There are various kinds of probability distributions, and each distribution shows the probability of different outcomes for a random experiment. In this section, we'll explore the various kinds of probability distributions.

A normal distribution

A normal distribution is the most common and widely used distribution in statistics. It is also called a "bell curve" and "Gaussian curve" after the mathematician Karl Friedrich Gauss. A normal distribution occurs commonly in nature. Let's take the height example we saw previously. If you have data for the height of all the people of a particular gender in Hong Kong city, and you plot a bar chart where each bar represents the number of people at this particular height, then the curve that is obtained will look very similar to the following graph. The numbers in the plot are the standard deviation numbers from the mean, which is zero. The concept will become clearer as we proceed through the chapter.

Also, if you take an...

A z-score, in simple terms, is a score that expresses the value of a distribution in standard deviation with respect to the mean. Let's take a look at the following formula that calculates the z-score:

Here, X is the value in the distribution, µ is the mean of the distribution, and σ is the standard deviation of the distribution

Let's try to understand this concept from the perspective of a school classroom.

A classroom has 60 students in it and they have just got their mathematics examination score. We simulate the score of these 60 students with a normal distribution using the following command:

>>> classscore
>>> classscore = np.random.normal(50, 10, 60).round()

[ 56.  52.  60.  65.  39.  49.  41.  51.  48.  52.  47.  41.  60.  54.  41.
  46.  37.  50.  50.  55.  47.  53.  38.  42.  42.  57.  40.  45.  35.  39.
  67.  56.  35.  45.  47.  52.  48.  53.  53.  50.  61.  60.  57.  53.  56.
  68.  43.  35.  45.  42.  33.  43.  49.  54.  45.  54.  48.  55.  56.  30...

A p-value is the probability of rejecting a null-hypothesis when the hypothesis is proven true. The null hypothesis is a statement that says that there is no difference between two measures. If the hypothesis is that people who clock in 4 hours of study everyday score more that 90 marks out of 100. The null hypothesis here would be that there is no relation between the number of hours clocked in and the marks scored.

If the p-value is equal to or less than the significance level (α), then the null hypothesis is inconsistent and it needs to be rejected.

Let's understand this concept with an example where the null hypothesis is that it is common for students to score 68 marks in mathematics.

Let's define the significance level at 5%. If the p-value is less than 5%, then the null hypothesis is rejected and it is not common to score 68 marks in mathematics.

Let's get the z-score of 68 marks:

>>> zscore = ( 68 - classscore.mean() ) / classscore.std()
>>> zscore
2.283

Now...

The example in the previous section was an instance of a one-tailed test where the null hypothesis is rejected or accepted based on one direction of the normal distribution.

In a two-tailed test, both the tails of the null hypothesis are used to test the hypothesis.

In a two-tailed test, when a significance level of 5% is used, then it is distributed equally in the both directions, that is, 2.5% of it in one direction and 2.5% in the other direction.

Let's understand this with an example. The mean score of the mathematics exam at a national level is 60 marks and the standard deviation is 3 marks.

The mean marks of a class are 53. The null hypothesis is that the mean marks of the class are similar to the national average. Let's test this hypothesis by first getting the z-score 60:

>>> zscore = ( 53 - 60 ) / 3.0
>>> zscore
-2.3333333333333335

The p-value would be:

>>> prob = stats.norm.cdf(zscore)
>>> prob

0.0098153286286453336...

Type 1 error is a type of error that occurs when there is a rejection of the null hypothesis when it is actually true. This kind of error is also called an error of the first kind and is equivalent to false positives.

Let's understand this concept using an example. There is a new drug that is being developed and it needs to be tested on whether it is effective in combating diseases. The null hypothesis is that it is not effective in combating diseases.

The significance level is kept at 5% so that the null hypothesis can be accepted confidently 95% of the time. However, 5% of the time, we'll accept the rejecttion of the hypothesis although it had to be accepted, which means that even though the drug is ineffective, it is assumed to be effective.

The Type 1 error is controlled by controlling the significance level, which is alpha. Alpha is the highest probability to have a Type 1 error. The lower the alpha, the lower will be the Type 1 error.

The Type 2 error is the kind...

A confidence interval is a type of interval statistics for a population parameter. The confidence interval helps in determining the interval at which the population mean can be defined.

Let's try to understand this concept by using an example. Let's take the height of every man in Kenya and determine with 95% confidence interval the average of height of Kenyan men at a national level.

Let's take 50 men and their height in centimeters:

>>> height_data = np.array([ 186.0, 180.0, 195.0, 189.0, 191.0, 177.0, 161.0, 177.0, 192.0, 182.0, 185.0, 192.0,
  173.0, 172.0, 191.0, 184.0, 193.0, 182.0, 190.0, 185.0, 181.0, 188.0, 179.0, 188.0,
  170.0, 179.0, 180.0, 189.0, 188.0, 185.0, 170.0, 197.0, 187.0, 182.0, 173.0, 179.0,
  184.0, 177.0, 190.0, 174.0, 203.0, 206.0, 173.0, 169.0, 178.0, 201.0, 198.0, 166.0,
  171.0, 180.0])

Plotting the distribution, it has a normal distribution:

>>> plt.hist(height_data, 30, normed=True)
>>> plt.show()

The mean of the...

In statistics, correlation defines the similarity between two random variables. The most commonly used correlation is the Pearson correlation and it is defined by the following:

The preceding formula defines the Pearson correlation as the covariance between X and Y, which is divided by the standard deviation of X and Y, or it can also be defined as the expected mean of the sum of multiplied difference of random variables with respect to the mean divided by the standard deviation of X and Y. Let's understand this with an example. Let's take the mileage and horsepower of various cars and see if there is a relation between the two. This can be achieved using the pearsonr function in the SciPy package:

>>> mpg = [21.0, 21.0, 22.8, 21.4, 18.7, 18.1, 14.3, 24.4, 22.8, 19.2, 17.8, 16.4, 17.3, 15.2, 10.4, 10.4, 14.7, 32.4, 30.4,
       33.9, 21.5, 15.5, 15.2, 13.3, 19.2, 27.3, 26.0, 30.4, 15.8, 19.7, 15.0, 21.4]
>>> hp = [110, 110, 93, 110, 175, 105, 245, 62, 95, 123...

We have already done a few Z-tests before where we validated our null hypothesis.

A T-distribution is similar to a Z-distribution—it is centered at zero and has a basic bell shape, but its shorter and flatter around the center than the Z-distribution.

The T-distributions' standard deviation is usually proportionally larger than the Z, because of which you see the fatter tails on each side.

The t distribution is usually used to analyze the population when the sample is small.

The Z-test is used to compare the population mean against a sample or compare the population mean of two distributions with a sample size greater than 30. An example of a Z-test would be comparing the heights of men from different ethnicity groups.

The T-test is used to compare the population mean against a sample, or compare the population mean of two distributions with a sample size less than 30, and when you don't know the population's standard deviation.

Let's do a T-test on two classes that are given...