This p-value calculator is a tool that can help you determine the p-value of a standard normal distribution. It is defined as the probability of obtaining a result equal to or more "extreme" than the observed one. Read on to learn how to find the p-value and what is the p-value formula.

Standard normal distribution

The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution. It is often used in the natural and social sciences.

A variant of normal distribution is the standard normal distribution. It is a special case, in which the mean of all values is equal to 0, and the standard deviation is equal to 1. The total area under the probability density function is also equal to 1. Hence, the area under this graph between any two x-values is equivalent to the probability that the experimental result is in this given range.

How to find the p-value?

p value The p-value is the probability that a result will be equal to or more extreme than the observed one. As probability is equivalent to the area under the graph, p-value is the area under the probability density function to the right of the observed value.

In the standard normal distribution, instead of using the value itself, you need to standarize it by calculating the z-score. It is defined as the number of standard deviations by which a data point is above the mean.

What is the p-value formula?

P-value is difficult to calculate manually. The most popular way to do it is to use a z-score table, in which the area under the probability density function is calculated for each value of z-score. You can also use an integral to calculate the area under the graph. The standard normal distribution function is

Φ(x) = 1/√(2π) * exp (-0.5 * x^2)

If you want to calculate the p-value, you have to find the integral with lower limit equal to the z-score and upper limit equal to infinity.

The easiest way to find the p-value, though, is to use our p-value calculator! Simply type in the value of the z-score and you will have the p-value calculated in no time!

What do the results mean?

Our p-value calculator gives you four different values. They are:

  • Right-tailed p-value (Z > z): the area under the probability density graph to the right of your z-score. This is the p-value "by definition".
  • Left-tailed p-value (Z < z): the area under the probability density graph to the left of your z-score. It is equal to the difference between 1 and the right-tailed p-value.
  • Two-tailed p-value: the area under the probability density graph to the right of your z-score, and to the left of the number opposite to your z-score.
  • Two-tailed confidence level: the area between your z-score and the opposite to this z-score.
p values

The benefits of using the p-value formula

You simply compare the p-value to the level of statistical significance of your choice.

There are three conventional levels: 0.1, 0.05, and 0.01. Sometimes, instead of reporting the p-values, you can find significant levels denoted by "stars" denoted by (), () and (**), respectively.

P-values and statistical inference

P-values provide by far the easiest way to answer the question if parameter estimates you obtain, say in a regression analysis, are statistically significant. Its is often the case that point estimates of a parameter.

Formally, test the hypotheses:

  • Null hypothesis: parameter equals zero
  • Alternative hypothesis: parameter value is different from zero

If the p-value you obtain is larger than your referential significance level, say 0.05, you cannot reject the null hypothesis.

However, when you get a p-value smaller that this, you can say that the estimate is actually different from zero. Hence, there is evidence of a dependence in the data you analyze.

Bogna Haponiuk

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