación
estándar para hallar el puntaje z y el percentil correspondiente — qué proporción de una
distribución normal cae en o por debajo de ese valor.
Un puntaje z de 0 significa que el valor es exactamente igual a la media. Los puntajes z
positivos están por encima de la media; los puntajes z negativos están por debajo. Debido a que
aproximadamente el 68% de una distribución normal cae dentro de 1 desviación estándar de la
media, el 95% dentro de 2, y el 99.7% dentro de 3, un puntaje z por sí solo da una idea inmediata
de cuán inusual es un valor.
La fórmula
Puntaje Z=Desviacioˊn estaˊndarValor−Media
El percentil proviene de la función de distribución acumulada de la distribución normal estándar
— la probabilidad de que un valor tomado al azar de esa distribución caiga en o por debajo del
puntaje z ingresado, calculada aquí mediante una aproximación polinómica bien establecida de la
función de error.
Ejemplo resuelto
Un puntaje de CI de 115, donde los puntajes de CI tienen una media de 100 y una
desviación estándar de 15:
Puntaje Z: 115−100=15, luego 15÷15=1.0.
Percentil: un puntaje z de exactamente 1 corresponde al percentil 84.13 — este valor es
más alto que aproximadamente el 84% de la distribución.
A z-score is how many standard deviations a value is from the mean. A z-score of 0 means the value equals the mean; positive z-scores are above it, negative z-scores are below it — the bigger the magnitude, the further (and more unusual) the value is.
What is the difference between a z-score and a percentile?
A z-score describes distance from the mean in standard deviations. A percentile converts that into 'what share of the distribution falls at or below this value' — a more intuitive number for most people, even though both describe the same underlying position.
Does this only work for normal (bell-curve) distributions?
The percentile conversion specifically assumes a normal distribution — the z-score itself (how many standard deviations from the mean) can be computed for any distribution, but the percentile figure only accurately reflects real-world data if that data is actually roughly normally distributed.
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