Probability of Continuous Random Variables (Normal Distribution)

The normal distribution is a type of discussion in statistics related to probability distribution, also known as probability distribution. Through the normal distribution table, it is often used for calculating various phenomena in everyday life, such as height measurement, blood pressure, error calculations, and IQ scores.

The term normal distribution is also referred to as Gaussian distribution, with one of the key equations being the density function. In probability theory, the normal distribution holds a crucial position and is present in various statistical analyses of the data obtained.

Definition of Normal Distribution

The normal distribution is a type of continuous random variable distribution characterized by a bell-shaped curve. This distribution, with its probability function, shows the variation or spread of the distribution, which can also be illustrated using a symmetric graph or bell curve.

The curve representing this distribution peaks at the center and slopes down on both sides, with equivalent values. The term “Gauss” refers to Carl Friedrich Gauss, a German mathematician who developed a distribution theory with an exponential function between 1794 and 1809.

With the distribution centered symmetrically around the mean of the entire population, it helps avoid biased or imbalanced assessments. This is also essential for determining the level of normality and central tendency, which are important and should not be overlooked.

By applying the theory of normal distribution, it becomes easier to determine the normality of data or central tendency. This application enhances the objectivity of assessments, which is particularly helpful in accurately placing members within specific groups.

Parameters of Normal Distribution

The application of this distribution theory is deemed important for several reasons, including enhancing the objectivity of assessments, aiding in placing the most suitable members in particular groups, and evaluating scores or grouping employees under the same criteria to avoid bias or skewed judgments.

A symmetric distribution centered around the overall population mean cannot avoid biased assessments. The normal distribution also assists in determining the level of central tendency normality. In probability statistics, the normality of data is an important aspect that should not be ignored.

In other statistical probability distribution theories, the curve shape and the probability values in the normal distribution table are determined by several parameters. This type of distribution has two types of parameters that serve as references: the mean and standard deviation, along with their brief explanations.

The mean is generally used as the center of the distribution or the spread of other values, which is then used to determine the peak point in a bell curve. Other values are intentionally spread around the mean.

The standard deviation measures variability and determines the width of the normal distribution curve. It calculates how far data tendencies extend from the mean, illustrating the general difference or distance between the mean and other tested data.

Population parameters versus sample estimates, with the mean and standard deviation being applicable parameters across the entire population. In a normal distribution, statisticians denote parameters using the Greek symbols μ (mu) for the population mean and σ (sigma) for the population standard deviation.

Typically, population parameters are unknown because measuring the entire population is generally impractical. Random samples are used to estimate parameters. In random samples, statisticians use sample estimates of parameters with x̅ for the sample mean and s for the sample standard deviation.

Characteristics of Normal Distribution

The theory of distribution indicates that the mean, median, and mode are the same, which is why this theory is often referred to as unimodal. The distribution curve can be symmetric in the shape of a bell curve. The peak of the curve occurs at the mean, which is located at the center of the curve, while the distribution data is situated around a straight line drawn down from the midpoint. The mean is the average value with standard deviation, which helps determine the location and shape of the distribution.

The total area under the normal curve equals 1, with ½ on the right side and ½ on the left. This applies to all continuous probability distributions. It is concluded that half of the population data will have values less than the average, while the other half will have values significantly greater. Each tail of the curve extends infinitely on both sides, and in some cases of distribution calculation, the tails may even intersect the horizontal axis.

Empirical Rule for Normal Distribution

Standard deviation is crucial for normally distributed data, and it is also used to determine the proportion of values within a specific number of standard deviations from the mean. In a normal distribution, approximately 68% of observations fall within ±1 standard deviation of the mean, 95% within ±2 standard deviations, and 99.7% within ±3 standard deviations from the mean.

This property is part of the empirical rule that explains data presentation, including a certain number of standard deviations from the mean in relation to the bell-shaped curve. It is essential to understand this rule, as the values derived from the mean are significantly useful in practice.

Standard Normal Distribution and Z-Scores

In terms of shape, the normal distribution can take on many different forms depending on parameter values. However, the standard normal distribution is a special case of the normal distribution, where the mean is zero and the standard deviation is one. This distribution is known as the Z distribution, with values in the standard normal distribution referred to as Z-scores.

A Z-score represents the number of standard deviations a particular observation is above or below the mean. For example, a Z-score of 1.5 indicates that the observation is 1.5 standard deviations above the mean. Conversely, a negative Z-score represents a value below the mean, with an average Z-score of 0.

Standardization and Calculation of Z-Score

The Z-score is an effective way to understand how far a specific observation is relative to the overall distribution. This allows for observations taken from a normally distributed population, with different methods and standard deviations, to be placed on a standard scale.

The standard scale also facilitates comparisons of observations that might otherwise seem difficult. This process is known as standardization and enables comparisons of observations and calculations of probabilities across different populations. To standardize data, it is necessary to convert raw measurements into Z-scores.

To calculate the Z-score of an observation, start by taking the raw measurement, subtracting the mean, and then dividing by the standard deviation. Mathematically, the formula for this calculation is as follows: Z = (x−μ)/σ. This normal distribution formula represents and is used for the desired measurement.

Z-Scores for Comparing Heights of Males and Females

When comparing heights, such as those of male and female students, the average height of a male is about 180 cm and for a female, it is 160 cm. It’s easy to see that male students tend to be taller than female students when comparing their raw values. However, to compare their Z-scores, a different method based on normal distribution must be employed.

This involves assuming properties of height distribution for both males and females, following a normal distribution with the following parameter values: Male height μ= 180, σ= 30; Female height μ=160, σ = 10. Here’s how to calculate the Z-scores in this example:

Z-score for males = (170−180)/30 = − 0.33
Z-score for females = (165−160)/10 = 0.5

A Z-score of -0.33 for males indicates that the male’s height is below the average. Conversely, the positive Z-score of 0.5 for females indicates that the female’s height is above the average, as the Z-score reflects their position in the standard normal distribution.

Finding the Area Under the Normal Distribution Curve

The normal distribution is equivalent to a probability distribution, where the proportion of the area under the curve lies between two points on the distribution plot, indicating the values that will fall within that interval. To understand this, it’s essential to first know what a probability distribution is. In practice, statistical software is typically used to find the area under the curve.

When working with the normal distribution and converting values to Z-scores, one can also calculate the area and find the Z-scores in a standard normal distribution table. Because there are infinitely many normal distributions, publishers cannot print a table for every possible distribution, which is a limitation encountered in this context.