a simple explanation of the P-value that used in the statistics!

1. 1
P-Value
Preface:
Before starting with p-value, one should be familiar with the basics of statistics, particularly hypothesis
testing. Hypothesis testing, which constitutes the core of application of statistics in research field, is a
procedure in which a perception about phenomenon under investigation is tested for validity. The
hypothesis which is tested is called the null hypothesis (H0) and it is considered the default hypothesis. The
reader is referred to chapter of Hypothesis testing, however briefly we go through the steps of hypothesis
testing for refreshment:
 Statement of Null hypothesis (h0) and alternative hypothesis (H1)
 Determination of the critical value or level of significance (α), commonly in biomedical
researches is 0.05.
 Calculation of test statistic e.g.: calculated t (t cal)
 Comparison between the test statistic value and the critical value
 Making decision regarding the null hypothesis (rejection = significance, or fail to reject H0 =
non significance)
 Put a conclusion regarding the phenomenon under investigation.
Introduction:
P-value is the level of significance at which the observed value of the test statistic (e.g. z, or t etc.) would
just be significant, that is, would just fall into the critical region. In other words the whole or part of critical
region could be p-value.
It is a value indicates the possibility of doing an error; the error in this context is called type-I error or α-
error. Type-I error is rejecting the null hypothesis when is true, in other words, it is the situation where there
is no significance in reality and we sustain the significance. Often there is believe that p-value means
probability, while the truth says that they are different things but they are related to each other.
Myths and facts in regard with the p-value:
Myths Facts
P-value is a probability of an event It is the portability of an event plus probability of other events
that have the same or less probabilities
p-value means critical value α Not true, it is either equal/less than α in case of significance or
more than α in case of insignificance.
p-value is always a point value
under the distribution curve
Since it includes several probabilities it represents area of density
under the distribution curve.

2. 2
Explanation:
Say we flip a fair coin (mind that we claim a null hypothesis considering the coin is fair. Fair coin means
that it has one face with head and the other face with tail, which means each two flips should result in 50%
to 50% probability of getting heads and tails. If you get some think else such as to successive repetition of
heads in several flips, that would mean something is not OK or an error… keep this in your mind! Back to
the example, we were saying that if we flip a fair coin two times, in the first flip there will be 50%
probability to get heads and 50% for tails. Again in the second flip there will be 50% probability for heads
and 50% for tails. Figure shows that the probability of getting heads in both successive flips equals ¼= 25%
and the probability of getting two tails is also 25%. On the other hand, the probability of getting heads and
tails together (no matter the order) in the two successive flips equals 2/4 = 50%.
What if we flip the coin 6 times, what would be the probability of getting heads in all 6 tosses? The
probability would be 0.56
= 0.015625 and we will rethink about fairness of the coin, we would say that the
coin is mostly unfair (the two face are heads) and we reject the null hypothesis with a confidence equals 1-
0.015625= 98.4375%. Compare this with the situation getting two heads or two tails from two flips where
the confidence is 75% that is not enough to reject the null hypothesis!
Instead of H meaning head assume it is one allele for a gene and T is the other allele for a gene. Consider
a Mather is heterozygote for this gene possessing H and T (half by half) and dad also is heterozygote as
well. So the outcomes of offspring after two pregnancies is the same as the example of flipping a coin two
times.

3. 3
If the Mather is homozygote (HH) and the father is heterozygote (HT) so the probability for getting
offspring with HH genotype is 50%, and the other 50% will go to HT while no probability for TT.
Now, let us go back to the original situation where both parents are heterozygote, by now we learned the
probability of HH. Lest define the p-value for HH.
Definition of P-value:
It is the probability that random chance generated the data (HH) or something else which is equal or
rarer. So from this definition we can understand that P-value for a data is not only the probability of
occurring that data only, it rather includes the probabilities of any data that have the same probability of
occurrence or less. That is why p-value is an area under the curve not a point and it is written in text as P ≤
0.05 in case of significance. P-value = probability of event (data) + any other equal probability in the data-
set + any lesser probability in the data-set.
In the example of the genotypes the P-value of HH consists of three parts: the probability of getting HH
offspring genotype, the probability of getting TT offspring genotype, which is equal to the probability of
HH, and the probability of any genotype which is rarer than HH. So p-value of HH = 0.25 + 0.25 + 0.0 =
0.5. Now, we can make out the P-value of HH that equals 50% is not the probability of HH which equals
25%.
If we flip fair coin 6 time (26
= 64), there will be 64 outcomes as shown in the figure. Let us calculate the
P-value of getting 6 head. As we know for the definition, P-value = probability of the event (6 heads) +
probability of the events that equal or less than the probability of the event.

4. 4
Probability of 6 heads (HHHHHH) =1/64 or 0.56
= 0.015625
Probability of equal event 6 tails (TTTTTT) = 1/64 or 0.56
= 0.01562
Probability of events that are less = 0.0 (nothing less than 1 in this circumstance)
So P-value of 6 heads = 0.01562 + 0.01562 = 0.03125
Statistically this finding means that the result is significance i.e. if we consider 0.05 is the critical point α,
and we would reject the null hypothesis that says the coin is fair!
Home work: Calculate: a) the p-value of getting 5 heads if you toss coin five times. b) Calculate the p-
value of getting 4 heads and 1 tails.
The answer will show that there is no statistical significance and you will fail to reject the null hypothesis
in both (a) and (b)!