P-value Calculator

Table of contents:
This calculator will enable you to find the probability that a hypothesis you might have holds true. An example is that is might help you find the probability that your next partner will be a certain height, given the heights of the population in your area, or the probability that you are going to pass a difficult test.

What is a p-value?

In statistics, we form a hypothesis, usually about a relationship between two or more variables, and then we run operations on a dataset to test whether the relationship we assume is true. We call this hypothesis testing. We usually start by testing a Null Hypothesis – which is that no relationship exists. What we’re actually doing is disproving that there is no relationship. If we find that there is significant evidence to disprove this hypothesis, then we know a relationship exists.

A p-value is a measure of probability, or significance. This means that if the result is found to be significant, and that the assumed relationship does exist, then the alternative hypothesis can be relied on, and the test is proof that the facts hold true. It helps us to use a sample of a broader population to make a conclusion about the population itself.

Tests of p-value, to test significance, can only be applied to data which follows the standard normal distribution – the bell curve. 
The bell curve of a normal distribution is a line graph of values – your dataset, or set of observations. A normal distribution is symmetrical around the mean value, and the mean, median and mode are equal, in the strictest sense. This means that the average value appears most frequently, and lies in the middle of the highest and lowest values in the dataset. The distribution is defined by the mean and the standard deviation from the mean. These values depict how steep or wide the curve is around the mean, but the rule applies. A standard normal distribution is a specific variation of this, which follows specific rules with regards to the standard deviation around the mean value. This is the only distribution for which the p-value approach to calculating probability applies.

The area under the graph represents probability, by differential calculus, we can calculate the probability that any value on this line will occur. The p-value is thus the area under the graph to right of the observed value (remember the p-value represents the probability that the value is equal to or greater than the one we have observed).
When we run the test of the relationship, or expected value of an outcome, and find that the relationship, or value is as expected, the p-value represents a probability (the p actually does stand for probability) that we will find this relationship, or value, or an even greater one, in similar cases, between the same tested variables. By sampling adequately, we can use the sample to draw conclusions about the population.

The line represents the distribution of observations, and the area between any two points on this line represents the probability that the result of your experiment will land between those two values. In the case of the standard normal distribution, we can use the z-statistic to find the p-value. The z-score is how far away from the mean we are in a given situation, the one we are testing. It helps to lay down a few parameters. It is calculated by seeing how many standard deviations we are away from the mean. The z-statistic is the difference between our sampled value, and the value assumed by the null hypothesis, divided by the standard deviation of our sample. The standard deviation is how much our variables differ from the mean – this gives us an idea of how spread out the data is from the average or mean value. The easiest way to calculate the p-value is using this calculator, but it helps to understand these basics.

How to use the test:

Someone told you it takes 5 minutes to get to the park from point A. You are sure it takes longer than that! You try the drive a few times and time it. Your null hypothesis in this case is that is takes 5 minutes. You alternative Hypothesis is that it takes longer then 5 minutes! These measures are the data points you need to run through the p-value calculator. Bear in mind, we have made some strong assumptions in this experiment!
Your Null Hypothesis is stated as follows:

H0 : The trip takes 5 minutes
H1: The trip does not take five minutes,

Interpreting the results:

We start by assuming that the null hypothesis is true, then the p-value is a calculation of the probability that we will get the value described by this hypothesis. SO if the probability is really small, then we can imagine that this hypothesis is not true – that is, that it does not, in fact, take five minutes in the example above. 
The p-value is a number between 0 and 1. This is because the area under the normal distribution is equal to 1, so given that the p value lies in this range, it cannot exceed 1. 

This is how we interpret a p-value:
	A p-value which is very small, p ≤ 0,05, means that the probability that the Null hypothesis is true is very low, and so we can reject the Null Hypothesis’
	A large p-value, p > 0,05 means we should not reject the Null Hypothesis.

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