6.2 Using the Normal Distribution
3 min read•june 25, 2024
The , with its iconic shape, is a powerful tool in statistics. It allows us to calculate probabilities for any normally distributed data by converting values to z-scores, which measure how far a data point is from the in terms of standard deviations.
Understanding the helps us interpret data across different scales. By using z-scores and the standard normal table, we can find probabilities for specific values or ranges, making it easier to compare and analyze data from various normal distributions.
The Standard Normal Distribution
Probabilities from standard normal distribution
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- with mean of 0 and of 1 (also known as the bell curve)
- Z-scores measure number of standard deviations from mean
- Positive z-scores are values above mean
- Negative z-scores are values below mean
- Standard table provides area () to left of given
- Table organized with z-scores in rows and additional decimal places in columns
- Find probability for specific by locating row with integer and first decimal place, then move to column with second decimal place
- Value at intersection is probability (area) to left of z-score
- For probabilities to right of z-score, subtract table value from 1
- For probabilities between two z-scores, find areas to left of each z-score and subtract smaller area from larger area
Conversion to z-scores
- converts values from any normal distribution to z-scores:
- is value being converted
- is mean of original distribution
- is standard deviation of original distribution
- Convert value to z-score by subtracting mean from value and dividing by standard deviation
- Z-scores allow comparison of values from different normal distributions on common scale
Interpretation of normal curve area
- Total area under normal curve equals 1, representing 100% probability
- Area between any two points on curve represents probability of value falling within that range
- In standard normal distribution:
- Approximately 68% of data falls within one standard deviation of mean ()
- Approximately 95% of data falls within two standard deviations of mean ()
- Approximately 99.7% of data falls within three standard deviations of mean ()
- Area to left of z-score represents probability of value being less than or equal to that z-score
- Area to right of z-score represents probability of value being greater than that z-score
Additional Properties of the Normal Distribution
- : The normal distribution is perfectly symmetrical about its mean
- : The normal distribution is a continuous function, allowing for infinite possible values
- : As sample size increases, the distribution of sample means approaches a normal distribution
- : The value below which a given percentage of observations fall in a normal distribution
- : Gives the probability that a random variable is less than or equal to a specific value
Key Terms to Review (26)
68-95-99.7 Rule: The 68-95-99.7 rule, also known as the empirical rule, is a statistical principle that describes the distribution of data in a normal or bell-shaped curve. It states that approximately 68% of the data falls within one standard deviation of the mean, 95% of the data falls within two standard deviations of the mean, and 99.7% of the data falls within three standard deviations of the mean. This rule is particularly useful in understanding the Standard Normal Distribution and in applying the Normal Distribution to real-world scenarios.
Bell Curve: The bell curve, also known as the normal distribution, is a symmetrical, bell-shaped probability distribution that describes how a set of data is distributed around the mean. It is a fundamental concept in statistics and probability theory, with applications across various fields, including 6.1 The Standard Normal Distribution, 6.2 Using the Normal Distribution, and 7.2 Using the Central Limit Theorem.
Central Limit Theorem: The central limit theorem is a fundamental concept in probability and statistics that states that the sampling distribution of the mean of a random variable will tend to a normal distribution as the sample size increases, regardless of the underlying distribution of the variable.
Continuous Probability Distribution: A continuous probability distribution is a probability distribution where the random variable can take on any value within a specified range, rather than being limited to discrete values. It is a fundamental concept in probability theory and statistics, with applications across various fields, including business, engineering, and the natural sciences.
Cumulative Distribution Function: The cumulative distribution function (CDF) is a fundamental concept in probability and statistics that describes the probability of a random variable taking a value less than or equal to a given value. It provides a comprehensive way to represent the distribution of a random variable and is closely related to other important statistical concepts such as probability density functions and probability mass functions.
Cumulative distribution function (CDF): A cumulative distribution function (CDF) represents the probability that a continuous random variable takes on a value less than or equal to a specific value. It is an integral of the probability density function (PDF).
Empirical Rule: The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical concept that describes the distribution of data in a normal distribution. It provides a general guideline for understanding the relationship between the standard deviation and the proportion of data that falls within certain ranges around the mean.
Equal standard deviations: Equal standard deviations, also known as homoscedasticity, occur when the variability within each group being compared is similar. This is an important assumption for performing One-Way ANOVA.
Expected mean: The expected mean in the context of linear regression is the average value of the response variable predicted by the regression equation for a given set of predictor variables. It represents the central tendency around which individual observations are expected to vary.
Mean: The mean, also known as the arithmetic mean, is a measure of central tendency that represents the average value in a dataset. It is calculated by summing up all the values in the dataset and dividing by the total number of values. The mean provides a summary statistic that describes the central or typical value in a distribution of data.
Normal distribution: A normal distribution is a continuous probability distribution that is symmetrical and bell-shaped, where most of the observations cluster around the central peak. It is characterized by its mean ($\mu$) and standard deviation ($\sigma$).
Normal Distribution: The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetrical and bell-shaped. It is one of the most widely used probability distributions in statistics and plays a crucial role in various statistical analyses and concepts covered in this course.
Percentile: A percentile is a statistical measure that indicates the relative position of a value within a distribution. It represents the percentage of values in a dataset that fall below a given value. Percentiles are particularly relevant in the context of the standard normal distribution and when using the normal distribution to make inferences about data.
Percentiles: Percentiles are measures that divide a dataset into 100 equal parts, indicating the relative standing of a value within the data. For example, the 25th percentile (first quartile) is the value below which 25% of the observations fall.
Probability: Probability is the measure of the likelihood of an event occurring. It quantifies the chance or odds of a particular outcome happening within a given set of circumstances or a defined sample space. Probability is a fundamental concept in statistics, as it provides the foundation for understanding and analyzing uncertainty, risk, and decision-making.
Sigma Notation (Σ): Sigma notation, denoted by the Greek letter Σ, is a concise way to represent the sum of a series of values or the application of a mathematical operation across multiple elements. It is a fundamental concept in statistics and various mathematical disciplines, allowing for the efficient expression and calculation of sums, means, and other statistical measures.
Standard Deviation: Standard deviation is a measure of the spread or dispersion of a set of data around the mean. It quantifies the typical deviation of values from the average, providing insight into the variability within a dataset.
Standard normal distribution: The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is used as a reference to transform any normal distribution into a standardized form for easier analysis.
Standard Normal Distribution: The standard normal distribution is a probability distribution that describes a normal distribution with a mean of 0 and a standard deviation of 1. It is a fundamental concept in statistics that is used to analyze and make inferences about data that follows a normal distribution.
Standardizing formula: Standardizing a formula involves converting data from different scales to a common scale using the Z-score. This allows for comparison across different datasets by transforming the data to have a mean of 0 and a standard deviation of 1.
Standardizing Formula: The standardizing formula is a mathematical expression used to convert a raw data value into a standardized score, often referred to as a z-score. This formula allows for the comparison of data points across different distributions by transforming them into a common scale with a mean of 0 and a standard deviation of 1. The standardizing formula is a crucial tool in various statistical analyses, including the use of the normal distribution, the central limit theorem, and hypothesis testing.
Symmetry: Symmetry refers to the balanced and proportional arrangement of elements or features around a central axis or point. It is a fundamental concept that is closely tied to the measures of center, skewness, and the normal distribution in statistics.
The Central Limit Theorem: The Central Limit Theorem (CLT) states that the distribution of the sample mean approaches a normal distribution as the sample size grows, regardless of the original population's distribution. This theorem is fundamental in inferential statistics because it allows for making predictions about population parameters.
Z-score: A z-score represents the number of standard deviations a data point is from the mean. It is used to determine how unusual or typical a value is within a normal distribution.
Z-Score: A z-score, also known as a standard score, is a statistical measure that expresses how many standard deviations a data point is from the mean of a dataset. It is a fundamental concept in probability and statistics that is widely used in various statistical analyses and hypothesis testing.
μ (Mu): Mu (μ) is a Greek letter commonly used in statistics to represent the population mean or average. It is a central parameter that describes the central tendency or typical value of a population distribution. Mu is a crucial concept in understanding various statistical measures and distributions covered in this course.