Frequency

Frequency is defined as the amount of times a value occurs. To illustrate this, let’s start by looking at a table, which shows the number of times someone bought a given phone model.

 

Model Count
A1 IIIII IIIII
S4 IIIII
D5 IIIII IIIII IIIII
F6 IIIII IIIII IIIII IIIII IIIII
J9 IIIII II

 

In order to calculate the frequency, you simply need to add all of the times the model has been bought, which can be seen in the ‘Count’ column. See the final frequencies below.

 

Model Frequency
A1 10
S4 5
D5 15
F6 25
J9 7

 

Frequencies are helpful because they can help us understand the distribution of the data.

 

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Frequency Distribution

Continuing from the previous example, a frequency distribution is a function or a graph that helps us see the distribution of a variable. A distribution is the way something is spread or shared. Take a look at the graph below, which shows the distribution of the phones that have been bought.

probability_example_1

Frequency distributions aren’t only applicable to categorical variables. One common example can be seen through heights. Take the table below, which has the heights of students in a university class.

 

Height Frequency
150 2
155 3
160 9
165 12
170 20
175 25
180 25
185 10
190 18
195 20
200 5

 

As you can see, displaying the frequency of more than a couple of categories or values. This is where a frequency distribution can come in handy. The image below shows a frequency distribution of the example heights.

probability_distribution

Without graphing the frequency distribution, we wouldn’t have been able to see the distribution is bimodal.

 

Graph Type Description
A Unimodal Has one peak, where values are cantered
B Bimodal Has two peaks, which illustrate the two centres in the data
C Multimodal Has multimodes, which illustrate multiple points around which values in the data centre

 

unimodal_distributionbimodal_distribution

 

Probability

The probability of an event is the likelihood of that event happening. This probability, however, depends on what type of event it is. The table below summarizes the formulas for the probability of different types of events.

 

Notation Formula
1 event P(A) \frac{\# \; of \; times \; something \; can \; occur}{ total \; \# \; of \; outcomes}
2 independent events P(A \cap B) P(A) * P(B)
2 dependent events P(A \cap B) P(A) * P(B|A)
2 mutually exclusive events P(A \cup B) P(A) + P(B)
2 not mutually exclusive events P(A \cup B) P(A) + P(B) - P(A \cap B)

 

Probability Distribution

A probability distribution is a formula or visualization that tells us the probability of an event occurring. The image below shows the probability distribution of landing on heads when flipping a coin 5 times.

coin_toss_heads

As you can see, this distribution tells us the probability of the number of heads in terms of all the possible combinations of heads. This information can also be summarised in a table, like the one below.

 

Heads Probability Probability
0 \frac{1}{16} 0.06
1 \frac{ 4}{16} 0.25
2 \frac{6 }{16} 0.38
3 \frac{4 }{16} 0.25
4 \frac{1 }{16} 0.06

 

While this information is easy to read in a table format, it can get messy when you’re dealing with more than five events or events with continuous distributions, which we often are. This is where probability distributions come in.

 

Types of Distributions

There are many types of probability distributions. The three most common ones can be seen in the images below.

distribution_example_1 poisson_distribution_example_1 distribution_example

As you can see, these distributions are similar to the one in the previous example. However, finding the probability of a single value can be performed using formulas instead of mechanically. These formulas are the following.

 

Mean Other Parameters Probability Equation
Standard normal Mean = \mu SD = \sigma, x = test value z = \frac{x-\mu}{\sigma}, probability found on z-table
Binomial n = # of trials p  = probability of success in each trial, q = 1-p, x = test value P_{x} = \binom{n}{x} p^{x} q^{n-x}
Poisson Mean = \lambda x = test value P = \frac{\lambda^{x} e^{-\lambda}}{x!}

 

Standard Normal Distributions

The most common type of distribution that you will encounter in statistics classes is the standard normal distribution. This distribution standardizes the values for a variable that has a normal distribution. It does this by taking the plain numbers, known as “raw scores,” and plugging them into the following formulas.

 

Population Parameters Sample Statistics
Formula \frac{x-\mu}{\sigma} \frac{x- \bar{x}}{s}

 

Z-score

Another name for the standardized value found from the standardisation formula above is a z-score. Z-scores are vital for finding the probability of a value in a standard normal distribution because they let us know how far away any given value is from the mean. Take a look at the standard normal distribution below.

standard_normal_probability

As you can see, the standard normal distribution tells us how many standard deviations away any number is from the mean. No matter what the standard deviation is, the z-score will always stay the same. That is to say, regardless if the SD is 3 or 100, the z score for a value 1 standard deviation away will always be the same.

 

Mean Standard Deviation Value 1 SD Above the Mean Z-score
30 4 30 + 4 \frac{30-34}{4} = 1
30 10 30 + 10 \frac{30-40}{10} = 1
30 100 30 + 100 \frac{30-130}{100} = 1

 

Z-table

Calculating the z-score is only one half of finding the probability. As you can see in the image above, each z-score corresponds to a probability, which is written down in a z-table. A z-table is either right or left tailed, which depends on whether or not you are finding the probability above, below or of an interval of any values.

right_left_tail_probability
When Graph
Right-tail Of or above a value C,D
Left-tail Of or below a value A,B

 

Standard Normal Example

You are interested in knowing the probability of getting 1000 points for an exam that has a mean of 850 and a standard deviation of 100. First, you find the z-score.

 

    \[ z-score = \frac{1000-850}{100} = 1.50 \]

 

Next, we look up this value in a left tail z-table. This is because we are scoring up to 1000 points, not above 1000.

z-table_step_by_step

This is 0.9332, which is about 93%.

 

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Danica

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