Probability

In order to understand what a normal distribution is, there are two important terms you should familiarize yourself with: probability and random variables. These are summarized in the table below.

Definition Notation Example
Probability The likelihood of some event occurring P(event) P(A) -> Probability of event A occurring
Random Variable A variable whose outcome is unknown until the random experiment has been run X,Y,Z,M, etc. X -> a coin flip

The reason why these two concepts are linked is because we typically calculate the probability of one or several outcomes of a random variable. Random variables are different from traditional variables in that you don’t know what the outcome is until the moment the event actually happens.

The outcome of a coin flip, for example, cannot be known until the coin lands on either side. This is the reason we use probability to model it’s outcomes.

Probability Distribution

A probability distribution is a visualization that plots possible outcomes on the x-axis and each of the probabilities associated with those outcomes on the y-axis. Let’s continue with the coin flip example to illustrate this. Let’s see the probability of getting heads for 3 coin flips.

Coin Flips Events Sample Space Possibilities Number Equivalents Probability of Heads
0 P(H) or P(T) S = {H,T} 0 0 P(H) = 0
1 P(H) or P(T) S = {H,T} H

T

1

0

P(0) = 1/2

P(1) = 1/2

2 P(H) or P(T) S = {H,T} HH

TT

HT

TH

2

0

1

1

P(0) = 1/4

P(1) = 2/4

P(2) = 1/4

3 P(H) or P(T) S = {H,T} HHH

TTT

HHT

HTH

HTT

THH

THT

TTH

3

0

2

2

1

2

1

1

P(0) = 1/8

P(1) = 3/8

P(2) = 3/8

P(3) = 1/8

As you can see, when we start with only 1 coin flip, it is easy to calculate the probabilities for each outcome because there are only two possibilities. As you increase the amount of flips you do, the possible outcomes also increase. We can graph each distribution on a graph.

coin_flip_probabilities

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Types of Distributions

There are many different kinds of distributions. The example that we gave above is one example of a very common distribution known as a binomial distribution. The table below lists the three common distributions, their properties and when they should be used.

Distribution Notation Definition Used When Example
Binomial/Bernoulli X ~ B(n,p) Distribution with parameters n and p There’s only two outcomes (heads/tails, pass/not pass, success/failure, etc.) Coin toss
Normal/ Standard Normal X ~ N(\mu,\sigma) Distribution with parameters \mu and \sigma The random variable X has a normal distribution IQ scores
Poisson X ~ P(\lambda) Distribution with parameter \lambda The random variable is related with time Number of hours a lightbulb works

Normal Distribution Properties

A normal distribution, which can also be transformed into a standard normal distribution, is used in many scenarios in probability. A normal distribution is usually called a “bell-shaped curve,” because the probability distribution is shaped like a bell. Take IQ scores of the population, for example.

normal_distribution

As you can see, IQ scores have a bell-shaped distribution. While visual confirmation is normally enough, we can also conduct statistical tests to make sure the distribution is normal or not. The IQ scores plotted above are in raw form. We can transform each IQ score by standardizing them, which we do by plugging in the raw score into the z-score formula below.

z-score_formula

This gives us the same distribution - only now, each IQ score is in terms of how many standard deviations away from the mean it is.

standard_normal_distribution

This gives us some special properties of a normal distribution, which we can see in the table below.

Standard Deviation Percentage Description
-1 \sigma, +1 \sigma 68% 68% of the data fall in the interval between -1 and +1 standard deviations. This means 65% of the population are within 1 standard deviation from the mean
-2, +2 95% 95% of the data fall within 2 sd from the mean
-3, +3 99.7% 99.7% of the data fall within 3 sd from the mean. After this point, it will be rare.

Right Tail Probability

We can calculate the probability of a given value occurring through the z-score formula because the properties of the normal distribution are always the same, regardless of what the mean or standard deviation are. Contrast this to all the work we did to calculate the probabilities of a coin toss by hand!

Right tail probabilities are when we want to know the probability of a value equal to or above a value. The image below illustrates.

right_tail_example

Once you find the z-score, you simply find it on the right-tail z-table. The probability in this table will be compared to the significance level you choose, which will tell you whether to reject or accept your hypothesis.

Left Tail Probability

In contrast, when we want to know the probability of a value equal to or less than a value, this would be a left tail probability. The image below shows what a left-tail probability looks like on a normal distribution.

left_tail_example

While you can find the left-tail z-table for this, you can also just use 1 minus the right tail probability.

Probability for an Interval

When you want to know the probability that a value will be in between two values, this is called an interval probability. Take a look at the image below.

interval_probabilities

While it may seem complicated to be able to distinguish between these three types of probabilities, the information is summarised below for your ease.

Type Notation Example Z-table Probability
Right Tail P(x > a) P(X > 140) Right tail, or (1-P_{left}) P_{right}, 1-P_{left}
Left Tail P(x < b) P(X < 110) Left tail, or (1-P_{right}) P_{left}, P_{right}
Interval P(a < x < b) P(120 < X < 135) Either P_{right1}-P_{right2},

P_{left1}-P_{left2}

Step-by-Step Example

The population average IQ score is 100 with a standard deviation of 15 points. We calculate the following probabilities, given an alpha of 0.05.

IQ Score Type Notation Z-score
Above 110 Right P(x>110) \frac{110-100}{15} = 0.67
Below 80 Left P(x<80) \frac{90-100}{15} = -1.33
Between 85 and 115 Interval P(85<x<115) \frac{85-100}{15} = -1, \frac{115-100}{15} - 1

Next, we look up the z-score in the z-table for each.

z-table

The p-value for a score of 110 is about 0.75, which is above our alpha of 0.05. This means this value is likely, or we ‘retain’ the null hypothesis if we were running a hypothesis test.

z-table_example

The p-value for a score of 80 is 0.092, which is above our alpha of 0.05. As you can see, both 110 and 80 are either below 1 standard deviation away from 100 or just a bit above 1 SD away.

To get the interval, we can look at the picture below.

interval_probability_example

left_tail_z-table

First we get the left tail probability of 0.15866.

right-tale_z_table

Next we get the right tail probability of 0.84134. Because the standard deviation is 1 and -1, we can see that the left tail probability at -1 is 0.15866, while the right tail probability at the same point would be 0.84134. To get the portion in the middle, we simply:

    \[ p-value = 0.8134-0.15866 = 0.68268 \]

Which we could have guessed, as the rules of a normal distribution dictate that between -1 and 1 is 68% of the population.

 
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Danica

Located in Prague and studying to become a Statistician, I enjoy reading, writing, and exploring new places.