NORMAL DISTRIBUTION
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This document discusses statistical concepts such as descriptive and inferential statistics, types of data, and properties of the normal distribution. It provides examples of how to calculate standardized scores (z-values) and probabilities using the normal distribution. Specifically, it shows how to find the probability of a student's test score falling within or exceeding certain ranges given the mean and standard deviation of scores from a sample of 120 students. Formulas and steps are outlined for standardizing values and determining the corresponding areas under the normal curve to obtain probabilities.
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STATISTICS: Normal Distribution
The document provides an outline and explanation of key concepts related to the normal distribution. It begins with an introduction to probability distributions for continuous random variables and the definition of a density curve. It then defines terms and symbols used in the normal distribution, including mean, standard deviation, and z-scores. The document explains the characteristics of the normal distribution graphically and provides examples of finding areas under the normal curve using z-tables. It concludes with examples of finding unknown z-values and calculating probabilities for specific scenarios involving the normal distribution.
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This document discusses the normal distribution and standard normal curve. It defines key properties of the normal distribution including that it is bell-shaped and symmetrical around the mean. The standard normal curve is introduced which has a mean of 0 and standard deviation of 1. The z-score is defined as a way to locate a value within a distribution based on its mean and standard deviation. Various probabilities are associated with areas under the normal curve based on z-scores.
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The central limit theorem states that the distribution of sample means approaches a normal distribution as sample size increases. It allows using a normal distribution for applications involving sample means. The mean of the sample means equals the population mean, and the standard deviation of sample means is the population standard deviation divided by the square root of the sample size. For samples larger than 30, the distribution of means can be approximated as normal, becoming closer for larger samples. If the population is already normal, the sample means will be normally distributed for any sample size.
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The document discusses the normal distribution and related concepts. It describes how normal distributions can vary in their mean and standard deviation. It then discusses key features of normal distributions including that they are symmetric, have equal mean, median and mode, and are denser in the center than the tails. Finally, it discusses related statistical concepts like kurtosis and skewness, describing how kurtosis measures the thickness of a distribution's tails and how skewness measures a distribution's asymmetry.
4 2 continuous probability distributionn
Here are the steps to solve this problem:
a) Find the z-score corresponding to 115 mm Hg: (115 - 85)/13 = 2.31
The proportion that is NOT severely hypertensive is 1 - P(Z >= 2.31) = 1 - 0.0103 = 0.9897
b) Find the z-score corresponding to 90 mm Hg: (90 - 85)/13 = 0.3846
The proportion that will be asked to consult a physician is P(Z >= 0.3846) = 0.6507
c) Find the z-scores corresponding to the mildly hypertensive range:
(90 - 85)/13 = 0.3846
(
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- Probability distributions describe the probabilities associated with random variables.
- Expected value refers to the mean or weighted average of a probability distribution.
- As the number of trials increases, the actual mean approaches the true mean due to the Law of Large Numbers.
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Normal distribution
The normal distribution is a continuous probability distribution that is symmetric and bell-shaped. It is defined by its mean and standard deviation. Many real-world variables are approximately normally distributed. The standard normal distribution refers to a normal distribution with a mean of 0 and standard deviation of 1. The normal distribution is commonly used to model variables and calculate probabilities related to areas under the normal curve.
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The document discusses discrete and continuous random variables. It defines discrete random variables as variables that can take on countable values, like the number of heads from coin flips. Continuous random variables can take any value within a range, like height. The document explains how to calculate and interpret the mean, standard deviation, and probabilities of events for both types of random variables using examples like Apgar scores for babies and heights of young women.
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This document discusses the normal distribution and standard normal curve. It defines key properties of the normal distribution including that it is bell-shaped and symmetrical around the mean. The standard normal curve is introduced which has a mean of 0 and standard deviation of 1. The z-score is defined as a way to locate a value within a distribution based on its mean and standard deviation. Various probabilities are associated with areas under the normal curve based on z-scores.
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Here are the steps to solve this problem:
a) Find the z-score corresponding to 115 mm Hg: (115 - 85)/13 = 2.31
The proportion that is NOT severely hypertensive is 1 - P(Z >= 2.31) = 1 - 0.0103 = 0.9897
b) Find the z-score corresponding to 90 mm Hg: (90 - 85)/13 = 0.3846
The proportion that will be asked to consult a physician is P(Z >= 0.3846) = 0.6507
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(90 - 85)/13 = 0.3846
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- A random variable is a numerical measure of outcomes from a random phenomenon.
- Probability distributions describe the probabilities associated with random variables.
- Expected value refers to the mean or weighted average of a probability distribution.
- As the number of trials increases, the actual mean approaches the true mean due to the Law of Large Numbers.
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Central limit theorem
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The document discusses the normal distribution and its key properties: bell-shaped and symmetrical around the mean, extending from negative to positive infinity with an area under the curve of 1. Approximately 95% and 99.9% of the distribution lies within 2 and 3 standard deviations of the mean, respectively. It also discusses how to calculate probabilities using the standard normal distribution where the mean is 0 and standard deviation is 1, and how to standardize other normal distributions.
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The document discusses the normal distribution and some of its key properties. It also discusses the central limit theorem and how the distribution of sample means approaches a normal distribution as the sample size increases. Additionally, it covers how to transform a normally distributed variable into a standard normal variable using z-scores and how the normal distribution can be used to approximate the binomial distribution through a correction for continuity.
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This document discusses estimating population parameters such as proportions, means, and standard deviations from sample data. It covers how to calculate confidence intervals for a population proportion based on a sample proportion. The key steps are to determine the sample proportion, calculate the margin of error using the sample size and a critical z-value, and use these to estimate the confidence interval. An example is provided to demonstrate calculating the confidence interval for a population proportion based on survey data. The summary accurately conveys the main topic and methods discussed in the document in under 3 sentences.
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This document provides an overview of descriptive statistics and related concepts. It begins with an introduction to descriptive analysis and then covers various types of variables and levels of measurement. It describes measures of central tendency including mean, median and mode. Measures of dispersion like range, standard deviation and normal distribution are also discussed. The document also covers measures of asymmetry, relationship and concludes with emphasizing the importance of statistical planning in research.
BMS.ppt
This document discusses various measures of central tendency and dispersion used in statistics. It defines average, mean, median and mode as the main measures of central tendency. It provides formulas and methods to calculate arithmetic mean, weighted arithmetic mean, median and mode for both discrete and continuous data sets. The document also introduces absolute measures of dispersion like range, mean deviation, quartile deviation, standard deviation and relative measures of dispersion like coefficient of range, coefficient of mean deviation and coefficient of variation.
The Standard Normal Distribution
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Chapter 6: Normal Probability Distribution
6.1: The Standard Normal Distribution
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The document discusses z-scores and the normal distribution. It defines z-scores as a measure of relative standing, and explains that z-scores represent distances from the mean measured in standard deviations. The document provides formulas for calculating z-scores from raw scores and details how z-scores correspond to specific areas under the normal curve and probabilities. An example demonstrates converting a raw test score to its z-score. In summary, the document outlines how z-scores transform raw scores to standardized values that can be positioned on the normal distribution curve.
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This document provides an overview of basic statistical concepts and terms used in 3D geological modeling and uncertainty analysis, including:
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2. It describes statistical measures like the mean, median, mode, and different types of averages (arithmetic, geometric, harmonic).
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This document summarizes various statistical measures used to analyze and describe data distributions, including measures of central tendency (mean, median, mode), dispersion (range, standard deviation, variance), skewness, and kurtosis. It provides formulas and methods for calculating each measure along with interpretations of the results. Measures of central tendency provide a single value to represent the center of the data set. Measures of dispersion describe how spread out or varied the data values are. Skewness and kurtosis measure the symmetry and peakedness of distributions compared to the normal curve.
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This document provides an outline for a statistical methods course. It covers topics including probability distributions, estimation, hypothesis testing, regression, analysis of variance, and statistical process control. Under probability distributions, it defines key concepts such as random variables, parameters, statistics, and the normal distribution. It also describes properties of the standard normal distribution and how to use the standard normal table to find probabilities and areas under the normal curve.
Describing quantitative data with numbers
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Unit 5 8614.pptx A_Movie_Review_Pursuit_Of_Happiness
This document discusses measures of central tendency and dispersion used in descriptive statistics. It defines mean, median, and mode as the three main measures of central tendency, and explains how to calculate and interpret each. Measures of dispersion discussed include range, variance, standard deviation, interquartile range, skewness, and kurtosis. The goals of these statistical measures are to condense data into single values, facilitate comparisons between data sets, and describe how dispersed or spread out the data are around the central tendency. Examples are provided to demonstrate calculating and applying these key statistical concepts.
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Slideshare notes about measures of central tendancy(mean,median and mode)
This presentation discusses various measures of central tendency including the mean, median, mode, harmonic mean and geometric mean. It provides definitions and formulas for calculating each measure along with their merits and demerits. The mean is the sum of all values divided by the number of values and can be affected by outliers. The median is the middle value when values are arranged from lowest to highest. The mode is the most frequent value. Harmonic and geometric means are other types of averages.
DESCRIPTIVE STATISTICS.pptx
SHORT LECTURE FOR;
Statistical Analysis in social sciences (PhD Education)
Research Methods in Social Sciences (M.Phil Education)
Testing and Measurement in Education (M.Phil Education)
Educational Measurement and Evaluation
(B.Ed / M.Ed/ M.A Education)
Normal Distribution – Introduction and Properties
In this video you can see Normal Distribution – Introduction and Properties.
Watch the video on above ppt
https://www.youtube.com/watch?v=ocTXHLWsec8&list=PLBWPV_4DjPFO6RjpbyYXSaZHiakMaeM9D&index=4
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This document discusses measures of central tendency and variability in descriptive statistics. It defines and provides formulas for calculating the mean, median, and mode as measures of central tendency. The mean is the most useful measure and is calculated by summing all values and dividing by the total number of observations. Variability refers to how spread out or clustered the data values are and is measured by calculations like the range, variance, and standard deviation. The standard deviation is specifically defined as the average deviation of the data from the mean and is considered the best single measure of variability.
Measures of dispersion
The document discusses different methods for measuring the dispersion or spread of data values in a data set, including range, mean deviation, standard deviation, and coefficient of variation. It defines each method and provides the relevant formulas. Range is the simplest measure but can be unreliable since it uses only the minimum and maximum values. Mean deviation measures the average distance from the data's average value. Standard deviation and variance measure how far the values are spread out from the mean, with standard deviation being the square root of variance. Coefficient of variation expresses standard deviation as a percentage of the mean.
8490370.ppt
The document discusses various measures of central tendency and variability used in descriptive statistics. It defines the mean as the sum of all values divided by the number of values. The median is the middle value when values are sorted in ascending order. The mode is the most frequently occurring value. Variability measures the dispersion of scores around the mean and includes the range, interquartile range, standard deviation, and variance. The interquartile range is the difference between the third and first quartiles. Covariance measures how two variables vary together and is used to calculate the correlation coefficient. Factors like extreme scores, sample size, stability under sampling, and open-ended distributions can affect measures of variability.
PSUnit_II_Lesson_2_Understanding_the_z-scores.pptx
This document discusses z-scores and the normal distribution. It defines z-scores as a measure of relative standing compared to the mean. The key points are:
1) z-scores represent the distance from the mean in standard deviation units and allow conversion between raw scores and a standard normal distribution.
2) The formula to calculate z-scores allows matching raw scores to their position relative to the mean.
3) Converting to z-scores allows using normal distribution tables to find the percentage or probability associated with a given raw score.
Stats chapter 2
This document discusses measures of relative standing and density curves in statistics. It introduces z-scores as a standardizing process that measures the relative distance of a data point from the mean of a distribution. It describes key properties of the normal distribution including that 68% of data lies within one standard deviation of the mean. It also provides methods for assessing whether a dataset can be approximated as normally distributed including constructing histograms, stem-plots, and normal probability plots.
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NORMAL DISTRIBUTION
- 1. STATISTICA L TREATMENT Using the appropriate statistical tool to arrive at accurate and definite interpretation of results.
- 2. STATISTICS It is a science of collecting, presenting, analyzing and interpreting data to arrive at an effective decision.
- 3. TYPES OF DATA 1. QUALITATIVE DATA ---- Non-numeric data 2. QUANTITATIVE DATA ---- Numeric data
- 4. CLASSIFICATION OF STATISTICS 1. DESCRIPTIVE STATISTICS ---is a manner of organizing, presenting or summarizing a set of data or observations in an informative way. 2. INFERENTIAL STATISTICS ---proceeds from conducting a study of a subset taken from a population.
- 6. PROPERTIES OF NORMAL DISTRIBUTION Bell-shaped The mean, median, and mode are all equal and are located at the center of the distribution. The distribution is symmetric. The total area under a normal curve is 1 or 100% The distribution is asymptotic. The location of the distribution is determined by the mean and the standard deviation determines dispersion of the distribution.
- 7. STANDARDIZED SCORE (Z-value) Formula: Z = 𝑥 − µ 𝛿 z = Normal Value X = value of any particular observation µ = mean of the distribution 𝛿 = standard deviation
- 8. The scores of 120 students in a stat preliminary examination show a bell-shaped distribution. The mean score is 29 and the standard deviation is 3.02. if a student is selected at random, find the probability of selecting a student whose score is a. Between 24 and 35? b. Between 33 and 37? c. Greater than 34? d. Less than 37?
- 9. SOLUTIONS: STEP 1: Standardize the given observation using the formula. Z = 𝑥 − µ 𝛿 STEP 2: Find the area of the standardized score using the areas under the normal curve.
- 11. STEP 3: Draw the curve and write the z-value along the horizontal line to where it should belong. Positive written to the right side of 0 and negative value is written to left side of 0. shade the corresponding area. STEP 4: Calculate the area. The shaded region serves as our guide on what we are going to do with the areas corresponding to their respective z-value.
- 12. Z-values Rules 1. The z-values are POSITIVE and NEGATIVE ADD the areas of the corresponding z values 2. Both z-values are POSITIVE or Both z-values are NEGATIVE In either case, SUBTRACT the smaller area from the bigger area. 3. To the right of a POSITIVE z- value or to the left of a NEGATIVE z-value SUBTRACT the area from 0.5 4. To the right of a NEGATIVE z value or to the left of a POSITIVE z-value ADD area to 0.5
- 13. THANK YOU! REFERENCE: Pagala, R. 2008; Statistics. Mindshapers Co., Inc.