The Definition of Statistical Measures – Central Tendency and Spread

A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. They are also categorized as summary statistics:

• Mean: The mean is the sum of all values pided by the total number of values.
• Median: The median is the middle value. It is the value that splits the dataset in half. To find the median, order your data from smallest to largest, and then find the data point that has an equal amount of values above and below it.
• Mode: The mode is the value that occurs the most frequently in your dataset. On a bar chart, the mode is the highest bar.

Generally, the mean is a better measure to use for symmetric data while the median is a better measure for data with a skewed (left- or right-heavy) distribution. For categorical data, you have to use the mode:

Figure 3.22: A screenshot of a curve showing the mean, median, and mode

The spread of the data is a measure of by how much the values in the dataset are likely to differ from the mean of the values. If all the values are close together, then the spread is low; on the other hand, if some or all of the values differ by a large amount from the mean (and each other), then there is a large spread in the data:

• Variance: This is the most common measure of spread. The variance is the average of the squares of the deviations from the mean. Squaring the deviations ensures that negative and positive deviations do not cancel each other out.
• Standard deviation: Because variance is produced by squaring the distance from the mean, its unit does not match that of the original data. Standard deviation is a mathematical trick that brings back parity. It is the positive square root of the variance.

Random Variables and Probability Distribution

A random variable is defined as the value of a given variable that represents the outcome of a statistical experiment or process.

Although it sounds very formal, pretty much everything around us that we can measure can be thought of as a random variable.

The reason behind this is that almost all natural, social, biological, and physical processes are the final outcome of a large number of complex processes, and we cannot know the details of those fundamental processes. All we can do is observe and measure the final outcome.

Typical examples of random variables that are around us are as follows:

• The economic output of a nation
• The blood pressure of a patient
• The temperature of a chemical process in a factory
• The number of friends of a person on Facebook
• The stock market price of a company

These values can take any discrete or continuous value and follow a particular pattern (although this pattern may vary over time). Therefore, they can all be classified as random variables.

What is a Probability Distribution?

A probability distribution is a mathematical function that tells you the likelihood of a random variable taking each different possible value. In other words, a probability distribution gives the probabilities of the different possible outcomes in a given situation.

Suppose you go to a school and measure the heights of students who have been selected randomly. Height is an example of a random variable here. As you measure height, you can create a distribution of height. This type of distribution is useful when you need to know which outcomes are the most likely to occur (that is, which heights are the most common), the spread of potential values, and the likelihood of different results.

The concepts of central tendency and spread are applicable to a distribution and are used to describe the properties and behavior of a distribution.

Statisticians generally pide all distributions into two broad categories:

• Discrete distributions
• Continuous distributions

Discrete Distributions

Discrete probability functions, also known as probability mass functions, can assume a discrete number of values. For example, coin tosses and counts of events are discrete functions. You can only have heads or tails in a coin toss. Similarly, if you're counting the number of trains that arrive at a station per hour, you can count 11 or 12 trains, but nothing in between.

Some prominent discrete distributions are as follows:

• Binomial distribution to model binary data, such as coin tosses
• Poisson distribution to model count data, such as the count of library book checkouts per hour
• Uniform distribution to model multiple events with the same probability, such as rolling a die

Continuous Distributions

Continuous probability functions are also known as probability density functions. You have a continuous distribution if the variable can assume an infinite number of values between any two values. Continuous variables are often measurements on a real number scale, such as height, weight, and temperature.

The most well-known continuous distribution is normal distribution, which is also known as Gaussian distribution or the bell curve. This symmetric distribution fits a wide variety of phenomena, such as human height and IQ scores.

Normal distribution is linked to the famous 68-95-99.7 rule, which describes the percentage of data that falls within 1, 2, or 3 standard deviations away from the mean if the data follows a normal distribution. This means that you can quickly look at some sample data, calculate the mean and standard deviation, and can have confidence (a statistical measure of uncertainty) that any future incoming data will fall within those 68%-95%-99.7% boundaries. This rule is widely used in industries, medicine, economics, and social science:

Figure 3.23: Curve showing the normal distribution of the famous 68-95-99.7 rule