9.3 Normal distribution
4 min read•july 30, 2024
The is a crucial concept in probability theory, shaping our understanding of data spread. It's characterized by its symmetrical , with two key parameters: the (μ) and (σ). These determine the curve's center and spread, respectively.
Standardizing normal variables transforms them into a with a mean of 0 and standard deviation of 1. This process, along with the , allows us to calculate probabilities for various scenarios, making the normal distribution a powerful tool in statistical analysis.
Normal distribution properties
Characteristics and parameters
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- Normal distribution, also known as , exhibits around its mean
- Two parameters characterize the normal distribution
- Mean (μ) determines the center of the distribution
- Standard deviation (σ) measures the spread of the distribution
- (PDF) for a normal distribution follows the equation:
- Bell-shaped curve represents the normal distribution
- Highest point occurs at the mean
- Probability decreases symmetrically as values move away from the mean
Distribution properties
- (68-95-99.7 rule) describes data distribution
- 68% of data falls within one standard deviation of the mean
- 95% of data falls within two standard deviations
- 99.7% of data falls within three standard deviations
- with mode, median, and mean equal and centered
- Total area under the normal distribution curve always equals 1
- Represents the sum of probabilities for all possible outcomes
Standardizing normal variables
Standardization process
- Standardization converts a normal random variable X to a standard normal variable Z
- Resulting Z has a mean of 0 and standard deviation of 1
- Formula for standardization:
- X represents the original value
- μ represents the mean
- σ represents the standard deviation
- Standard normal distribution (z-distribution) results from standardization
- Special case of normal distribution with μ = 0 and σ = 1
Using the standard normal table
- Z-table (standard normal table) provides cumulative probabilities for the standard normal distribution
- Steps to find probabilities using the z-table:
- Standardize the given value
- Locate the corresponding probability in the table
- Z-table typically gives area to the left of a given
- Can be used to find areas to the right or between two z-scores through calculations
- Interpolation may be necessary for z-scores falling between provided table values
- Example: For z-score 1.234, interpolate between values for 1.23 and 1.24
Probabilities for normal distributions
Calculating probabilities
- Determine probabilities for general normal distributions by standardizing values and using the z-table
- (CDF) gives probability that X ≤ x for a random variable X
- Find probabilities between two values by calculating the difference between their CDFs
- Use symmetric intervals around the mean with the formula:
- Φ represents the standard normal CDF
- Example: Probability of X falling within 2σ of the mean is 2Φ(2) - 1 ≈ 0.9545
Determining quantiles
- Calculate (percentiles, quartiles) using inverse standardization and the z-table
- Formula for finding a quantile:
- Z represents the z-score corresponding to the desired percentile
- (IQR) for a normal distribution approximately equals 1.34σ
- Useful for identifying potential outliers
- Example: In a normal distribution with σ = 10, IQR ≈ 13.4
Normal approximation of binomial distributions
Conditions for approximation
- Normal distribution approximates when:
- Sample size n is large
- Probability p is not too close to 0 or 1
- Rule of thumb for using
- Both np and n(1-p) should be ≥ 5 or 10, depending on desired accuracy
- Example: For n = 100 and p = 0.3, np = 30 and n(1-p) = 70, satisfying the condition
Applying the approximation
- Approximating normal distribution parameters:
- Mean = np
- Standard deviation = √(np(1-p))
- Apply when using normal approximation
- Add or subtract 0.5 to the value of interest
- Depends on calculating "less than" or "greater than" probability
- Accuracy improves as n increases and p approaches 0.5
- Useful for large n values where direct binomial calculation becomes computationally intensive
- Recognize limitations and use exact binomial probabilities when high precision is required
- Example: Medical studies often require exact probabilities rather than approximations
Key Terms to Review (25)
Bell curve: A bell curve is a graphical representation of a normal distribution, characterized by its symmetrical shape resembling a bell. It illustrates how values are distributed around the mean, with most values clustering around the average and fewer values appearing as you move away from the center. This shape is crucial in statistics as it reflects the probability distribution of many natural phenomena and helps to understand the spread and likelihood of different outcomes.
Binomial Distribution: The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is crucial for analyzing situations where there are two outcomes, like success or failure, and is directly connected to various concepts such as discrete random variables and probability mass functions.
Central Limit Theorem: The Central Limit Theorem (CLT) states that, regardless of the original distribution of a population, the sampling distribution of the sample mean will approach a normal distribution as the sample size increases. This is a fundamental concept in statistics because it allows for making inferences about population parameters based on sample statistics, especially when dealing with larger samples.
Confidence Interval: A confidence interval is a range of values derived from sample data that is likely to contain the true population parameter with a specified level of confidence, usually expressed as a percentage. This concept is essential for understanding the reliability of estimates made from sample data, highlighting the uncertainty inherent in statistical inference. Confidence intervals provide a way to quantify the precision of sample estimates and are crucial for making informed decisions based on statistical analyses.
Continuity correction: Continuity correction is an adjustment made when using a continuous probability distribution to approximate a discrete distribution. This is important because discrete data consists of distinct, separate values, while continuous distributions represent an unbroken range of values. The correction typically involves adding or subtracting 0.5 to the value being approximated, ensuring a more accurate representation when calculating probabilities, particularly when using the normal distribution to approximate binomial or Poisson distributions.
Cumulative Distribution Function: The cumulative distribution function (CDF) of a random variable is a function that describes the probability that the variable will take a value less than or equal to a specific value. The CDF provides a complete description of the distribution of the random variable, allowing us to understand its behavior over time and its potential outcomes in both discrete and continuous contexts.
Empirical Rule: The empirical rule, also known as the 68-95-99.7 rule, is a statistical guideline that describes how data is distributed in a normal distribution. It states that approximately 68% of the data falls within one standard deviation from the mean, about 95% falls within two standard deviations, and around 99.7% lies within three standard deviations. This rule helps in understanding the spread and behavior of data in normally distributed sets, providing valuable insights for analysis.
Gaussian distribution: A Gaussian distribution, also known as a normal distribution, is a continuous probability distribution characterized by its symmetric bell-shaped curve. It is defined by its mean and standard deviation, with most of the data points clustering around the mean and fewer points appearing as you move away in either direction. This distribution is vital in statistics because it describes how values are distributed in many natural phenomena and allows for the application of various statistical methods.
Interquartile Range: The interquartile range (IQR) is a measure of statistical dispersion that represents the difference between the first quartile (Q1) and the third quartile (Q3) in a data set. This range effectively captures the middle 50% of data points, offering insights into the spread and variability within a dataset, while being resistant to outliers and extreme values.
Kurtosis: Kurtosis is a statistical measure that describes the shape of a distribution's tails in relation to its overall shape. It indicates how much of the data is in the tails and can highlight whether data points are heavy-tailed or light-tailed compared to a normal distribution. This property is important for understanding variability and the likelihood of extreme values occurring in continuous random variables and normal distributions.
Mean: The mean is a measure of central tendency that represents the average value of a set of numbers. It is calculated by summing all values in a dataset and then dividing by the total number of values. This concept plays a crucial role in understanding various types of distributions, helping to summarize data and make comparisons between different random variables.
Normal approximation: Normal approximation refers to the use of the normal distribution as an approximation for the distribution of a sum or average of a large number of independent random variables. This concept is pivotal when dealing with binomial or Poisson distributions, as the normal distribution simplifies complex calculations and provides a way to estimate probabilities. The normal approximation becomes increasingly accurate as the sample size grows, leveraging the Central Limit Theorem, which states that the sampling distribution of the mean approaches a normal distribution as sample size increases.
Normal distribution: Normal distribution is a continuous probability distribution that is symmetric around its mean, showing that data near the mean are more frequent in occurrence than data far from the mean. This bell-shaped curve is crucial in statistics because it describes how many real-valued random variables are distributed, allowing for various interpretations and applications in different areas.
Normality test: A normality test is a statistical procedure used to determine whether a given dataset follows a normal distribution. This concept is crucial because many statistical methods assume that the underlying data is normally distributed, impacting the validity of the results obtained. Normality tests help in assessing the goodness of fit of a dataset to the normal distribution, which can influence decisions regarding the choice of statistical analysis techniques.
Null hypothesis: The null hypothesis is a statement that indicates there is no effect or no difference in a given situation, serving as a starting point for statistical testing. It is essential in determining whether observed data deviates significantly from what would be expected under this assumption. The null hypothesis is often denoted as H0 and provides a foundation for conducting various statistical analyses, such as determining relationships or differences among groups, assessing probabilities, and making predictions about population parameters.
Probability Density Function: A probability density function (PDF) describes the likelihood of a continuous random variable taking on a particular value. Unlike discrete variables, which use probabilities for specific outcomes, a PDF represents probabilities over intervals, making it essential for understanding continuous distributions and their characteristics.
Quantiles: Quantiles are values that divide a dataset into equal intervals, with each interval containing the same proportion of the data. They are essential for understanding the distribution of data and are often used to summarize and describe characteristics of datasets. Key quantiles include median, quartiles, and percentiles, which help in analyzing and interpreting the spread and central tendency of a normal distribution.
Sampling distribution: A sampling distribution is the probability distribution of a statistic obtained through repeated sampling from a population. It describes how the values of a statistic, like the sample mean, vary from sample to sample, and helps in understanding the behavior of estimates as sample sizes change. This concept connects deeply with ideas about normal distributions, central limit theorem, and statistical inference, illustrating how sample statistics can be used to make inferences about the population parameters.
Standard Deviation: Standard deviation is a statistic that measures the dispersion or variability of a set of values around their mean. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation suggests that the values are spread out over a wider range. This concept is crucial in understanding the behavior of both discrete and continuous random variables, helping to quantify uncertainty and variability in data.
Standard normal distribution: The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. This distribution is crucial in statistics because it provides a way to standardize different normal distributions, making it easier to compare and analyze data. The area under the curve of the standard normal distribution represents probabilities, allowing statisticians to make inferences about populations based on sample data.
Symmetry: Symmetry refers to a balanced and proportional arrangement of elements, where one part mirrors or corresponds to another part. In probability, symmetry is often seen in distributions where the values on one side of a central point are identical in shape and distribution to those on the opposite side, indicating that outcomes are evenly distributed. This feature is crucial in understanding properties of certain probability distributions and how different variables interact with each other.
Type I Error: A Type I error occurs when a null hypothesis is incorrectly rejected, indicating that a supposed effect or difference exists when, in reality, it does not. This error is significant in statistical testing as it can lead to false conclusions about the data being analyzed, impacting decisions based on those findings. The implications of a Type I error can be particularly critical in various real-world applications, influencing areas such as medicine, quality control, and social sciences.
Unimodal distribution: A unimodal distribution is a type of probability distribution that has a single peak or mode, indicating that most of the data points cluster around one particular value. This characteristic shape makes it easier to analyze and interpret, as the majority of observations are concentrated in one area, with symmetry on either side if it follows a normal distribution. The unimodal feature plays a crucial role in various statistical analyses, particularly in understanding how data behaves and the likelihood of different outcomes.
Z-score: A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It indicates how many standard deviations an element is from the mean, allowing for comparison between different data sets. Z-scores are essential for understanding how individual data points relate to the overall distribution and are particularly useful in the context of normal distributions and when dealing with sampling distributions.
Z-table: A z-table is a mathematical table used in statistics to determine the probability of a standard normal distribution. It provides the area under the curve of a standard normal distribution for given z-scores, which represent the number of standard deviations a data point is from the mean. This tool is essential for making inferences about populations based on sample data, especially when dealing with normally distributed variables.